Rudolf Gorenflo Anatoly A. Kilbas Francesco Mainardi Sergei V

Springer Monographs in Mathematics

Rudolf Gorenflo Anatoly A. Kilbas Francesco Mainardi Sergei V. Rogosin Mittag-Leffler Functions, Related Topics and Applications Springer Monographs in Mathematics More information about this series at http://www.springer.com/series/3733 Rudolf Gorenﬂo • Anatoly A. Kilbas • Francesco Mainardi • Sergei V. Rogosin

Mittag-Lefﬂer Functions, Related Topics and Applications

123 Rudolf Gorenﬂo Anatoly A. Kilbas (July 20, 1948 - June 28, Free University Berlin Mathematical 2010) Institute Belarusian State University Department Berlin of Mathematics and Mechanics Germany Minsk Belarus

Francesco Mainardi Sergei V. Rogosin University of Bologna Department Belarusian State University Department of Physics of Economics Bologna Minsk Italy Belarus

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-662-43929-6 ISBN 978-3-662-43930-2 (eBook) DOI 10.1007/978-3-662-43930-2 Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014949196

Mathematics Subject Classiﬁcation: 33E12, 26A33, 34A08, 45K05, 44Axx, 60G22

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Springer is part of Springer Science+Business Media (www.springer.com) To the memory of our colleague and friend Anatoly Kilbas

Preface

The study of the Mittag-Leffler function and its various generalizations has become a very popular topic in mathematics and its applications. However, during the twentieth century, this function was practically unknown to the majority of scientists, since it was ignored in most common books on special functions. As a noteworthy exception the handbook Higher Transcendental Functions,vol.3,byA.Erdelyi et al. deserves to be mentioned. Now the Mittag-Leffler function is leaving its isolated role as Cinderella (using the term coined by F.G. Tricomi for the incomplete gamma function). The recent growing interest in this function is mainly due to its close relation to the Fractional Calculus and especially to fractional problems which come from applications. Our decision to write this book was motivated by the need to fill the gap in the literature concerning this function, to explain its role in modern pure and applied mathematics, and to give the reader an idea of how one can use such a function in the investigation of modern problems from different scientific disciplines. This book is a fruit of collaboration between researchers in Berlin, Bologna and Minsk. It has highly profited from visits of SR to the Department of Physics at the University of Bologna and from several visits of RG to Bologna and FM to the Department of Mathematics and Computer Science at Berlin Free University under the European ERASMUS exchange. RG and SR appreciate the deep scientific atmosphere at the University of Bologna and the perfect conditions they met there for intensive research. We are saddened that our esteemed and always enthusiastic co-author Anatoly A. Kilbas is no longer with us, having lost his life in a tragic accident on 28 June 2010 in the South of Russia. We will keep him, and our inspiring joint work with him, in living memory.

Berlin, Germany Rudolf Gorenﬂo Bologna, Italy Francesco Mainardi Minsk, Belarus Sergei Rogosin March 2014 vii

Contents

1 Introduction ...... 1 2 Historical Overview of the Mittag-Leffler Functions ...... 7 2.1 A Few Biographical Notes on Gösta Magnus Mittag-Leffler ...... 7 2.2 The Contents of the Five Papers by Mittag-Leffler on New Functions ...... 9 2.3 Further History of Mittag-Leffler Functions ...... 12 3 The Classical Mittag-Leffler Function...... 17 3.1 Definition and Basic Properties ...... 17 3.2 Relations to Elementary and Special Functions ...... 19 3.3 Recurrence and Differential Relations ...... 21 3.4 Integral Representations and Asymptotics ...... 23 3.5 Distribution of Zeros ...... 30 3.6 Further Analytic Properties ...... 35 3.7 The Mittag-Leffler Function of a Real Variable ...... 39 3.7.1 Integral Transforms ...... 39 3.7.2 The Complete Monotonicity Property ...... 46 3.7.3 Relation to Fractional Calculus...... 48 3.8 Historical and Bibliographical Notes ...... 51 3.9 Exercises ...... 53 4 The Two-Parametric Mittag-Leffler Function ...... 55 4.1 Series Representation and Properties of Coefficients ...... 56 4.2 Explicit Formulas: Relations to Elementary and Special Functions ...... 57 4.3 Differential and Recurrence Relations ...... 58 4.4 Integral Relations and Asymptotics...... 60 4.5 The Two-Parametric Mittag-Leffler Function as an Entire Function ...... 65 4.6 Distribution of Zeros ...... 67

ix x Contents

4.7 Computations with the Two-Parametric Mittag-Leffler Function ... 74 4.8 Extension for Negative Values of the First Parameter ...... 80 4.9 Further Analytic Properties ...... 83 4.10 The Two-Parametric Mittag-Leffler Function of a Real Variable ... 84 4.10.1 Integral Transforms of the Two-Parametric Mittag-Leffler Function ...... 84 4.10.2 The Complete Monotonicity Property ...... 85 4.10.3 Relations to the Fractional Calculus ...... 86 4.11 Historical and Bibliographical Notes ...... 88 4.12 Exercises ...... 91 5 Mittag-Leffler Functions with Three Parameters ...... 97 5.1 The Prabhakar (Three-Parametric Mittag-Leffler) Function...... 97 5.1.1 Definition and Basic Properties ...... 97 5.1.2 Integral Representations and Asymptotics...... 100 5.1.3 Integral Transforms of the Prabhakar Function...... 101 5.1.4 Fractional Integrals and Derivatives of the Prabhakar Function ...... 103 5.1.5 Relations to the Wright Function, H-Function and Other Special Functions...... 105 5.2 Generalized (Kilbas–Saigo) Mittag-Leffler Type Functions...... 106 5.2.1 Definition and Basic Properties ...... 106 5.2.2 The Order and Type of the Entire Function E˛;m;l .z/ ...... 107 5.2.3 Recurrence Relations for E˛;m;l .z/ ...... 110 5.2.4 Connection of En;m;l .z/ with Functions of Hypergeometric Type ...... 112 5.2.5 Differentiation Properties of En;m;l .z/ ...... 114 5.2.6 Fractional Integration of the Generalized Mittag-Leffler Function ...... 119 5.2.7 Fractional Differentiation of the Generalized Mittag-Leffler Function ...... 121 5.3 Historical and Bibliographical Notes ...... 125 5.4 Exercises ...... 126 6 Multi-index Mittag-Leffler Functions ...... 129 6.1 The Four-Parametric Mittag-Leffler Function: Luchko–Kilbas–Kiryakova’s Approach ...... 129 6.1.1 Definition and Special Cases ...... 129 6.1.2 Basic Properties ...... 130 6.1.3 Integral Representations and Asymptotics...... 132 6.1.4 Extended Four-Parametric Mittag-Leffler Functions ...... 134 6.1.5 Relations to the Wright Function and the H-Function ..... 135 6.1.6 Integral Transforms of the Four-Parametric Mittag-Leffler Function ...... 136 Contents xi

6.1.7 Integral Transforms with the Four-Parametric Mittag-Leffler Function in the Kernel...... 138 6.1.8 Relations to the Fractional Calculus ...... 142 6.2 Mittag-Leffler Functions with 2n Parameters ...... 145 6.2.1 Definition and Basic Properties ...... 145 6.2.2 Representations in Terms of Hypergeometric Functions... 148 6.2.3 Integral Representations and Asymptotics...... 149 6.2.4 Extension of the 2n-Parametric Mittag-Leffler Function ...... 150 6.2.5 Relations to the Wright Function and to the H-Function ...... 152 6.2.6 Integral Transforms with the Multi-parametric Mittag-Leffler Functions...... 153 6.2.7 Relations to the Fractional Calculus ...... 157 6.3 Historical and Bibliographical Notes ...... 159 6.4 Exercises ...... 163 7 Applications to Fractional Order Equations ...... 165 7.1 Fractional Order Integral Equations ...... 165 7.1.1 The Abel Integral Equation ...... 165 7.1.2 Other Integral Equations Whose Solutions Are Represented via Generalized Mittag-Leffler Functions ...... 170 7.2 Fractional Ordinary Differential Equations...... 171 7.2.1 Fractional Ordinary Differential Equations with Constant Coefficients...... 171 7.2.2 Ordinary FDEs with Variable Coefficients ...... 179 7.2.3 Other Types of Ordinary Fractional Differential Equations ...... 182 7.3 Differential Equations with Fractional Partial Derivatives ...... 183 7.3.1 Cauchy-Type Problems for Differential Equations with Fractional Partial Derivatives ...... 184 7.3.2 The Cauchy Problem for Differential Equations with Fractional Partial Derivatives ...... 186 7.4 Historical and Bibliographical Notes ...... 187 7.5 Exercises ...... 197 8 Applications to Deterministic Models ...... 201 8.1 Fractional Relaxation and Oscillations ...... 201 8.1.1 Simple Fractional Relaxation and Oscillation ...... 202 8.1.2 The Composite Fractional Relaxation and Oscillations .... 211 8.2 Examples of Applications of the Fractional Calculus in Physical Models...... 219 8.2.1 Linear Visco-Elasticity ...... 219 8.2.2 Other Deterministic Fractional Models ...... 224 xii Contents

8.3 Historical and Bibliographical Notes ...... 227 8.3.1 General Notes ...... 227 8.3.2 Notes on Fractional Differential Equations ...... 228 8.3.3 Notes on the Fractional Calculus in Linear Viscoelasticity ...... 228 8.4 Exercises ...... 230 9 Applications to Stochastic Models ...... 235 9.1 Introduction ...... 235 9.2 The Mittag-Leffler Process According to Pillai ...... 236 9.3 Elements of Renewal Theory and Continuous Time Random Walk (CTRW) ...... 238 9.3.1 Renewal Processes ...... 238 9.3.2 Continuous Time Random Walk (CTRW) ...... 239 9.3.3 The Renewal Process as a Special CTRW ...... 243 9.4 The Poisson Process and Its Fractional Generalization (the Renewal Process of Mittag-Leffler Type) ...... 245 9.4.1 The Mittag-Leffler Waiting Time Density ...... 245 9.4.2 The Poisson Process ...... 246 9.4.3 The Renewal Process of Mittag-Leffler Type...... 247 9.4.4 Thinning of a Renewal Process...... 251 9.5 The Fractional Diffusion Process ...... 253 9.5.1 Renewal Process with Reward ...... 253 9.5.2 Limit of the Mittag-Leffler Renewal Process ...... 253 9.5.3 Subordination in the Space-Time Fractional Diffusion Equation ...... 258 9.5.4 The Rescaling and Respeeding Concept Revisited: Universality of the Mittag-Leffler Density ...... 262 9.6 Historical and Bibliographical Notes ...... 264 9.7 Exercises ...... 266

A The Eulerian Functions ...... 269 A.1 The Gamma Function ...... 269 A.1.1 Analytic Continuation ...... 270 A.1.2 The Graph of the Gamma Function on the Real Axis ...... 271 A.1.3 The Reﬂection or Complementary Formula ...... 273 A.1.4 The Multiplication Formulas ...... 273 A.1.5 Pochhammer’s Symbols ...... 274 A.1.6 Hankel Integral Representations ...... 274 A.1.7 Notable Integrals via the Gamma Function ...... 276 A.1.8 Asymptotic Formulas ...... 277 A.1.9 Inﬁnite Products...... 277 A.2 The Beta Function ...... 278 A.2.1 Euler’s Integral Representation...... 278 A.2.2 Symmetry ...... 278 Contents xiii

A.2.3 Trigonometric Integral Representation...... 278 A.2.4 Relation to the Gamma Function ...... 279 A.2.5 Other Integral Representations ...... 279 A.2.6 Notable Integrals via the Beta Function ...... 280 A.3 Historical and Bibliographical Notes ...... 282 A.4 Exercises ...... 282

B The Basics of Entire Functions ...... 285 B.1 Deﬁnition and Series Representations ...... 285 B.2 Growth of Entire Functions: Order, Type and Indicator Function... 286 B.3 Weierstrass Canonical Representation: Distribution of Zeros ...... 287 B.4 Entire Functions of Completely Regular Growth ...... 289 B.5 Historical and Bibliographical Notes ...... 290 B.6 Exercises ...... 292

C Integral Transforms ...... 299 C.1 Fourier Type Transforms ...... 299 C.2 The Laplace Transform...... 304 C.3 The Mellin Transform ...... 307 C.4 Simple Examples and Tables of Transforms of Basic Elementary and Special Functions ...... 311 C.5 Historical and Bibliographical Notes ...... 313 C.6 Exercises ...... 314

D The Mellin–Barnes Integral ...... 319 D.1 Deﬁnition: Contour of Integration ...... 319 D.2 Asymptotic Methods for the Mellin–Barnes Integral ...... 322 D.3 Historical and Bibliographical Notes ...... 324 D.4 Exercises ...... 325

E Elements of Fractional Calculus ...... 327 E.1 Introduction to the Riemann–Liouville Fractional Calculus...... 327 E.2 The Liouville–Weyl Fractional Calculus ...... 330 E.3 The Abel–Riemann Fractional Calculus...... 334 E.3.1 The Abel–Riemann Fractional Integrals and Derivatives...... 334 E.4 The Caputo Fractional Calculus ...... 336 E.4.1 The Caputo Fractional Derivative ...... 336 E.5 The Riesz–Feller Fractional Calculus ...... 338 E.5.1 The Riesz Fractional Integrals and Derivatives ...... 339 E.5.2 The Feller Fractional Integrals and Derivatives ...... 341 E.6 The Grünwald–Letnikov Fractional Calculus ...... 344 E.6.1 The Grünwald–Letnikov Approximation in the Riemann–Liouville Fractional Calculus ...... 345 xiv Contents

E.6.2 The Grünwald–Letnikov Approximation in the Riesz–Feller Fractional Calculus ...... 348 E.7 Historical and Bibliographical Notes ...... 348

F Higher Transcendental Functions ...... 351 F.1 Hypergeometric Functions ...... 351 F.1.1 Classical Gauss Hypergeometric Functions ...... 351 F.1.2 Euler Integral Representation: Mellin–Barnes Integral Representation ...... 353 F.1.3 Basic Properties of Hypergeometric Functions...... 354 F.1.4 The Hypergeometric Differential Equation ...... 355 F.1.5 Kummer’s and Tricomi’s Confluent Hypergeometric Functions ...... 356 F.1.6 Generalized Hypergeometric Functions and Their Properties ...... 360 F.2 Wright Functions ...... 361 F.2.1 The Classical Wright Function ...... 361 F.2.2 Mellin–Barnes Integral Representation and Asymptotics ...... 362 F.2.3 The Bessel–Wright Function: Generalized Wright Functions and Fox–Wright Functions ...... 363 F.3 Meijer G-Functions...... 365 F.3.1 Definition via Integrals: Existence ...... 365 F.3.2 Basic Properties of the Meijer G-Functions ...... 366 F.3.3 Special Cases ...... 367 F.3.4 Relations to Fractional Calculus ...... 368 F.3.5 Integral Transforms of G-Functions ...... 369 F.4 Fox H-Functions ...... 370 F.4.1 Definition via Integrals: Existence ...... 370 F.4.2 Series Representations and Asymptotics: Recurrence Relations ...... 374 F.4.3 Special Cases ...... 377 F.4.4 Relations to Fractional Calculus ...... 379 F.4.5 Integral Transforms of H-Functions ...... 381 F.5 Historical and Bibliographical Notes ...... 382 F.6 Exercises ...... 385

References...... 389

Index ...... 441 Chapter 1 Introduction

The book is devoted to an extended description of the properties of the Mittag-Leffler function, its numerous generalizations and their applications in different areas of modern science. The function E˛.z/ is named after the great Swedish mathematician Gösta Magnus Mittag-Leffler (1846–1927) who defined it by a power series

X1 zk E .z/ D ;˛2 C; Re ˛>0; (1.0.1) ˛ .˛kC 1/ kD0 and studied its properties in 1902–1905 in five subsequent notes [ML1, ML2, ML3, ML4, ML5-5] in connection with his summation method for divergent series. This function provides a simple generalization of the exponential function because of the replacement of kŠ D .k C 1/ by .˛k/Š D .˛k C 1/ in the denominator of the power terms of the exponential series. During the first half of the twentieth century the Mittag-Leffler function remained almost unknown to the majority of scientists. They unjustly ignored it in many treatises on special functions, including the most common (Abramowitz and Stegun [AbrSte72] and its novel version “NIST Handbook of Mathematical Functions” [NIST]). Furthermore, there appeared some relevant works where the authors arrived at series or integral representations of this function without recognizing it, e.g. (Gnedenko and Kovalenko [GneKov68]), (Balakrishnan [BalV85]) and (Sanz-Serna [San88]). A description of the most important properties of this function is present in the third volume [Bat-3] of the Handbook on Higher Transcendental Functions of the Bateman Project, (Erdelyi et al.). In it, the authors have included the Mittag-Leffler functions in their Chapter XVIII devoted to the so-called miscellaneous functions. The attribution of ‘miscellaneous’ to the Mittag- Leffler function is due to the fact that it was only later, in the 1960s, that it was recognized to belong to a more general class of higher transcendental functions, known as Fox H-functions (see, e.g., [MatSax78, KilSai04, MaSaHa10]). In fact,

© Springer-Verlag Berlin Heidelberg 2014 1 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__1 2 1 Introduction this class was well-established only after the seminal paper by Fox [Fox61]. A more detailed account of the Mittag-Leffler function is given in the treatise on complex functions by Sansone and Gerretsen [SanGer60]. However, the most specialized treatise, where more details on the functions of Mittag-Leffler type are given, is surely the book by Dzherbashyan [Dzh66], in Russian. Unfortunately, no official English translation of this book is presently available. Nevertheless, Dzherbashyan has done a lot to popularize the Mittag-Leffler function from the point of view of its special role among entire functions of a complex variable, where this function can be considered as the simplest non-trivial generalization of the exponential function. Successful applications of the Mittag-Leffler function and its generalizations, and their direct involvement in problems of physics, biology, chemistry, engineering and other applied sciences in recent decades has made them better known among scientists. A considerable literature is devoted to the investigation of the analyticity properties of these functions; in the references we quote several authors who, after Mittag-Leffler, have investigated such functions from a mathematical point of view. At last, the 2000 Mathematics Subject Classification has included these functions in item 33E12: “Mittag-Leffler functions and generalizations”. Starting from the classical paper of Hille and Tamarkin [HilTam30]inwhichthe solution of Abel integral equation of the second kind

Zx .t/ .x/ dt D f.x/;0<˛<1; 0

Chapter 3 is devoted to the classical Mittag-Leffler function (1.0.1). We collect here the main results on the function which were discovered during the century following Mittag-Leffler’s definition. These are of an analytic nature, comprising rules of composition and asymptotic properties, and its character as an entire function of a complex variable. Special attention is paid to integral transforms related to the Mittag-Leffler function because of their importance in the solution of integral and differential equations. We point out its role in the Fractional Calculus and its place among the whole collection of higher transcendental functions. In Chap. 4 we discuss questions similar to those of Chap. 3. This chapter deals with the simplest (and for applications most important) generalizations of the Mittag- Leffler function, namely the two-parametric Mittag-Leffler function

X1 zk E .z/ D ;˛;ˇ2 C; Re ˛>0; (1.0.3) ˛;ˇ .˛kC ˇ/ kD0 which was deeply investigated independently by Humbert and Agarval in 1953 [Hum53, Aga53, HumAga53] and by Dzherbashyan in 1954 [Dzh54a, Dzh54b, Dzh54c] (but formally appeared first in the paper by Wiman [Wim05a]). Chapter 5 presents the theory of two types of three-parametric Mittag-Leffler function. First of all the three-parametric Mittag-Leffler function (or Prabhakar function) introduced by Prabhakar [Pra71]

X1 ./ E .z/ D k zk;˛;ˇ; 2 C; Re ˛; > 0; (1.0.4) ˛;ˇ kŠ .˛k C ˇ/ kD0

.Ck/ where ./k D . C 1/:::. C k 1/ D ./ is the Pochhammer symbol (see (A.1.17)). This function is now widely used for different applied problems. Another type of three-parametric Mittag-Lefﬂer function is not as well-known as the Prabhakar function (1.0.4). It was introduced and studied by Kilbas and Saigo [KilSai95b] in connection with the solution of a new type of fractional differential equation. This function (the Kilbas–Saigo function) is deﬁned as follows

X1 k E˛;m;l .z/ D ckz .z 2 C/; (1.0.5) kD0 where

kY1 .˛Œim C l C 1/ c D 1; c D .k D 1; 2; /; ˛ 2 C; Re ˛>0: 0 k .˛Œim C l C 1 C 1/ iD1 (1.0.6) Some basic results on this function are also included in Chap. 5. 1 Introduction 5

By introducing additional parameters one can discover new interesting properties of these functions and extend their range of applicability. This is exactly the case with the generalizations described in this chapter. Together with some appendices, Chaps. 2–5 constitute a short course on the Mittag-Leffler function and its generalizations. This course is self-contained and requires only a basic knowledge of Real and Complex Analysis. Chapter 6 is rooted deeper mathematically. The reader can find here a number of modern generalizations. The ideas leading to them are described in detail. The main focus is on four-parametric Mittag-Leffler functions (Dzherbashyan [Dzh60]) and 2n-parametric Mittag-Leffler functions (Al-Bassam and Luchko [Al-BLuc95] and Kiryakova [Kir99]). Experts in higher transcendental functions and their applications will find here many interesting results, obtained recently by various authors. These generalizations will all be labelled by the name Mittag-Leffler, in spite of the fact that some of them can be considered for many values of parameters as particular cases of the general class of Fox H-functions. These H-functions offer a powerful tool for formally solving many problems, however by inserting relevant parameters one often arrives at functions whose behaviour is easier to handle. This is the case for the Mittag-Leffler functions, and so these functions are often more appropriate for applied scientists who prefer direct work to a detour through a wide field of generalities. The last three chapters deal with applications of the functions treated in the pre- ceding chapters. We start (Chap. 7) with the “formal” (or mathematical) applications of Mittag-Leffler functions. The title of the chapter is “Applications to fractional order equations”. By fractional order equations we mean either integral equations with weak singularities or differential equations with ordinary or partial fractional derivatives. The collection of such equations involving Mittag-Leffler functions in their analysis or explicit solution is fairly big. Of course, we should note that a large number of fractional order equations arise in certain applied problems. We would like to separate the questions of mathematical analysis (solvability, asymptotics of solutions, their explicit presentation etc.) from the motivation and description of those models in which such equations arise. In Chap. 7 we focus on the development of a special “fractional” technique and give the reader an idea of how this technique can be applied in practice. Further applications are presented in the two subsequent chapters devoted to mathematical modelling of special processes of interest in applied sciences. Chapter 8 deals mainly with the role of Mittag-Leffler functions in discovering and analyzing deterministic models based on certain equations of fractional order. Special attention is paid to fractional relaxation and oscillation phenomena, to fractional diffusion and diffusive wave phenomena and to hereditary phenomena in visco-elasticity and hydrodynamics. These are models in physics, chemistry, biology etc., which by adopting a macroscopic viewpoint can be described without using probabilistic ideas and machinery. 6 1 Introduction

In contrast, in Chap. 9 we describe the role of Mittag-Leffler functions in models involving randomness. We explain here the key role of probability distributions of Mittag-Leffler type which enter into a variety of stochastic processes including fractional Poisson processes and transition from continuous time random walk to fractional diffusion. Our six appendices can be divided into two groups. First of all we present here some basic facts from certain areas of analysis. Such appendices are useful additions to the course of lectures which can be extracted from Chaps. 3 to 5. The second type of appendices constitute those which can help the reader to understand modern results in the areas in which the Mittag-Leffler function is essential and important. They serve to make the book self-contained. The book is addressed to a wide audience. Special attention is paid to those topics which are accessible for students in Mathematics, Physics, Chemistry, Biology and Mathematical Economics. Also in our audience are experts in the theory of the Mittag-Leffler function and its applications. We hope that they will find the technical parts of the book and the historical and bibliographical remarks to be a source of new ideas. Lastly, we have to note that our main goal, which we always had in mind during the writing of the book, was to make it useful for people working in different areas of applications (even those far from pure mathematics). Chapter 2 Historical Overview of the Mittag-Leffler Functions

2.1 A Few Biographical Notes on Gösta Magnus Mittag-Lefﬂer

Gösta Magnus Mittag-Leffler was born on March 16, 1846, in Stockholm, Sweden. His father, John Olof Leffler, was a school teacher, and was also elected as a member of the Swedish Parliament. His mother, Gustava Vilhelmina Mittag, was a daughter of a pastor, who was a person of great scientific abilities. At his birth Gösta was given the name Leffler and later (when he was a student) he added his mother’s name “Mittag” as a tribute to this family, which was very important in Sweden in the nineteenth century. Both sides of his family were of German origin. At the Gymnasium in Stockholm Gösta was training as an actuary but later changed to mathematics. He studied at the University of Uppsala, entering it in 1865. In 1872 he defended his thesis on applications of the argument principle and in the same year was appointed as a Docent (Associate Professor) at the University of Uppsala. In the following year he was awarded a scholarship to study and work abroad as a researcher for 3 years. In October 1873 he left for Paris. In Paris Mittag-Leffler met many mathematicians, such as Bouquet, Briot, Chasles, Darboux, and Liouville, but his main goal was to learn from Hermite. However, he found the lectures by Hermite on elliptic functions difficult to understand. In spring 1875 he moved to Berlin to attend the lectures by Weierstrass whose research and teaching style was very close to his own. From Weierstrass’ lectures Mittag-Leffler learned many ideas and concepts which would later become the core of his scientific interests. In Berlin Mittag-Leffler received news that professor Lorenz Lindelöf (Ernst Lindelöf’s father) had decided to leave a chair at the University of Helsingfors (now Helsinki). At the same time Weierstrass requested from the ministry of education the installation of a new position at his institute and suggested Mittag-Leffler for the

© Springer-Verlag Berlin Heidelberg 2014 7 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__2 8 2 Historical Overview of the Mittag-Leffler Functions position. In spite of this, Mittag-Leffler applied for the chair at Helsingfors. He got the chair in 1876 and remained at the University of Helsingfors for the next 5 years. In 1881 the new University of Stockholm was founded, and Gösta Mittag-Leffler was the first to hold a chair in Mathematics there. Soon afterwards he began to organize the setting up of the new international journal Acta Mathematica. In 1882 Mittag-Leffler founded Acta Mathematica and served as the Editor-in-Chief of the journal for 45 years. The original idea for such a journal came from Sophus Lie in 1881, but it was Mittag-Leffler’s understanding of the European scene, together with his political skills, that ensured the success of the journal. Later he invited many well-known mathematicians (Cantor, Poincaré and many others) to submit papers to this journal. Mittag-Leffler was always a good judge of the quality of the work submitted to him for publication. In 1882 Gösta Mittag-Leffler married Signe af Linfors and they lived together until the end of his life. Mittag-Leffler made numerous contributions to mathematical analysis, particularly in the areas concerned with limits, including calculus, analytic geometry and probability theory. He worked on the general theory of functions, studying relationships between independent and dependent variables. His best known work deals with the analytic representation of a single-valued complex function, culminating in the Mittag-Leffler theorem. This study began as an attempt to generalize results from Weierstrass’s lectures, where Weierstrass had described his theorem on the existence of an entire function with prescribed zeros each with a specified multiplicity. Mittag-Leffler tried to generalize this result to meromorphic functions while he was studying in Berlin. He eventually assembled his findings on generalizing Weierstrass’ theorem to meromorphic functions in a paper which he published (in French) in 1884 in Acta Mathematica. In this paper Mittag-Leffler proposed a series of general topological notions on infinite point sets based on Cantor’s new set theory. With this paper Mittag-Leffler became the sole proprietor of a theorem that later became widely known and so he took his place in the circle of internationally known mathematicians. Mittag-Leffler was one of the first mathematicians to support Cantor’s theory of sets but, one has to remark, a consequence of this was that Kronecker refused to publish in Acta Mathematica. Between 1899 and 1905 Mittag- Leffler published a series of papers which he called “Notes” on the summation of divergent series. The aim of these notes was to construct the analytical continuation of a power series outside its circle of convergence. The region in which he was able to do this is now called Mittag-Leffler’s star. Andre Weyl in his memorial [Weil82]says:“A well-known anecdote has Oscar Wilde saying that he had put his genius into his life; into his writings he had put merely his talent. With at least equal justice it may be said of Mittag-Leffier that the Acta Mathematica were the product of his genius, while nothing more than talent went into his mathematical contributions. Genius transcends and defies analysis; but this may be a fitting occasion for examining some of the qualities involved in the creating and in the editing of a great mathematical journal.” 2.2 The Contents of the Five Papers by Mittag-Leffler on New Functions 9

In the same period Mittag-Leffler introduced and investigated in five subsequent papers a new special function, which is now very popular and useful for many applications. This function, as well as many of its generalizations, is now called the “Mittag-Leffler” function.1 His contribution is nicely summed up by Hardy [Har28a]: “Mittag-Leffler was a remarkable man in many ways. He was a mathematician of the front rank, whose contributions to analysis had become classical, and had played a great part in the inspiration of later research; he was a man of strong personality, fired by an intense devotion to his chosen study; and he had the persistence, the position, and the means to make his enthusiasm count.” Gösta Mittag-Leffler passed away on July 7, 1927. During his life he received many honours. He was an honorary member or corresponding member of almost every mathematical society in the world including the Accademia Reale dei Lincei, the Cambridge Philosophical Society, the Finnish Academy of Sciences, the London Mathematical Society, the Moscow Mathematical Society, the Netherlands Academy of Sciences, the St.-Petersburg Imperial Academy, the Royal Institution, the Royal Belgium Academy of Sciences and Arts, the Royal Irish Academy, the Swedish Academy of Sciences, and the Institute of France. He was elected a Fellow of the Royal Society of London in 1896. He was awarded honorary degrees from the Universities of Oxford, Cambridge, Aberdeen, St. Andrews, Bologna and Christiania (now Oslo).

2.2 The Contents of the Five Papers by Mittag-Lefﬂer on New Functions

Let us begin with a description of the ideas which led to the introduction by Mittag- Leffler of a new transcendental function. In 1899 Mittag-Leffler began the publication of a series of articles under the common title “Sur la représentation analytique d’une branche uniforme d’une fonction monogène” (“On the analytic representation of a single-valued branch of a monogenic function”) published mainly in Acta Mathematica ([ML5-1, ML5-2, ML5-3,ML5-4,ML5-5,ML5-6]). The first articles of this series were based on three reports presented by him in 1898 at the Swedish Academy of Sciences in Stockholm. His research was connected with the following question: Let k0;k1;:::be a sequence of complex numbers for which

1= 1 lim jkj D 2 RC !1 r

1Since it is the subject of this book we will give below a wider discussion of these five papers and of the role of the Mittag-Leffler functions. 10 2 Historical Overview of the Mittag-Leffler Functions is finite. Then the series

2 FC.z/ WD k0 C k1z C k2z C ::: is convergent in the disk Dr Dfz 2 C Wjzj r. It determines a single-valued analytic function in the disk Dr . The questions discussed were: 1. To determine the maximal domain on which the function FC.z/ possesses a single-valued analytic continuation; 2. To ﬁnd an analytic representation of the corresponding single-valued branch. Abel [Abe26a] had proposed (see also [Lev56]) to associate with the function FC.z/ the entire function

k z k z2 k z X1 k z F .z/ WD k C 1 C 2 C :::C C :::D : 1 0 1Š 2Š Š Š D0

This function was used by Borel (see, e.g., [Bor01]) to discover that the answer to the above question is closely related to the properties of the following integral (now called the Laplace–Abel integral): Z 1 ! e F1.!z/d!: (2.2.1) 0

An intensive study of these properties was carried out at the beginning of twentieth century by many mathematicians (see, e.g., [ML5-3,ML5-5] and references therein). Mittag-Lefﬂer introduced instead of F1.z/ a one-parametric family of (entire) functions

k z k z2 X1 k z F .z/ WD k C 1 C 2 C :::D ;.˛>0/; ˛ 0 .1 ˛ C 1/ .2 ˛ C 1/ . ˛ C 1/ D0 and studied its properties as well as the properties of the generalized Laplace–Abel integral Z Z 1 1 !1=˛ 1=˛ ! ˛ e F˛.!z/d! D e F˛.! z/d!: (2.2.2) 0 0

2The notation FC.z/ is not described in Mittag-Lefﬂer’s paper. The letter “C” probably indicates the word ‘convergent’ in order to distinguish this function from its analytic continuation FA.z/ (see discussion below). 2.2 The Contents of the Five Papers by Mittag-Lefﬂer on New Functions 11

The main result of his study was: in a maximal domain A (star-like with respect to origin) the analytic representation of the single-valued analytic continuation FA.z/ of the function FC.z/ can be represented in the following form Z 1 ! ˛ FA.z/ D lim e F˛.! z/d!: (2.2.3) ˛!1 0

For this reason analytic properties of the functions F˛.z/ become highly important. Due to this construction Mittag-Lefﬂer decided to study the most simple function of the type F˛.z/, namely, the function corresponding to the unit sequence k.This function

z z2 X1 z E .z/ WD 1 C C C ::: D ; (2.2.4) ˛ .1 ˛ C 1/ .2 ˛ C 1/ . ˛ C 1/ D0 was introduced and investigated by G. Mittag-Leffler in five subsequent papers [ML1, ML2, ML3, ML4, ML5-5] (in particular, in connection with the above formulated questions). This function is known now as the Mittag-Leffler function. In the first paper [ML1], devoted to his new function, Mittag-Leffler discussed the relation of the function F˛.z/ with the above problem on analytic continuation. In particular, he posed the question of whether the domains of analyticity of the function Z 1 ! ˛ lim e F˛.! z/d! ˛#1 0 and the function (introduced and studied by Le Roy [LeR00])

X1 .˛C 1/ lim kz ˛#1 .C 1/ D0 coincide. In the second paper [ML2] the new function (i.e., the Mittag-Leffler function) appeared. Its asymptotic properties were formulated. In particular, Mittag-Leffler z1=˛ ˛ ˛ showed that E˛.z/ behaves as e in the angle 2 < arg z < 2 and is bounded ˛ 3 for values of z with 2 < jarg zjÄ. In the third paper [ML3] the asymptotic properties of E˛.z/ were discussed more carefully. Mittag-Leffler compared his results with those of Malmquist [Mal03], Phragmén [Phr04] and Lindelöf [Lin03] which they obtained for similar functions (the results form the background of the classical Phragmén–Lindelöf theorem [PhrLin08]).

3 ˛ The behaviour of E˛.z/ on critical rays jarg zjD˙ 2 was not described. 12 2 Historical Overview of the Mittag-Lefﬂer Functions

The fourth paper [ML4] was completely devoted to the extension of the function E˛.z/ (as well as the function F˛.z/) to complex values of the parameter ˛. Mittag-Lefﬂer’s most creative paper on the new function E˛.z/ is his ﬁfth paper [ML5-5]. There he:

(a) Found an integral representation for the function E˛.z/; (b) Described the asymptotic behaviour of E˛.z/ in different angle domains; (c) Gave the formulas connecting E˛.z/ with known elementary functions; (d) Provided the asymptotic formulas for Z 1 1 !1=˛ d! E˛.z/ D e 2i L ˛ ! z

by using the so-called Hankel integration path; (e) Obtained detailed asymptotics of E˛.z/ for negative values of the variable, i.e. for z Dr; (f) Compared in detail his asymptotic results for E˛.z/ with the results obtained by Malmquist; (g) Found domains which are free of zeros of E˛.z/ in the case of “small” positive values of parameter, i.e. for 0<˛<2;˛6D 1; (h) Applied his results on E˛.z/ to answer the question of the domain of analyticity of the function FA.z/ and its analytic representation (see formula (2.2.3)).

2.3 Further History of Mittag-Lefﬂer Functions

The importance of the new function was understood as soon as the first analytic results for it appeared. First of all, it is a very simple function playing the key role in the solution of a general problem of the theory of analytic functions. Secondly, the Mittag-Leffler function can be considered as a direct generalization of the exponential function, preserving some of its properties. Furthermore, E˛.z/ has some interesting properties which later became essential for the description of many problems arising in applications. After Mittag-Leffler’s introduction of the new function, one of the first results on it was obtained by Wiman [Wim05a]. He used Borel’s method of summation of divergent series (which Borel applied to the special case of the Mittag-Leffler function, namely, for ˛ D 1,see[Bor01]). Using this method, Wiman gave a new proof of the asymptotic representation of E˛.z/ in different angle domains. This representation was obtained for positive rational values of the parameter ˛. He also noted4 that analogous asymptotic results hold for the two-parametric generalization E˛;ˇ.z/ of the Mittag-Leffler function (see (1.0.3)). Applying the

4But did not discuss in detail. 2.3 Further History of Mittag-Leffler Functions 13 obtained representation Wiman described in [Wim05b] the distribution of zeros of the Mittag-Leffler function E˛.z/. The main focus was on two cases – to the case of real values of the parameter ˛ 2 .0; 2; ˛ 6D 1, and to the case of complex values of ˛,Re˛>0. In [Phr04] Phragmén proved the generalization of the Maximum Modulus Principle for the case of functions analytic in an angle. For this general theorem the Mittag-Leffler function plays the role of the key example. It satisfies the inequality jzj jE˛.z/j

X z ; .1C a / where the sequence a tends to zero as !1. The particular goal was to construct a simple example of an entire function which tends to zero along almost all rays when jzj!1.Suchanexample

X z G.z/ D ; 0<˛<1; (2.3.1) .1C / .log /˛ was constructed [Mal05] and carefully examined by using the calculus of residues for the integral representation of G.z/ (which is also analogous to that for E˛). At the beginning of the twentieth century many mathematicians paid great attention to obtaining asymptotic expansions of special functions, in particular, those of hypergeometric type. The main reason for this was that these functions play an important role in the study of differential equations, which describe different phenomena. In the fundamental paper [Barn06] Barnes proposed a unified approach to the investigation of asymptotic expansions of entire functions defined by Taylor series. This approach was based on the previous results of Barnes [Barn02]and Mellin [Mel02]. The essence of this approach is to use the representation of the quotient of the products of Gamma functions in the form of a contour integral which is handled by using the method of residues. This representation is now known as the Mellin–Barnes integral formula (see Appendix D). Among the functions which were treated in [Barn06] was the Mittag-Leffler function. The results of Barnes were further developed in his articles, including applications to the theory of differential equations, as well as in the articles by Mellin (see, e.g., [Mel10]). In fact, the idea of employing contour integrals involving Gamma functions of the variable in the subject of integration is due to Pincherle, whose suggestive paper [Pin88](seealso 14 2 Historical Overview of the Mittag-Leffler Functions

[MaiPag03]) was the starting point of Mellin’s investigations (1895), although the type of contour and its use can be traced back to Riemann, as Barnes wrote in [Barn07b, p. 63]. Generalizations of the Mittag-Lefﬂer function are proposed among other generalizations of the hypergeometric functions. For them similar approaches were used. Among these generalizations we should point out the collection of Wright functions, ﬁrst introduced in 1934, see [Wri34],

X1 zn X1 zn .zI ; ˇ/ WD D I (2.3.2) .nC 1/ . n C ˇ/ nŠ . n C ˇ/ nD0 nD0 the collection of generalized hypergeometric functions, ﬁrst introduced in 1928, see [Fox28],

X1 k .˛1/k .˛2/k .˛p/k z 5 pFq .z/ D pFq ˛1;˛2;:::;˛pI ˇ1;ˇ2;:::;ˇqI z D I .ˇ1/k .ˇ2/k .ˇq/k kŠ kD0 (2.3.3) the collection of Meijer G-functions introduced in 1936, see [Mei36], and inten- sively treated in 1946, see [Mei46], Â ˇ Ã ˇ m;n ˇ a1;:::;ap Gp;q z ˇ b1;:::;bq

Qm Qn Z .bi C s/ .1 ai s/ 1 iD1 iD1 s D Qq Qp z ds; (2.3.4) 2i T .1 bi s/ .ai C s/ iDmC1 iDnC1 and the collection of more general Fox H-functions introduced in 1961 [Fox61] Â ˇ Ã ˇ m;n ˇ .a1;˛1/;:::;.ap;˛p/ Hp;q z ˇ .b1;ˇ1/;:::;.bq;ˇq /

Qm Qn Z .bi C ˇi s/ .1 ai ˛i s/ 1 iD1 iD1 s D Qq Qp z ds: (2.3.5) 2i T .1 bi ˇi s/ .ai C ˛i s/ iDmC1 iDnC1

Some generalizations of the Mittag-Lefﬂer function appeared as a result of developments in integral transform theory. In this connection in 1953 Agarval and

5 Here ./k is the Pochhammer symbol, see (A.1.17). 2.3 Further History of Mittag-Lefﬂer Functions 15

Humbert (see [Hum53,Aga53,HumAga53]) and independently in 1954 Djrbashyan (see [Dzh54a,Dzh54b,Dzh54c]) introduced and studied the two-parametric Mittag- Lefﬂer function (or Mittag-Lefﬂer type function)

X1 z E .z/ WD : (2.3.6) ˛;ˇ . ˛ C ˇ/ D0

We note once more that, formally, the function (2.3.6) first appeared in the paper of Wiman [Wim05a], who did not pay much attention to its extended study. In 1971 Prabhakar [Pra71] introduced the three-parametric Mittag-Leffler function (or generalized Mittag-Leffler function,orPrabhakar function)

X1 . / z E .z/ WD : (2.3.7) ˛;ˇ . ˛ C ˇ/ D0

This function appeared in the kernel of a first order integral equation which Prabhakar treated by using fractional calculus. Recently, other three-parametric Mittag-Leffler functions (also called generalized Mittag-Leffler functions or Mittag-Leffler type functions,orKilbas–Saigo functions) were introduced by Kilbas and Saigo (see, e.g., [KilSai95a])

X1 n E˛;m;l .z/ WD cnz ; (2.3.8) nD0 where

nY1 Œ˛.im C l/ C 1 c D 1; c D : 0 n Œ˛.im C l C 1/ C 1 iD0

These functions appeared in connection with the solution of new types of integral and differential equations and with the development of the fractional calculus. R 2 2 C For real ˛1;˛2 2 .˛1 C ˛2 ¤ 0/ and complex ˇ1;ˇ2 2 the following function was introduced by Dzherbashian (=Djrbashian) [Dzh60] in the form of the series (in fact only for ˛1;˛2 >0)

X1 k z C E˛1;ˇ1I˛2;ˇ2 .z/ Á .z 2 /: (2.3.9) .˛1k C ˇ1/ .˛2k C ˇ2/ kD0

Generalizing the four-parametric Mittag-Leffler function (2.3.9) Al-Bassam and Luchko [Al-BLuc95] introduced the following Mittag-Leffler type function 16 2 Historical Overview of the Mittag-Leffler Functions

X1 zk N E..˛; ˇ/mI z/ D Qm .m 2 / (2.3.10) kD0 .˛j k C ˇj / j D1 with 2m real parameters ˛j >0I ˇj 2 R .j D 1;:::;m/ and with complex z 2 C: In [Al-BLuc95] an explicit solution to a Cauchy type problem for a fractional differential equation is given in terms of (2.3.10). The theory of this class of functions was developed in the series of articles by Kiryakova et al. [Kir99, Kir00, Kir08, Kir10a, Kir10b]. In the last several decades the study of the Mittag-Leffler function has become a very important branch of Special Function Theory. Many important results have been obtained by applying integral transforms to different types of functions from the Mittag-Leffler collection. Conversely, Mittag-Leffler functions generate new kinds of integral transforms with properties making them applicable to various mathematical models. A number of more general functions related to the Mittag-Leffler function will be discussed in Chap. 6 below. Nowadays the Mittag-Leffler function and its numerous generalizations have acquired a new life. The recent notable increased interest in the study of their relevant properties is due to the close connection of the Mittag-Leffler function to the Fractional Calculus and its application to the study of Differential and Integral Equations (in particular, of fractional order). Many modern models of fractional type have recently been proposed in Probability Theory, Mechanics, Mathematical Physics, Chemistry, Biology, Mathematical Economics etc. Historical remarks concerning these subjects will be presented at the end of the corresponding chapters of this book. Chapter 3 The Classical Mittag-Leffler Function

In this chapter we present the basic properties of the classical Mittag-Leffler function E˛.z/ (see (1.0.1)). The material can be formally divided into two parts. At first, starting from the basic definition of the Mittag-Leffler function in terms of a power series, we discover that for parameter ˛ with positive real part the function E˛.z/ is an entire function of the complex variable z. Therefore we discuss in the first part the (analytic) properties of the Mittag-Leffler function as an entire function. Namely, we calculate its order and type, present a number of formulas relating it to elementary and special functions as well as recurrence relations and differential formulas, introduce some useful integral representations and discuss the asymptotics and distribution of zeros of the classical Mittag-Leffler function. It is well-known that current applications mostly use the properties of the Mittag- Leffler function with real argument. Thus, in the second part (Sect. 3.7), we collect results of this type. They concern integral representations and integral transforms of the Mittag-Leffler function of a real variable, the complete monotonicity property and relations to fractional calculus. On first reading, people working in applications can partly omit some of the deeper mathematical material (say, that from Sects. 3.4 to 3.6).

3.1 Deﬁnition and Basic Properties

Following Mittag-Leffler’s classical definition we consider the one-parametric Mittag-Leffler function as defined by the power series

X1 zk E .z/ D .˛ 2 C/: (3.1.1) ˛ .˛kC 1/ kD0

© Springer-Verlag Berlin Heidelberg 2014 17 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__3 18 3 The Classical Mittag-Leffler Function

Although information on this function is widely spread in the literature (see, e.g. [Dzh66, GupDeb07, MatHau08, HaMaSa11]), we think that a fairly complete presentation here will help the reader to understand the ideas and results presented later this book. 1 Applying to the coefﬁcients c WD of the series (3.1.1) the Cauchy– k .˛kC 1/ Hadamard formula for the radius of convergence

jckj R D lim supk!1 ; (3.1.2) jckC1j and the asymptotic formula [Bat-1, 1.18(4)] Ä Â Ã .z C a/ .a b/.a b 1/ 1 D zab 1 C C O .z !1; jarg zj </; .z C b/ 2z z2 (3.1.3) one can see that the series (3.1.1) converges in the whole complex plane for all Re ˛>0.ForallRe˛<0it diverges everywhere on C nf0g.ForRe˛ D 0 the radius of convergence is equal to

jIm ˛j R D e 2 :

In particular, for ˛ 2 RC tending to 0 one obtains the following relation:

X1 1 E .˙z/ D .˙1/k zk D ; jzj <1: (3.1.4) 0 1 z kD0

In the most interesting case, Re ˛>0, the Mittag-Lefﬂer function is an entire function. Moreover, it follows from the Cauchy inequality for the Taylor coefﬁcients and simple properties of the Gamma function that there exists a number k 0 and a positive number r.k/ such that

rk ME˛ .r/ WD maxjzjDr jE˛.z/j < e ; 8r>r.k/: (3.1.5)

This means that E˛.z/ is an entire function of finite order (see, e.g., [Lev56]). For ˛>0by Stirling’s asymptotic formula [Bat-1, 1.18(3)] p ˛kC 1 .˛kC 1/ D 2 .˛k/ 2 e˛k .1 C o.1//; k !1; (3.1.6) one can see that the Mittag-Leffler function satisfies for ˛>0the relations

klog k klog k 1 lim sup D limk!1 D ; k!1 log 1 logj.˛kC 1/j ˛ jck j 3.2 Relations to Elementary and Special Functions 19 and s ! p Á Á 1 1 1 e ˛ lim sup k k jc j D lim k k D : k!1 k k!1 j.˛kC 1/j ˛

If Re ˛>0,andIm˛ 6D 0, the corresponding result is valid too. This follows from formula (3.1.3) which in particular means ˇ ˇ ˇ ˇ ˇ .˛kC 1/ ˇ 00;the Mittag-Lefﬂer 1 function (3.1.1) is an entire function of order D and type D 1. Re ˛

In a certain sense each E˛.z/ is the simplest entire function among those having the same order (see, e.g., [Phr04]and[GoLuRo97]). The Mittag-Leffler function also furnishes examples and counter-examples for the growth and other properties of entire functions of finite order (see, e.g., [Buh25a]). One can also observe that from the definition of the order and the type of an entire function (see formulas (B.2.3)and(B.2.4)) and from the above Proposition 3.1 it follows that the function

X1 . ˛ z/k E . ˛ z/ D ; >0; ˛ .˛kC 1/ kD0

1 has order D and type . Re ˛

3.2 Relations to Elementary and Special Functions

The Mittag-Leffler function plays an important role among special functions. First of all it is not difficult to obtain a number of its relations to elementary and special functions. The simplest relation is formula (3.1.4) representing E0.z/ as the sum of a geometric series. We collect in the following proposition other relations of this type. Proposition 3.2 (Special cases). For all z 2 C the Mittag-Leffler function satisfies the following relations 20 3 The Classical Mittag-Leffler Function

X1 .˙1/kzk E .˙z/ D D e˙z; (3.2.1) 1 .kC 1/ kD0

X1 .1/kz2k E .z2/ D D cos z; (3.2.2) 2 .2kC 1/ kD0

X1 z2k E .z2/ D D cosh z; (3.2.3) 2 .2kC 1/ kD0

1 k h i X k 2 1 .˙1/ z z 1 z 1 E 1 .˙z 2 / D D e 1 C erf.˙z 2 / D e erfc.z 2 /; (3.2.4) 2 .1 k C 1/ kD0 2 where erf (erfc) denotes the error function (complementary error function) Z z 2 2 erf.z/ WD p eu du; erfc.z/ WD 1 erf.z/; z 2 C; 0

1 and z 2 means the principal branch of the corresponding multi-valued function deﬁned in the whole complex plane cut along the negative real semi-axis. A more general formula for the function with half-integer parameter is valid Â Ã 1 2 p 1 z2 E p .z/ D0Fp1 I ; ;:::; I 2 p p p pp

C Â Ã p 1 2 2 2 z p C 2 p C 3 3p z C p 1F2p1 1I ; ;:::; I ; (3.2.5) pŠ 2p 2p 2p pp where pFq is the .p; q/-hypergeometric function

X1 k .a1/k.a2/k :::.ap/k z pFq.z/ Dp Fq.a1;a2;:::;apI b1;b2;:::;bq I z/ D : .b1/k.b2/k :::.bq/k kŠ kD0 (3.2.6)

G The formulas (3.2.1)–(3.2.3) follow immediately from the deﬁnition (3.1.1). Let us prove formula (3.2.4). We ﬁrst rewrite the series representation (3.1.1) 1 assuming that z 2 is the principal branch of the corresponding multi-valued function 1 and substituting z in place of z 2 :

X1 z2m X1 z2mC1 E 1 .z/ D C D u.z/ C v.z/: (3.2.7) 2 .mC 1/ 3 mD0 mD0 .mC 2 / 3.3 Recurrence and Differential Relations 21

The sum u.z/ is equal to ez2 . To obtain the formula for the remaining function v one can use the series representation of the error function as in [Bat-1]

X1 m 2 2 2 erf.z/ D p ez z2mC1; z 2 C: (3.2.8) .2m C 1/ŠŠ mD0

An alternative proof can be obtained by a term-wise differentiation of the second series in (3.2.7). It follows that v.z/ satisﬁes the Cauchy problem for the ﬁrst order differential equation in C. Ä 1 v0.z/ D 2 p C zv.z/ ; v.0/ D 0:

Representation (3.2.4) follows from the solution of this problem Z z 2 2 2 2 v.z/ D ez p eu du D ez erf.z/: 0

To prove (3.2.5) one can simply use the definitions of the Mittag-Leffler function (3.1.1) with ˛ D p=2 and the generalized hypergeometric function (3.2.6) and compare the coefficients at the same powers in both sides of (3.2.5). F For an interesting application of the function E1=2 see [Gor98, Gor02].

3.3 Recurrence and Differential Relations

Proposition 3.3 (Recurrence relations). The following recurrence formulas relating the Mittag-Lefﬂer function for different values of parameters are valid:

Xq1 1 2li E .z/ D E .z1=q e q /; q 2 N: (3.3.1) p=q q 1=p lD0 " # Xq1 .1 m ; z/ 1 z q E 1 .z q / D e 1 C ;q D 2;3;:::; (3.3.2) q .1 m / mD0 q R z u a1 1=q where .a;z/ WD 0 e u du denotes the incomplete gamma function, and z means the principal branch of the corresponding multi-valued function. G To prove relation (3.3.1) we use the well-known identity (discrete orthogonality relation) 22 3 The Classical Mittag-Lefﬂer Function

Xp1 2lki p; if k Á 0.mod p/; e p D (3.3.3) 0; if k 6Á 0.mod p/: lD0

This together with deﬁnition (3.1.1) of the Mittag-Lefﬂer function gives

Xp1 2li p p E˛.ze / D pE˛p.z /; p 1: (3.3.4) lD0

˛ 1 Substituting for ˛ and z p for z we arrive at the desired relation (3.3.1) after setting p ˛ D p=q. We mention the following “symmetric” variant of (3.3.4):

Xm Á 1 1 2li E .z/ D E z 2mC1 e 2mC1 ;m 0: (3.3.5) ˛ 2m C 1 ˛=.2mC1/ lDm

Relation (3.3.2) follows by differentiation, valid for all p; q 2 N (see below). F Proposition 3.4 (Differential relations). Â Ã d p E .zp/ D E .zp/; (3.3.6) dz p p

q1 dp X zkp=q E zp=q D E zp=q C ;qD 2;3;:::: (3.3.7) dzp p=q p=q .1 kp=q/ kD1

G These formulas are simple consequences of deﬁnition (3.1.1).F Let p D 1 in (3.3.7). Multiplying both sides of the corresponding relation by ez we get

q1 d X zk=q ezE z1=q D ez : dz 1=q .1 k=q/ kD1

By integrating and using the deﬁnition of the incomplete gamma function we arrive at the relation (3.3.2). The relation (3.3.2) shows that the Mittag-Lefﬂer functions of rational order can be expressed in terms of exponentials and the incomplete gamma function. In particular, for q D 2 we obtain the relation Ä 1=2 z 1 E1=2.z / D e 1 C p .1=2;z/ : (3.3.8)

2 .z/ D .1=2;p z / This is equivalent to relation (3.2.4) by the formula erf . 3.4 Integral Representations and Asymptotics 23

3.4 Integral Representations and Asymptotics

Many important properties of the Mittag-Lefﬂer function follow from its integral representations. Let us denote by ."I a/ ." > 0; 0 < a Ä / a contour oriented by non-decreasing arg consisting of the following parts: the ray arg Da, j j", the arc a Ä arg Ä a, j jD", and the ray arg D a, j j".If0

• Let ˛ D 2. Then the Mittag-Lefﬂer function E2 can be represented in the form

Z 1 2 1 e ./ E2.z/ D d ; z 2 G ."I /I (3.4.4) 4i ."I/ z

Z 1 1 2 1 z 2 1 e .C/ E2.z/ D e C d ; z 2 G ."I /: (3.4.5) 2 4i ."I/ z

1 1 In (3.4.2)–(3.4.5) the function z ˛ (or ˛ ) means the principal branch of the corresponding multi-valued function determined in the complex plane C cut along the negative semi-axis which is positive for positive z (respectively, ). G We use in the proof the Hankel integral representation for the reciprocal of the Euler Gamma function (see formula (A.1.19a)) 24 3 The Classical Mittag-Lefﬂer Function Z 1 1 D euusdu;">0;

Formula (3.4.6) is also valid for a D 2 ; Re s>0,i.e. Z 1 1 D euusdu;">0;Re s>0: (3.4.7) .s/ 2i ."I 2 /

We now rewrite formulas (3.4.6)and(3.4.7) in a slightly modiﬁed form. Let us begin with the integral representation (3.4.6). After the change of variables u D 1=˛ (in the case 1 Ä ˛<2we only consider the contours ."I / with 2 .=2; =˛/) we arrive at Z C 1 1 1=˛ s 1 1 D e ˛ d ; ˛=2<ˇÄ min f; ˛g : (3.4.8) .s/ 2i ˛ ."Iˇ/

Similarly, using the change of variables u D 1=2 in (3.4.7)1 we have Z C 1 1 1=2 s 1 D e 2 d ; Re s>0: (3.4.9) .s/ 4i ."I/

Let us begin with the case ˛<2. First let jzj <". In this case

sup jz 1j <1: 2."Iˇ/

It now follows from the integral representation (3.4.9) and the deﬁnition (3.1.1)of the function E˛.z/ that for 0<˛<2; jzj <";

1 Z X C 1 1=˛ ˛k 1 1 1 k E˛.z/ D e ˛ d z 2i ˛ I kD0 ." ˇ/ ( ) Z X1 1 1=˛ 1 D e .z 1/k d 2i ˛ I ." ˇ/ kD0 Z 1 e 1=˛ D d : 2i ˛ ."Iˇ/ z

The last integral converges absolutely under condition (3.4.1) and represents an analytic function of z in each of the two domains: G./."I ˇ/ and G.C/."I ˇ/.

1Since " is assumed to tend to zero, in the following we will retain the same notation ."I / for the path which appears after the change of variable. 3.4 Integral Representations and Asymptotics 25

On the other hand, the disk jzj <"is contained in the domain G./."I ˇ/ for any ˇ. It follows from the Analytic Continuation Principle that the integral representation (3.4.2) holds for the whole domain G./."I ˇ/. .C/ ./ Let now z 2 G ."I ˇ/. Then for any "1 > jzj we have z 2 G ."1I ˇ/,and using formula (3.4.2) we arrive at

Z 1=˛ 1 e E˛.z/ D d : (3.4.10) 2i ˛ ."1Iˇ/ z

On the other hand, for "

.C/ The representation (3.4.3) of the function E˛.z/ in the domain G . I ˇ/ now follows from (3.4.10)and(3.4.11). To prove the integral representations (3.4.4)and(3.4.5)for˛ D 2 we argue analogously to the case 0<˛<2using the representation (3.4.9). Recall that there is no need to revise formula (3.4.5)for˛ D 2 sincewehaveexact representations (3.2.2)and(3.2.3) in this case. F It should be noted that integral representations (3.4.2)–(3.4.3) can be used for the representation of the function E˛.z/; 0 < ˛ < 2; at any point z of the complex plane. To obtain such a representation it is sufﬁcient to consider contours ."I ˇ/ and ."I / with parameter "

Then we have the following asymptotics for formulas in which p is an arbitrary positive integer2:

p 1 X zk E .z/ D exp.z1=˛/ C O jzj1p ; jzj!1; jargzjÄ ; ˛ ˛ .1 ˛k/ kD1 (3.4.14) p X zk E .z/ D C O jzj1p ; jzj!1; ÄjargzjÄ: ˛ .1 ˛k/ kD1 (3.4.15) For the case ˛ 2, we have

X Á Xp k 1 1 2i z 1p E .z/ D exp z ˛ e ˛ C O jzj ; ˛ ˛ .1 ˛k/ kD1 jzj!1; jargzj <; (3.4.16)

˛ where the ﬁrst sum is taken over all integers such that j2 C argzjÄ 2 ; i.e., ˛ 2 A.z/ Dfn W n 2 Z; jargz C 2njÄ g (3.4.17) 2 and where argz can take any value between and C inclusively. G 10. Let us ﬁrst prove asymptotic formula (3.4.14). Let ˇ be chosen so that ˛ < <ˇÄ min f; ˛g : (3.4.18) 2

Substituting the expansion

p 1 X k1 p D C ;p 1; (3.4.19) z zk zp. z/ kD1 into formula (3.4.3) with " D 1, we get the representation of the function E˛.z/ in the domain G.C/.1I ˇ/: Xp Â Z Ã 1 z1=˛ 1 1=˛ k1 k E˛.z/ D e e d z ˛ 2i ˛ I kD1 .1 ˇ/ Z 1 e 1=˛ p C p d : (3.4.20) 2i ˛ z .1Iˇ/ z

2We adopt here and in what follows the empty sum convention: if the upper limit is smaller than the lower limit in a sum, then this sum is empty, i.e. has to be omitted. In particular, it can said that (3.4.14)and(3.4.15) also hold for p D 0. 3.4 Integral Representations and Asymptotics 27

Now Hankel’s formula (3.4.6) yields Z 1 1=˛ 1 e k1 d D ;k 1: 2i ˛ .1Iˇ/ .1 k˛/

Using this formula and (3.4.20) we arrive under conditions (3.4.18)at

Xp k 1 1=˛ z E .z/ D ez ˛ ˛ .1 k˛/ kD1 Z 1 e 1=˛ p C ; j zjÄ ; jzj >1: p d arg (3.4.21) 2i ˛z .1Iˇ/ z

We denote the last term in formula (3.4.21)byIp.z/ and estimate it for sufﬁciently large jzj and jargzjÄ :In this case we have

min j zjDjzj sin.ˇ / 2.1Iˇ/ and, consequently, Z 1p jzj 1=˛ p jIp.z/jÄ je jj jjd j : (3.4.22) 2 ˛ sin.ˇ / .1Iˇ/

Note that the integral in the right-hand side of (3.4.22) converges since the contour .1I ˇ/ consists of two rays arg D˙ˇ; j j1;on which we have

˚ « ˇ jexp 1=˛ jDexp cos j j1=˛ ; arg D˙ˇ; j j1; ˛ and cos ˇ=˛ < 0 due to condition (3.4.18). The asymptotic formula (3.4.14)now follows from the representation (3.4.21) and the estimate (3.4.22). 20. To prove (3.4.15) let us choose a number ˇ satisfying ˛ <ˇ<

p Z 1=˛ X zk 1 e p E .z/ D C d z 2 G./.1I ˇ/: ˛ .1 k˛/ 2i ˛ zp z kD1 .1Iˇ/ (3.4.24) 28 3 The Classical Mittag-Lefﬂer Function

If ÄjargzjÄ condition (3.4.23) gives, for sufﬁciently large jzj,

min j zjDjzj sin . ˇ/: 2.1Iˇ/

For ˇ chosen as in (3.4.23) the domain ÄjargzjÄ is contained in the domain G./.1I ˇ/, and thus the result follows from representation (3.4.24) and the estimate ˇ ˇ Z ˇ Xp k ˇ 1p ˇ z ˇ jzj 1=˛ ˇ ˇ p ˇE˛.z/ C ˇ Ä je jj jjd j ; (3.4.25) .1 k˛/ 2 ˛ sin. ˇ/ I kD1 .1 ˇ/ is valid for sufﬁciently large jzj and ÄjargzjÄ: 30. To prove (3.4.16) we note ﬁrst that formula (3.3.5)istrueforany˛>0and p 0. Fixing ˛; ˛ 2, we can always choose an integer m 1 such that ˛ ˛ D <2and, consequently, we can use (3.4.14)–(3.4.15)for any term 1 2m C 1 of the sum in the right-hand side of formula (3.3.5). ˛ ˛ Let 1 <

X 2i 1 2m C 1 1 e ˛ E .z/ D ez ˛ ˛ 2m C 1 ˛ 2B.z/ 0 1 ( ) q C 1 Xm Xq k 2ki 1 z 2mC1 e 2mC1 B C C O @jzj 2m C 1 A ; (3.4.26) 2m C 1 k˛ Dm kD1 1 2mC1 where ˇ Â Ãˇ ˇ 2ni ˇ ˇ 1 e 2mC1 ˇ B.z/ D n W n 2 Z; ˇarg z 2mC1 ˇ Ä : (3.4.27)

The last inequality can be rewritten in the form jargz C 2njÄ.2m C 1/ . ˛ ˛ Let us ﬁx some z.If 0 > and the difference 0 is small enough, then 2 2 ˛ the inequalities jargz C 2njÄ and jargz C 2njÄ 0 havethesamesetof 2 solutions with respect to n 2 Z. ˛ Since the number .2m C 1/ > can be chosen in an arbitrary small neigh- 2 ˛ bourhood of ,formula(3.4.27) can be rewritten in the form 2 3.4 Integral Representations and Asymptotics 29

X 1 1 2i E .z/ D ez ˛ e ˛ ˛ ˛ 2A.z/ ( ) q k m Á X 2mC1 X C 1 z 2ki q 1 Â Ã e 2mC1 C O jzj 2mC1 ; (3.4.28) 2m C 1 k˛ kD1 1 Dm 2m C 1 where the summation is taken over the set A.z/ described in (3.4.17). Formula (3.4.28) has been proved for any integer q 1. To get from here the representation of the form (3.4.16) let us ﬁx any p 1 and choose q D (2m +1) .p C 1/ 1. Using formula (3.4.28) and the discrete orthogonality relation

2ki Xm 2m C 1; if k Á 0.mod .2m C 1//; e 2m C 1 D 0; if k 6Á 0.mod .2m C 1//; Dm we ﬁnally arrive at formula (3.4.16). F As a simple consequence of Proposition 3.6 we have ˛ Corollary 3.7. Let 0<˛<2and < 0:

1 M jE .z/jÄM eRe z ˛ C 2 : (3.4.29) ˛ 1 1 Cjzj

20.If ÄjargzjÄ and jzj0:

M jE .z/jÄ 2 : (3.4.30) ˛ 1 Cjzj

Here M1 and M2 are constants not depending on z. Corollary 3.8. Let 0<˛<2and jzjDr>0. Then the following relations hold: 10.

1 ˇ ˇ r ˛ ˇ i ˇ lim e E˛.re / D 0; j j <; (3.4.31) r!C1

and the limit in (3.4.31) is uniform with respect to . 30 3 The Classical Mittag-Lefﬂer Function

20.

1 r ˛ 1 lim e jE˛.r/j D : (3.4.32) r!C1 ˛

The last results were obtained by G. Mittag-Lefﬂer [ML3].

3.5 Distribution of Zeros

In this section we consider the problem of the distribution of zeros of the Mittag- Lefﬂer function E˛.z/. The following lemma presents two situations in which this distribution is easily described.

Lemma 3.9. (i) The Mittag-Lefﬂer function E1.z/ has no zero in C. (ii) All zeros of the function E2.z/ are simple and are situated on the real negative semi-axis. They are given by the formula Á 2 z D C k ;k2 N Df0; 1; 2; : : :g: (3.5.1) k 2 0 G The ﬁrst statement follows immediately from formula (3.2.1)

z E1.z/ D e ; z 2 C; and properties of the exponential function. As for zeros of the function E2.z/ one can use one of the representations (3.2.2) or (3.2.3). Hence the zeros are described by formula (3.5.1).Theyaresimple(i.e. p are of ﬁrst order) by the differentiation formulas for the function cos z. F The following fact is commonly used:

Corollary 3.10. The exponential function E1.z/ is the only Mittag-Lefﬂer function which has no zeros in the whole complex plane. All other functions E˛.z/; Re ˛> 0;˛6D 1; have inﬁnitely many zeros in C. This fact is partly a simple consequence of Proposition 3.1, which states that for 1 Re ˛>0the function E .z/ is an entire function of order D . Then for ˛ Re ˛ 1 each ˛; Re ˛ 6D ;n 2 N, the order of E .z/ is a positive non-integer. Then, for n ˛ these values of the parameter ˛, the statement follows from the general theory of entire functions (see, e.g., Levin [Lev56]). The statement is still valid for all ˛ 6D 1 including when ˛ is reciprocal to a natural number, but in this case the argument is much more delicate. We return to the proof later. Next we consider another simple case, Re ˛ 2. 3.5 Distribution of Zeros 31

Lemma 3.11. For each value ˛, Re ˛ 2, the Mittag-Leffler function E˛.z/ has infinitely many zeros lying on the real negative semi-axis. There exists only finitely many zeros of the function E˛.z/; Re ˛ 2; which are not negative real (i.e., zeros belonging to C n .1;0).

G If Re ˛>2then by Proposition 3.1 the function E˛.z/ is an entire function of 1 1 order D < . The general theory of entire functions says (see, e.g., [Lev56]) Re ˛ 2 that in this case the set of zeros of such a function is denumerable. Therefore it suffices to determine only the location of these zeros. Let us study for simplicity only the case of positive ˛>2. Consider the asymptotic formula (3.4.16). For each fixed z 6Dx;x 0; the index set A.z/ in the sum on the right-hand side of (3.4.16) consists of a finite number of elements (see definition (3.4.17)). Moreover, straightforward calculations show that the modulus of each term in this sum is equal to

Á ˇ Áˇ 1 C ˇ 1 2i ˇ ˛ argz 2 jzj cos ˛ '.z/ D ˇexp z ˛ e ˛ ˇ D e :

Comparing the values of the functions ' for a ﬁxed z but for different 2 A.z/ one can conclude that the function '0 is the maximal one. More precisely, if we consider any angle jargzj < " with sufﬁciently small ">0, then there exists ı>0and r0 >0such that

1 r ˛ .cos " ı/ jE˛.z/j > e ˛ ; jzjDr>r0; jargzj < ":

Both statements of the lemma follow since ">0is an arbitrary small number. F We also mention a result which was stated in [Wim05b], but was only proved in [OstPer97] (see the comments in [PopSed11, p. 56]).

Proposition 3.12. All zeros of the classical Mittag-Lefﬂer function E˛.z/ with ˛>2are simple and negative. These zeros zn;n D 1;2;:::;satisfy the following inequalities: Â Ã Â Ã n ˛ .n 1/ ˛ < zn < : (3.5.2) sin ˛ sin ˛

The most interesting case of the distribution of zeros of E˛.z/ is that for 0< ˛< 2;˛6D 1. Let us first introduce some notation and a few simple facts. It follows from Proposition 3.6 (see formulas (3.4.14)–(3.4.15)) that the zeros of the function E˛.z/ (if any) with sufficiently large modulus are situated in two angular domains n ˇ ˇ o ˇ ˛ˇ ˝.˙/ D z 2 C W ˇargz ˇ <ı ; ı 2 32 3 The Classical Mittag-Leffler Function ˚ « ˛ ˛ where ı is an arbitrary positive number, ı 2 0; min 2 ; 2 . Let us denote those zeros of E˛.z/ which belong to the upper half-plane (lower half-plane) by .C/ ./ zk (respectively by zk ) ordering each of these collections in increasing order by .C/ .C/ ./ ./ modulus, i.e. jzk jÄjzkC1j (jzk jÄjzkC1j). We also have to note that since for a real value of the parameter ˛ the function E˛.z/ has the symmetry property E˛.z/ D E˛.z/, we have the following connection between “upper” and “lower” zeros:

./ .C/ zk D zk ;kD 1;2;:::: (3.5.3)

The distribution of zeros of the function E˛.z/ in the considered case 0< ˛< 2;˛6D 1; can be described in the following form:

Proposition 3.13. All zeros of the function E˛;0<˛<2;˛6D 1; with sufﬁciently large modulus are simple, and the following asymptotic formula for its zeros holds: Â Ã .˙/ ˙i ˛ ˛ log k z D e 2 .2k/ 1 C O ;k!1: (3.5.4) k k

G Due to the symmetry property (3.5.3) it suffices to consider only zeros lying in the upper half-plane. To do this we use the asymptotic formula (3.4.14). The reason for using an asymptotic description of the zeros of the function E˛.z/ is two-fold. First, we select a part of the right-hand side of the asymptotic formula (3.4.14) and find its zeros with sufficiently large modulus. We also derive the asymptotic behaviour of these auxiliary zeros. This helps us in the second step to approximate the zeros of the right-hand side of (3.4.14) (i.e., of the function E˛.z/), and to find the asymptotics for them. Denote by

˛ sin ˛ c WD D .1C ˛/ (3.5.5) ˛ .1 ˛/ the real constant which is positive for 0<˛<1and negative for 1<˛<2.We also introduce the following function:

1 z ˛ e˛ WD z e c˛: (3.5.6)

Then formula (3.4.14) can be rewritten in the form Â Ã 1 ˛ z E .z/ D e .z/ C O ; jzj!1: (3.5.7) ˛ ˛ z 3.5 Distribution of Zeros 33

Let us ﬁrst solve the equation

1 z ˛ z e D c˛: (3.5.8)

Let us denote by L0 the set of all z for which the modulus of the left- and right-hand sides of (3.5.8) coincide

1 z ˛ L0 WD z 2 C Wjz e jDjc˛j :

i It is equivalent to say either z D r e 2 L0 or Â Ã 1 1 r ˛ cos Dlog r C log jc j: (3.5.9) ˛ ˛

.C/ ./ Hence L0 is a (continuous) curve consisting of two branches L0 ;L0 (symmetric with respect to the real axis) whose equations can be written for large enough r in the form ( !) 3 .˙/ ˛ log r log jc˛j log r L0 W arg z WD D˙ C ˛ 1 C ˛ 1 C O 1 : 2 r ˛ r ˛ r ˛ (3.5.10) In order to compare the arguments of the left- and right-hand sides of (3.5.8)we 1 calculate the argument of z ez ˛ Â Ã Â Ã 1 z ˛ 1 arg z e D C r ˛ sin : ˛

.C/ Then for all z 2 L0 with sufﬁciently large r Djzj we have from (3.5.10) Â Ã ! 2 1 1 log r ˛ ˛ r sin D r C O 1 ;r!1: (3.5.11) ˛ r ˛

i The zeros of the function e˛ in the upper half-plane are those points r e on the .C/ N curve L0 which satisfy for certain k 2 the corresponding equality for arguments

1 z ˛ argz e D arg c˛ C 2k; or Â Ã 1 C r ˛ sin D argc C 2k: (3.5.12) ˛ ˛ 34 3 The Classical Mittag-Lefﬂer Function

Since the left-hand side of the last equation satisfies the asymptotic formula (3.5.11), then for sufficiently large r there exists a denumerable collection of natural numbers k for which equality (3.5.12) holds. These values of k determine zeros of the i function e˛ with sufficiently large modulus. Denote these zeroes by k WD k e k and recall the equations for them 8 Á 1 ˆ ˛ k < k cos ˛ Dlog k C log jc˛j; Á (3.5.13) ˆ 1 : ˛ k k C k sin ˛ D arg c˛ C 2k:

It follows from the second relation that k !1for k !1. More precisely 0 1

1 ˛ @ 1 A k D 2k C O 1 : ˛ k

Therefore the asymptotic formulas (3.5.10)–(3.5.11) lead us to the following asymptotics for k and k with respect to k: 8 Á ˆ ˛ log k < k D 2 C O k ;k!1; ˆ h Ái (3.5.14) ˆ 2 : ˛ log k k D .2k/ 1 C O k2 ;k!1:

These relations determine the behaviour of the zeros k of the function e˛.z/ lying outside a certain disk. The formulas (3.5.14) show in particular that such zeros are simple. .C/ The continuity of the function e˛.z/ on L0 implies that for each pair of i k .C/ successive zeros k, kC1 there exists a point wk D !k e on L0 such that

1 w ˛ argwk e k D arg c˛ C 2k C ; and accordingly

e˛.wk / Dc˛:

.C/ Let us introduce the domain k./ encircled by two curves L0 .˙/ determined by the equations: Â Ã .C/ 1 L .˙/ W r ˛ cos Dlog r C log jc j˙; (3.5.15) 0 ˛ ˛ 3.6 Further Analytic Properties 35 with a fixed sufficiently small >0and two circular arcs lj WD fz 2 C WjzjD!j g, j D k 1; k. From the definition of these curves we obtain

je˛.z/j > jc˛j; z 2 lk;k>N0; (3.5.16)

˙ .C/ je˛.z/jjc˛jje 1j; z 2 L0 .˙/: (3.5.17)

The right-hand sides of inequalities (3.5.16)–(3.5.17) are constants not depending on k. Therefore one can apply to (3.5.7) Rouché’s theorem. This implies that the function E˛.z/ has the same number of zeros in the domains k with k N1 N0 as the function e˛.z/ has. On the other hand, according to the construction of the domain k, the function e˛.z/ has in this domain exactly one zero k. Consequently, .C/ the function E˛.z/ has exactly one simple zero zk inside of k;k N1,and

.C/ zk D k C ˛k ;˛k D O.dk/; (3.5.18) where dk is the diameter of the domain k. ˛1 It can easily be shown that the perimeter of the contour @k has order O k ˛1 and, consequently, dk D O k . This, together with the asymptotics of k in (3.5.14) and the representation (3.5.18), gives the following asymptotical formula .C/ for zk : Ä Â Ã .C/ i ˛ ˛ log k z D e 2 .2k/ 1 C O ;k!1: (3.5.19) k k

If z0 is a zero of the function E˛.z/ in the upper half-plane with large enough modulus then by using formula (3.5.7) we arrive at the inequalities Á 1 argz0 log jz jClog jc j < jz j ˛ cos < log jz jClog jc jC : 0 ˛ 2 0 ˛ 0 ˛ 2

This means that all zeros of the function E˛.z/ in the upper half-plane with sufﬁciently large modulus are contained in the curvilinear strip between the curves .C/ L0 .˙/. The latter inequalities also give the asymptotic representation (3.5.4)for .C/ the zeros zk of the function E˛.z/. F We note that the construction proposed in the proof of Proposition 3.13 also completes the proof of Corollary 3.10.

3.6 Further Analytic Properties

In this section we present some additional analytic results for the Mittag-Leffler function. They mostly deal with integral properties of the Mittag-Leffler function and its relation to some special functions. 36 3 The Classical Mittag-Leffler Function

The first property describes the relation between the Mittag-Leffler function and the generalized Wright function (this relation is also known as the Euler transform of the Mittag-Leffler function , see, e.g., [MatHau08,p.84]).Let˛; ; 2 C, >0 and Re ˛>0,Re >0, then the following representation holds:

Z1 Ä ˇ ˇ 1 1 . ; /; .1; 1/ ˇ t .1 t/ E˛.xt /dt D . /22 ˇ x ; (3.6.1) .1; ˛/; . C ; / 0 where 22 is a special case of the generalized Wright function pq (see formula (F.2.12)): Ä ˇ ˇ X1 k . ; /; .1; 1/ ˇ . C k/ .1 C k/ x 22.z/ WD 22 ˇ z D : .1; ˛/; . C ; / .1C ˛k/ . C C k/ kŠ kD0

Formula (3.6.1) follows from the series representation of the Mittag-Lefﬂer function (3.1.1) and simple calculations involving properties of the Beta function (see formula (A.2.4)). Next, we obtain the Mellin–Barnes integral representation for the Mittag-Lefﬂer function (see, e.g., [MatHau08, p. 88] and also Appendix D).

Lemma 3.14. Let ˛>0. Then the Mittag-Lefﬂer function E˛.z/ has the Mellin– Barnes integral representation Z 1 .s/.1 s/ E .z/ D .z/s ds; jarg zj <; (3.6.2) ˛ 2i .1 ˛s/ Lic where the contour of integration Lic is a straight line which starts at c i1 and ends at c C i1, .0

X1 X1 k .s C k/.s/.1 s/ s .1/ .1C k/ k D lim .z/ D .z/ D E˛.z/: s!k .1 ˛s/ kŠ .1 C ˛k/ kD0 kD0 F Two simple consequences of the representation (3.6.2) are the relations of the Mittag-Lefﬂer function E˛.z/ to the generalized Wright function (see formula (F.2.12)) 3.6 Further Analytic Properties 37 Ä ˇ ˇ .1; 1/ ˇ E˛.z/ D 11 ˇ z ; (3.6.3) .1; ˛/ and to the Fox H-function (see formulae (F.4.1)–(F.4.4)) Ä ˇ ˇ 1;1 ˇ .0; 1/ E˛.z/ D H z ˇ : (3.6.4) 1;2 .0; 1/; .0; ˛/

These formulas follow immediately from the Mellin–Barnes representations of the generalized Wright function and Fox H-function (see representations (F.2.17) and (F.4.1), respectively). 1 Further we also obtain an integral representation for the Cauchy kernel .Let z us ﬁrst introduce some deﬁnitions and notation. Let < Ä and ˛>0.By i 1 .e /˛ we denote the branch of the corresponding multi-valued function having 1 positive values expf ˛ log j jg when arg D and by L˛. I /;˛ > 0; 0,we denote the curve 8 <ˆ ; if ˛ 2; i 1 Re.e /˛ D ; jarg jÄ (3.6.5) :ˆ ˛ ; if 0<˛Ä 2: 2

The equation of the curve L˛. I / can be written in polar coordinates as

˛ 1 ˛ ; if ˛ 2; r cos . / D ; j jÄ ˛ (3.6.6) ˛ 2 ; if 0<˛Ä 2:

It follows from this equation that the curve L . I / is bounded and closed when ˛>2and has two unbounded branches when 0<˛Ä 2. Consequently, the -plane is divided by the curve L˛. I / into two complementary simply-connected domains D˛ . I / and D˛. I / containing, respectively, intervals 02;D 0 when the domain D˛. I / becomes the whole -plane without the point D 0). The following properties of these domains are easily veriﬁed: 0 1 .If 2 D˛. I /,then

i 1 Re .e /˛ >: (3.6.7)

0 2 . The domain D˛. I /; > 0; ˛ Ä 2, is contained in the angular domain n o ˛ . I ˛/ D Wjarg j < ; (3.6.8) 2 38 3 The Classical Mittag-Lefﬂer Function

and D˛. I 0/ D . I ˛/. The domain D˛ . I 0/ coincides with the angular domain complementary to . I ˛/: n o ˛ D. I 0/ D . I ˛/ D W < jarg jÄ : (3.6.9) ˛ 2 The following lemma gives an integral representation of the Cauchy kernel using the Mittag-Lefﬂer function. Lemma 3.15. Let ˛>0;>0and < Ä be some ﬁxed parameters. 0 1 .Ifz2 D˛ . I / and 2 D˛. I / then Z C1 1 1 1 ˛ .e i / ˛ t i ˛ i e E˛.e zt /dt D e ˛ : (3.6.10) 0 z

20. The integral (3.6.10) converges absolutely and uniformly with respect to the two variables z and if

z 2 G˛ . I /; 2 D˛. I /; (3.6.11)

where G˛ . I / is any bounded and closed subdomain of the domain D˛ . I /. G It sufﬁces to prove the statements of the lemma under the assumption

˛ jzjÄ1 ; 2 D˛. I /; (3.6.12) where 1 is any number from the interval .0; /. Let us choose for a given value of 1;0<1 <, a number ; 0 < < , Á ˛ q D 1 <1: (3.6.13) Then we arrive at the formula Â Ã k˛ k˛ max ft k˛e. /t gD ek˛; 0Ät

k˛C1 1 k˛ .1C k˛/>.k˛/ 2 e follows easily from Stirling’s formula. Hence, when q is chosen as in (3.6.13)and ˛ jzjÄ1 ,wehave ˇ ˇ ˇ i k k˛ ˇ ˇ.e z/ t . /t ˇ 1 k max ˇ e ˇ Ä .k ˛/ 2 q ;k k0: 0Ät

It follows that the expansion

X1 .ei z/kt k˛ e. /t E .ei zt˛/ D e. /t (3.6.14) ˛ .1C k˛/ kD0

˛ converges uniformly with respect to z and t when jzjÄ1 and 0 Ä tand assumption (3.6.12)theseries in (3.6.14) can be integrated term-by-term with respect to t along the semi-axis Œ0; C1/. Using the known formula Z C1 1 .e i / ˛ t k˛ .1C k˛/ i 1 e t dt D ; Re .e /˛ >0;k 0; kC 1 0 .ei / ˛ we arrive at Z C1 1 .e i / ˛ t i ˛ e E˛.e zt / dt 0 Z X1 i k C1 .e z/ i 1 D e.e / ˛ t t k˛dt .1C k˛/ kD0 0 Â Ã 1 k 1 X 1 ˛ i 1 z i D .e / ˛ D e ˛ ; z kD0 since

1 i ˛ ˛ ˛ ˛ j jjRe.e / j >1 jzj:

Thus we have proved formula (3.6.10) under the condition (3.6.12). F

3.7 The Mittag-Lefﬂer Function of a Real Variable

3.7.1 Integral Transforms

Let us recall a few basic facts about the Laplace transform (a more detailed discussion can be found in Appendix C, see also [BatErd54a,Wid46]). The classical Laplace transform is deﬁned by the following integral formula

Z1 .Lf/.s/D estf.t/dt; (3.7.1)

0 40 3 The Classical Mittag-Lefﬂer Function provided that the function f (the Laplace original) is absolutely integrable on the semi-axis .0; C1/. In this case the image of the Laplace transform (also called the Laplace image), i.e., the function

F.s/ D .Lf/.s/ (3.7.2)

(sometimes denoted as F.s/ D f.s/Q ) is deﬁned and analytic in the half-plane Re s>0. It may happen that the Laplace image can be analytically continued to the left of the imaginary axis Re s D 0 into a larger domain, i.e., there exists a non-positive real number s (called the Laplace abscissa of convergence) such that F.s/ D f.s/Q is analytic in the half-plane Re s s . Then the following inverse Laplace transform can be introduced Z 1 L1F .t/ D estF.s/ds; (3.7.3) 2i Lic where Lic D .c i1;c C i1/, c> s, and the integral is usually understood in the sense of the Cauchy principal value, i.e.,

Z cZCiT estF.s/ds D lim estF.s/ds: T !C1 Lic ciT

If the Laplace transform (3.7.2) possesses an analytic continuation into the half- plane Re s s and the integral (3.7.3) converges absolutely on the line Re s D c> s , then at any continuity point t0 of the original f the integral (3.7.3)givesthe value of f at this point, i.e., Z 1 est0 f.s/Q ds D f.t /: (3.7.4) 2i 0 Lic

Thus, under these conditions, the operators L and L1 constitute an inverse pair of operators. Correspondingly, the functions f and F D fQ constitute a Laplace transform pair.3 The following notation is used to denote this fact Z 1 st f.t/ f.s/Q D e f.t/dt; Re s> s ; (3.7.5) 0 where s is the abscissa of convergence. Here the sign denotes the juxtaposition of a function (depending on t 2 RC) with its Laplace transform (depending on s 2 C).

3It is easily seen that these properties do not depend on the choice of the real number c. 3.7 The Mittag-Lefﬂer Function of a Real Variable 41

In the following the conjugate variables ft;sg may be given in another notation, e.g. fr; sg, and the abscissa of the convergence may sometimes be omitted. Furthermore, throughout our analysis, we assume that the Laplace transforms obtained by our formal manipulations are invertible by using the standard Bromwich formula. For the Mittag-Lefﬂer function we have the Laplace integral relation Z 1 x ˛ 1 e E˛ .x z/ dx D ;˛ 0: (3.7.6) 0 1 z

This integral was evaluated by Mittag-Lefﬂer who showed that the region of convergence of the integral contains the unit disk and is bounded by the curve Re z1=˛ D 1. a The Laplace transform of E˛ .˙t / can be obtained from (3.7.6) by putting x D st and x˛ z D˙t ˛ ;weget Z 1 s˛1 L ŒE .˙t ˛ / WD st E .˙t a/ t D : ˛ e ˛ d ˛ (3.7.7) 0 s 1

This result was used by Humbert [Hum53] to obtain a number of functional relations satisﬁed by E˛.z/.Formula(3.7.7) can also be obtained by Laplace transforming the series (3.1.1) term-by-term, and summing the resulting series. Recalling the scale property of the Laplace transform

qf.qt/ F.s=q/;N 8q>0; we have

s˛1 E Œ˙.qt/˛ ; 8q>0: (3.7.8) ˛ s˛ q˛

As an exercise we can invert the r.h.s. of (3.7.8) either by means of the expansion method (ﬁnd the series expansion of the Laplace transform and then invert term- by-term to get the series representation of the l.h.s.) or by the Bromwich inversion formula (deform the Bromwich path into the Hankel path and, by means of an appropriate change of variable, obtain the integral representation of the left-hand side, see, e.g. [CapMai71a]). Using the asymptotic behaviour of the function E˛.z/ we will investigate further the Mellin integral transform of the function E˛.z/ and its properties. The Mellin integral transform is deﬁned by the formula

Z1 .Mf/.p/D f .p/ WD f.t/tp1dt.p2 C/ (3.7.9)

0 provided that the integral on the right-hand side exists. 42 3 The Classical Mittag-Lefﬂer Function

In many cases, the function E˛.z/ does not satisfy the convergence condition for the standard Mellin transform (see Theorem C.5). Therefore we ﬁnd the Mellin transform of a slightly different function. A good candidate for applying the general theory is the function

1 e .xI / D fE .x˛/ 1g ; (3.7.10) ˛ x ˛ where x>0and 6D 0 is any complex number. It happens that the function e˛.xI / satisﬁes, up to rotation, the above given convergence condition for certain values of parameters. 1 ˛ Lemma 3.16. Fix an ˛ in the interval . ;2. Then for each '; Ä ' Ä 2 2 2 ˛ ; the function e .xI ei'/ is square-integrable on the positive semi-axis: 2 ˛

i' e˛.xI e / 2 L2.0; C1/: (3.7.11)

G If 00such that

i' ˛1 je˛.xI e /jÄC1x ;0

1 Thus with the condition ˛> 2 we have

i' e˛.xI e / 2 L2.0; 1/; 0 Ä ' Ä 2: (3.7.12) ˛ ˛ Consider now the behaviour of the function e .xI ei'/ with Ä ' Ä 2 ˛ 2 2 for x 2 .1; C1/. In the case ˛ D 2 we have ' D . It follows then from (3.2.2)that Â Ã 1 1 cos x 1 e .xI1/ D E .x2/ 1 D D O ;x!1: 2 x 2 x x

Therefore

e2.xI1/ 2 L2.1; 1/: (3.7.13)

1 ˛ ˛ If <˛<2and Ä ' Ä 2 then one can use a variant of 2 2 2 the asymptotic estimate (3.4.30) for the Mittag-Lefﬂer function E˛.z/, namely the estimate

i' ˛ ˛ jE˛.e x /jÄC2x ;Ä ' Ä 2 ; 1 Ä x

˛ where C >0is a constant not depending on ',and <

i˛ e˛.xI e / 2 L2.1; C1/; Ä ˛ Ä 2 : (3.7.14) ˛ ˛ The same result follows for Ä ' Ä or 2 Ä ' Ä 2 from the 2 2 asymptotic formula (3.4.14). The lemma is proven. F For further considerations we need the Mellin integral transforms of some elementary functions, related to the Mittag-Lefﬂer function (cf., e.g., [AbrSte72, Mari83], and [NIST]). Lemma 3.17. If 0

The following lemma describes asymptotic properties of the function

Z 2 pei Ci.s1/ Up.s/ D e d ;0

s1 2 lim p Up.s/ D : (3.7.19) p!C1 .2 s/

G We can represent the function Up.s/ as a Taylor series in p in a neighbourhood of p D 0:

1 Z 1 X k 2 X k p i.k1Cs/ p sin.k 1 C s/ 2 Up.s/ D e d D 2 : kŠ kŠ k 1 C s kD0 2 kD0 44 3 The Classical Mittag-Lefﬂer Function

Here the series on the right-hand side can be rewritten as the sum of two series with even and odd indices of summation, respectively. Thus we arrive at cos s X1 p2k sin k C s X1 p2kC1 sin k C s U .s/D2 2 C2 2 2 C2 2 p 1 s .2k/Š 2k 1 C s .2k C 1/Š 2k C s kD1 kD0 cos s s X1 .1/kp2k s X1 .1/kp2kC1 D 2 2 2 cos C 2 sin 1 s 2 .2k/Š.2k 1 C s/ 2 .2k C 1/Š.2k C s/ kD1 kD0 Z Z cos s s p cos x 1 s p sin x D2 2 2p1s cos xs1 dx C2p1s sin xs1 dx: 1 s 2 0 x 2 0 x (3.7.20) Since 0

The formula (3.7.19) and the statement of the lemma now follow from the last formula and the well-known formula for the Euler Gamma function (formula (A.1.13)): .s/.1 s/ D :F sin.s/

In order to obtain the Mellin transform of the function connected with the Mittag- Lefﬂer function E˛.z/ we need one more auxiliary result. It is in a certain sense a special reﬁnement of the Jordan lemma (see, e.g., [AblFok97]). ˛ Let us draw a cut in the z-plane along the ray arg z D '; Ä ' Ä 2 ˛ 1 2 ; <˛Ä 2. Consider in the cut z-plane that branch of the function 2 2 .sC˛1/ ˛ 1 1 z ; Re s D 2 , which takes on the semi-axis 0

The proof follows from the asymptotic representations of the Mittag-Lefﬂer function and the above proved Lemmas 3.17 and 3.18. 3.7 The Mittag-Lefﬂer Function of a Real Variable 45

Finally we have the following: 1 Proposition 3.20. Let <˛Ä 2 be a ﬁxed value of the parameter. Then the 2 formula

Z .'/ C1 i' C i ˛ .sC˛1/ E˛.re / 1 .s ˛ 1/ 1 e 1 r ˛ dr D ; Res D ; rei' .2 s/ .sC˛1/ 2 0 sin. ˛ / (3.7.22) h i ˛ ˛ is valid for any ' 2 ;2 . 2 2 E .z/1 .sC˛1/ ˛ ˛ 1 1 G We consider the function z h z ,Rei;s D 2 ,inthez-plane which is ˛ ˛ cut along the ray argz D ', ' 2 ;2 . We denote by L.RI "/ aclosed 2 2 positive oriented contour consisting of two circles l" Dfz WjzjD"g, lR Dfz W jzjDRg;0<"

This formula can be rewritten as Z R i' C i .2 '/ .sC˛1/ E˛.re / 1 .s ˛ 1/ 1 ˛ r ˛ r e i' d " re Z Z C R i' C E˛.z/ 1 .s ˛ 1/ 1 i ' .sC˛1/ E˛.re / 1 .s ˛ 1/ 1 C z ˛ z ˛ r ˛ r d e i' d l z " re R Z C E˛.z/ 1 .s ˛ 1/ 1 C z ˛ dz D 0: l" z (3.7.23) If ">0is small enough, we have

E .z/ 1 2 max j ˛ jÄ : jzjD" z .C 1/

1 For such " and Re s D 2 we get the estimate ˇZ ˇ ˇ C ˇ ˇ E˛.z/ 1 .s ˛ 1/ 1 ˇ 2 ˛ 1=2 ˇ z ˛ dzˇ Ä 2" ˛ : l" z .˛C 1/

1 Since ˛>2 , we conclude that 46 3 The Classical Mittag-Lefﬂer Function Z C E˛.z/ 1 .s ˛ 1/ 1 lim z ˛ dz D 0: (3.7.24) "!0 l" z

Passing to the limit in the identity (3.7.23) with " ! 0 and using the formula (3.7.24) we arrive at the identity Z R i' C i ' .sC˛1/ i 2 .sC˛1/ E˛.re / 1 .s ˛ 1/ 1 ˛ f ˛ 1g r ˛ r e e i' d 0 re Z (3.7.25) C E˛.z/ 1 .s ˛ 1/ 1 D z ˛ dz; lR z where the integral in the left-hand side converges absolutely for any ﬁnite R,since 1 1 Re s D 2 and ˛>2 . .sC˛1/ .sC˛1/ 1 i2 ˛ If Re s D 2 then Re f ˛ gD˛ 1=2˛ < 1. Consequently, e 1 6D 0 in this case. Thus the identity (3.7.25) can be rewritten in the form Z R i' C E˛.re / 1 .s ˛ 1/ 1 r ˛ r i' d 0 re .'/ Z (3.7.26) i ˛ .sC˛1/ C e E˛.z/ 1 .s ˛ 1/ 1 1 D z ˛ dz; Re s D : .sC˛1/ z 2 sin. ˛ / lR

Passing to the limit with R !1in the last identity and using the formula (3.7.21) we ﬁnally obtain the formula (3.7.22). F 1 Corollary 3.21. Let 2 <˛Ä 2. Then the Mellin transform of the function i' e˛.xI e / (see (3.7.10)) exists and the following representation is valid

Â Ã Z1 E .t ˛ei'/ 1 M ˛ .s/ D e .tI ei'/t s1dt tei' ˛ 0 (3.7.27) i . '/ .sC˛1/ e ˛ 1 D ; Res D ; ˛ .2 s/ .sC˛1/ 2 sin. ˛ / h i ˛ ˛ where ' 2 ;2 . 2 2

3.7.2 The Complete Monotonicity Property

Deﬁnition 3.22. A function f W .0; 1/ ! R is called completely monotonic if it possesses derivatives f .n/.x/ of any order n D 0; 1; : : :, and the derivatives are alternating in sign, i.e. 3.7 The Mittag-Lefﬂer Function of a Real Variable 47

.1/nf .n/.x/ 0; 8x 2 .0; 1/: (3.7.28)

The above property (see, e.g., [Wid46, p. 161]), which is equivalent to the existence of a representation of the function f in the form of a Laplace–Stieltjes integral with non-decreasing density and non-negative measure d

Z1 f.x/ D extd.t/: (3.7.29)

Proposition 3.23 ([Poll48]). The Mittag-Lefﬂer function of negative argument E˛.x/ is completely monotonic for all 0 Ä ˛ Ä 1. x G Since E0.x/ D 1=.1 C x/ and E1.x/ D e there is nothing to be proved in these cases. Let 0

C1: the line y D .tan /x from x DC1to x D %, %>0; C2: an arc of the circle jzjD% sec , Ä argz Ä ; C3: the reﬂection of C1 in the x-axis. We assume > =˛>=2whileZ % is arbitrary but ﬁxed. 1 Let us replace .x C t/1 by e.xCt/ du in (3.7.30). The resulting double 0 integral converges absolutely, so that one can interchange the order of integration to obtain Z Z 1 1 xu t 1=˛ tu E˛.x/ D e du e e dt: (3.7.31) 2i˛ 0 L

It remains to compute the function Z 1 t 1=˛ tu F˛.u/ D e e dt (3.7.32) 2ia L and to prove it is non-negative when u 0 (see the remark concerning representation (3.7.29)). An integration by parts in (3.7.32) yields Z Â Ã 1 tu 1 1=˛1 t 1=˛ F˛.u/ D e t e dt: (3.7.33) 2i˛u L ˛ 48 3 The Classical Mittag-Lefﬂer Function

Now let tu D z˛.Then Z 11=˛ u 1 z˛ zu1=˛ F˛.u/ D e e dz ; (3.7.34) ˛ 2i L0 where L0 is the image of L under the mapping. Now consider the function Z 1 z˛ zt ˚˛.t/ D e e dz : (3.7.35) 2i L0

This is known to be the inverse Laplace transform of Z 1 z˛ zt e D e ˚˛.t/ dt; 0 which is completely monotonic [Poll48]. Hence

u11=˛ F .u/ D ˚ .u1=˛/ 0: ˛ ˛ ˛

The proof will be completed if we can show the existence of any derivative of F˛.u/ for all u 0. From the explicit series representation [Poll48] for the function ˚˛.t/ we deduce that

1 X1 .1/k1 F .u/ D sin.˛k/ .˛k C 1/ uk1 ; (3.7.36) ˛ ˛ kŠ 1 so that F˛.u/ is an entire function.F Note that it is also possible to obtain (3.7.34) directly from (3.7.36). From Proposition 3.23 it follows, in particular, that E˛.x/ has no real zeros when 0 Ä ˛ Ä 1.

3.7.3 Relation to Fractional Calculus

Let us recall a few deﬁnitions concerning fractional integrals and derivatives. The left- and right-sided Riemann–Liouville fractional integrals on any ﬁnite interval .a; b/ are given by the formulas (see, e.g., [SaKiMa93, p. 33])

Zx 1 '.t/ I ˛ ' .x/ D dt; x > a; (3.7.37) aC .˛/ .x t/1˛ a 3.7 The Mittag-Lefﬂer Function of a Real Variable 49

Zb 1 '.t/ I ˛ ' .x/ D dt; x < b: (3.7.38) b .˛/ .t x/1˛ x

The right-sided fractional integral on a semi-axis (also called the right-sided Liouville fractional integral) is deﬁned via the formula (see, e.g. [SaKiMa93,p.94])

Z1 1 '.t/ I ˛' .x/ D dt; 1

By simple calculation one can obtain the following values for the above integrals of power-type functions (see [SaKiMa93, p. 40])

.ˇ/ I ˛ .t a/ˇ1 .x/ D .x a/˛Cˇ1;x>a; (3.7.40) aC .˛C ˇ/

.ˇ/ I ˛ .b t/ˇ1 .x/ D .b x/˛Cˇ1;x

.1 ˛ C ˇ/ I ˛t ˇ1 .x/ D x˛ˇ1; 1

The last integral can be calculated using the following property of the fractional integrals Â Â ÃÃ Â Ã 1 1 I ˛' .x/ D x˛1 t ˛1I ˛ '.t/ : t 0C x

The values of fractional integrals of the Mittag-Lefﬂer function (or, in other words, the composition of the fractional integrals with the Mittag-Lefﬂer function) can be calculated by using the following auxiliary result. Lemma 3.24. Let ˛>0and suppose 2 C is not an eigenvalue of the Abel integral operator. Then

Zx E .t ˛/ ˛ dt D E .x˛/ 1: (3.7.43) .˛/ .x t/1˛ ˛ 0

G The proof follows from the Taylor expansion of E˛ andbyterm-by-term integration using formula (3.7.40). F 50 3 The Classical Mittag-Lefﬂer Function

We present here the above-mentioned composition formulas only for the left- and right-sided Riemann–Liouville fractional integrals and right-sided Liouville fractional integral. The formulas for other types of fractional integrals and derivatives (see Appendix E) can be obtained in a similar way. Proposition 3.25. Let Re ˛>0, then the following formulas are satisﬁed

1 I ˛ .E ..t a/˛// .x/ D fE ..x a/˛/ 1g ;6D 0: (3.7.44) aC ˛ ˛ 1 I ˛ .E ..b t/˛// .x/ D fE ..b x/˛/ 1g ;6D 0: (3.7.45) b ˛ ˛

x˛1 I ˛ t ˛1E .t ˛/ .x/ D fE .x˛/ 1g ;6D 0: (3.7.46) ˛ ˛ The result follows immediately from Lemma 3.24. The left- and right-sided fractional derivative of a non-integer order ˛ (m 1< ˛

Proposition 3.26. Let 0

.x a/˛ D˛ .E ..t a/˛// .x/ D C E ..x a/˛/; a 6D 0: (3.7.49) aC ˛ .1 ˛/ ˛

.b x/˛ D˛ .E ..b t/˛// .x/ D C E ..b x/˛ /; a 6D 0: (3.7.50) b ˛ .1 ˛/ ˛

Since the Mittag-Lefﬂer function is inﬁnitely differentiable, then one can use in the considered case (0

3.8 Historical and Bibliographical Notes

Historical remarks regarding early results on the classical Mittag-Leffler function as well as certain direct generalizations have already been presented in the Introduction and in Sect. 2.3. We also mention here the treatise on complex functions by Sansone and Gerretsen [SanGer60], where a detailed account of these functions is given. However, the most specialized treatise, where more details on the functions of Mittag-Leffler type are given, is surely that by Dzherbashyan [Dzh66], in Russian. We also mention another book by this author from 1993 [Djr93], in English, where a brief description of the theory of Mittag-Leffler functions is given. For the basic facts on the Mittag-Leffler function presented in Sects. 3.4–3.6 we have followed the monograph [Dzh66] and the survey paper [PopSed11]. In this section we have also mentioned more recent results related to the development of the theory of the Mittag-Leffler function E˛.z/. We note that many mathematicians, not only in Mittag-Leffler’s time, see, e.g. Phragmén [Phr04], Malmquist [Mal03], Wiman [Wim05a, Wim05b], have recognized its importance, providing interesting results and applications, which unfortunately are not as well known as they deserve to be, see, e.g. Hille and Tamarkin [HilTam30], Pollard [Poll48], Humbert and Agarwal [Aga53, Hum53, HumAga53], or the short reviews in books such as Buhl [Buh25b], Davis [Dav36], Evgrafov [Evg78]. We briefly outline some important results. The first asymptotic result which was obtained using Mittag-Leffler’s method was that of Malmquist [Mal05]. Phragmén [Phr04] studied the Mittag-Leffler function as an example of an entire function of finite order satisfying his theorem generalizing the maximum modulus principle for analytic functions. In the same direction Buhl [Buh25a] showed that the Mittag-Leffler function furnishes examples and counterexamples for the growth and other properties of entire functions of finite order (see also [GoLuRo97]). The asymptotic expansions of E˛.z/ were generalized by Wiman [Wim05a](see also the papers by Barnes [Barn06] and Mellin [Mel10], in which the complete theory of the Mellin–Barnes integral representation is developed). Wright [Wri34] introduced and discussed the integral representation of the function .˛;ˇI z/ (called the Wright function). He also found [Wri40a] the connection of .˛;ˇI z/ with the so-called generalized Bessel functions of order greater than one, which was a prototype of the two-parametric Mittag-Leffler function E˛;ˇ.z/. Humbert and Agarwal [Aga53, Hum53, HumAga53] introduced the two- parametric Mittag-Leffler function E˛;ˇ.z/ (mentioned for the first time in the article by Wiman [Wim05a]) and investigated its properties (see also [HumDel53] for the generalized Mittag-Leffler function of two variables). ˛ The Laplace transform of E˛.t / was used by Humbert [Hum53] to obtain a number of functional relations satisfied by the Mittag-Leffler function. 52 3 The Classical Mittag-Leffler Function

Feller (see, e.g., [Fel71], see also [GrLoSt90, Ch. 5]) conjectured the complete monotonicity of the Mittag-Leffler function E˛.x/ and proved it for 0 Ä ˛ Ä 1 by using methods of Probability Theory. An analytic proof of this result was obtained by Pollard [Poll48]. The result presented in Sect. 3.7.2 is due to him. For pioneering works on the mathematical applications of the Mittag-Leffler function we refer to Hille and Tamarkin [HilTam30] and Barrett [Barr54]. The 1930 paper by Hille and Tamarkin deals with the solution of the Abel integral equation of the second kind (a particular fractional integral equation). The 1954 paper by Barrett concerns the general solution of the linear fractional differential equation with constant coefficients. Concerning earlier applications of the Mittag-Leffler function in physics, we refer to the contributions by K.S. Cole, see [Col33] (mentioned in the book by Davis [Dav36, p. 287]), in connection with nerve conduction, and by de Oliveira Castro [Oli39], and Gross [Gro47], in connection with dielectrical and mechanical relaxation, respectively. Subsequently, Caputo and Mainardi [CapMai71a, CapMai71b], have proved that the Mittag-Leffler type functions appear whenever derivatives of fractional order are introduced in the constitutive equations of a linear viscoelastic body. Since then, several other authors have pointed out the relevance of such functions for fractional viscoelastic models. In recent times the attention of mathematicians towards the Mittag-Leffler function has increased from both the analytical and numerical point of view, overall because of its relation with the fractional calculus and its applications, see Diethelm et al. [Die10], Hilfer and Seybold [HilSey06], Luchko et al. [LucGor99, LucSri95], Gorenflo et al. [GoLoLu02], Samko et al. [SaKiMa93], Gorenflo and Vessella [GorVes91], Miller and Ross [MilRos93], Kiryakova [Kir94], Gorenflo and Rut- man [GorRut94], Mainardi [Mai96, Mai97], Podlubny [Pod99], Kilbas and Saigo [KilSai04], Blank [Bla97], Gorenflo and Mainardi [GorMai97], Schneider [Sch90]. In fact there has been a revived interest in the fractional calculus because of its applications in different areas of physics and engineering. In addition to the books and papers already quoted, here we would like to draw the reader’s attention to some relevant papers, in alphabetical order of the first author, Al Saqabi and Tuan [Al-STua96], Brankov and Tonchev [BraTon92], Berberan-Santos [Ber-S05a, Ber-S05b , Ber-S05c], and others (see Mainardi’s book [Mai10]andthe references therein). This list, however, is not exhaustive. Details on Mittag-Leffler functions can also be found in some treatises devoted to the theory and/or applications of special functions, integral transforms and fractional calculus, e.g. Davis [Dav36], Marichev [Mari83], Gorenflo and Vessella [GorVes91], Samko, Kilbas and Marichev [SaKiMa93], Kiryakova [Kir94], Carpinteri and Mainardi [CarMai97], Podlubny [Pod99], Hilfer [Hil00], West, Bologna and Grigolini [WeBoGr03], Kilbas, Srivastava and Trujillo [KiSrTr06], Magin [Mag06], Debnath-Bhatta [DebBha07], Mathai and Haubold [MatHau08]. 3.9 Exercises 53

To the best of the authors’ knowledge, earlier plots of the Mittag-Leffler functions are found (presumably for the first time in the literature of fractional calculus and special functions) in the 1971 paper by Caputo and Mainardi [CapMai71b]. More precisely, these authors provided plots of the function E.t / for some values of 2 .0; 1, adopting linear-logarithmic scales, in the framework of fractional relaxation for viscoelastic media. At the time, not only were such functions still almost ignored, but also the fractional calculus was not yet well accepted by the community of physicists. Recently, numerical routines for functions of the Mittag-Leffler type have been provided, see e.g. Gorenflo, Loutchko and Luchko [GoLoLu02] (with MATHE- MATICA), Podlubny [Pod06] (with MATLAB) and Seybold and Hilfer [SeyHil05]. Furthermore, in the NASA report by Freed, Diethelm and Luchko [FrDiLu02], an appendix is devoted to the table of Padè approximants for the Mittag-Leffler function E˛.x/. Since the fractional calculus has actually attracted wide interest in different areas of the applied sciences, we think that the Mittag-Leffler function is now leaving its isolated life as Cinderella (using the term coined by F.G. Tricomi in the 1950s for the incomplete gamma function). We like to refer to the classical Mittag-Leffler function as the Queen function of fractional calculus, and to consider all the related functions as her court, see [MaiGor07]. For the first reading we recommend Chaps. 2–5 in which the basic theory of the Mittag-Leffler function in one variable with one, two and three parameters is discussed. A short account of the further mathematical development of this theory is presented in Chap. 6.

3.9 Exercises

3.9.1. Prove that the following relations hold: " p !# 1 1 1 z 3 1 z 3 3 1 E .z/ D e C 2e 2 cos z 3 ; z 2 C; 3 2 2 h Á Ái 1 1 1 E .z/ D cos z 4 C cosh z 4 ; z 2 C: 4 2 3.9.2. Deduce the following duplication formula

1 E .z2/ D ŒE .z/ C E .z/ ; z 2 C .Re ˛>0/: 2˛ 2 ˛ ˛ 54 3 The Classical Mittag-Lefﬂer Function

3.9.3. Evaluate the Laplace transform of the Mittag-Lefﬂer function

X1 zk E .z/ D : ˛ .˛kC 1/ kD0

3.9.4. Prove that ([MatHau08,p.90])

Zx E .t ˛/ ˛ dt D E .x˛/ 1; Re ˛>0: .˛/ .x t/1˛ ˛ 0

3.9.5. Prove the following integral representation of the Mittag-Lefﬂer function for x 2 R ([HaMaSa11])

Z1 2 ˛ t ˛1 cos xt E .x˛/ D t; ˛>0I ˛ sin ˛ ˛ 2˛ d Re 2 1 C 2t cos 2 C t 0 Z1 1 t ˛1 1 E .x/ D sin ˛ etx ˛ tdt; Re ˛>0I ˛ 1 C 2t˛ cos ˛ C t 2˛ 0 Z1 Ä 1 ˛ 1 x ˛ t C cos ˛ 1 E .x/ D 1 C arctan etx ˛ tdt; Re ˛>0: ˛ 2˛ sin ˛ 0

3.9.6. Prove the following integro-functional relation for the Mittag-Lefﬂer function ([GorMai97, Ber-S05b])

Z1 2x E .t 2/ E .x/ D 2˛ dt; 0 Ä Re ˛ Ä 1: ˛ x2 C t 2 0

3.9.7. Represent the Mittag-Leffler function in the case of a positive integer parameter ˛ in the form of a G-function. 3.9.8. Represent the Mittag-Leffler function in the form of an H-function. Chapter 4 The Two-Parametric Mittag-Leffler Function

In this chapter we present the basic properties of the two-parametric Mittag-Leffler function E˛;ˇ.z/ (see (1.0.3)), which is the most straightforward generalization of the classical Mittag-Leffler function E˛.z/ (see (3.1.1)). As in the previous chapter, the material can be formally divided into two parts. At first, starting from the basic definition of the Mittag-Leffler function as a power series, we discover that, for the first parameter ˛ with positive real part and any complex value of the second parameter ˇ, the function E˛;ˇ.z/ is an entire function of the complex variable z. Therefore we discuss in the first part the (analytic) properties of the two-parametric Mittag-Leffler function as an entire function. Namely, we calculate its order and type, present a number of formulas relating the two-parametric Mittag-Leffler function to elementary and special functions as well as recurrence relations and differentiation formulas, introduce some useful integral representations and discuss asymptotics and the distribution of zeros of the considered function. An extension of the two-parametric Mittag-Leffler function to values of the first parameter ˛ with non-positive real part is given here too. It is well-known that in current applications the properties of the two-parametric Mittag-Leffler function of a real variable are often used. Thus, we collect in the second part (Sect. 4.10) results of this type. They concern integral representations and integral transforms of the two-parametric Mittag-Leffler function of a real variable, the complete monotonicity property, and relations to the fractional calculus. People working in applications can, at first reading, omit some of the deeper mathematical material (that from Sects. 4.4–4.9,say).

© Springer-Verlag Berlin Heidelberg 2014 55 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__4 56 4 The Two-Parametric Mittag-Leffler Function

4.1 Series Representation and Properties of Coefﬁcients

The Mittag-Lefﬂer type function (or the two-parametric Mittag-Lefﬂer function)

X1 zk E .z/ D .˛ > 0; ˇ 2 C/ (4.1.1) ˛;ˇ .˛kC ˇ/ kD0 generalizes the classical Mittag-Lefﬂer function

X1 zk E .z/ D .˛ > 0/: ˛ .˛kC 1/ kD0

We have E˛;1.z/ D E˛.z/.Forany˛; ˇ 2 C,Re˛>0, the function (4.1.1)isan entire function of order D 1=.Re ˛/ and type 1. Indeed, let us consider the slightly more general function

X1 . ˛ z/k E . ˛ z/ D ; (4.1.2) ˛;ˇ .˛kC ˇ/ kD0 i.e. take the coefﬁcients in the form

˛k c D .k D 0; 1; 2; : : :/; (4.1.3) k .˛kC ˇ/ where 0

1 and, consequently, for the sequence fckg0 we immediately obtain

k log k 1 =k lim D D ; lim kjckj D e : (4.1.5) k!1 log 1 .Re ˛/ k!1 jck j

According to a well known theorem in the theory of entire functions (see, e.g., formulas (B.2.3)and(B.2.4)), the function (4.1.2) has order D 1=˛ and type for any ˇ. Therefore, the two-parametric Mittag-Lefﬂer function (4.1.1) has order D 1=.Re ˛/ and type 1 foranyvalueoftheparameterˇ 2 C. 4.2 Explicit Formulas: Relations to Elementary and Special Functions 57

4.2 Explicit Formulas: Relations to Elementary and Special Functions

Using deﬁnition (4.1.1) we obtain a number of formulas relating the two-parametric Mittag-Lefﬂer function E˛;ˇ to elementary functions (see, e.g. [HaMaSa11])

ez 1 E .z/ D ez;E.z/ D ; (4.2.1) 1;1 1;2 z p p sinh z E .z/ D cosh z;E.z/ D p : (4.2.2) 2;1 2;2 z

Some extra formulas, for other special values of parameters, are presented as exercises at the end of the chapter. Here we also mention two recurrence relations for the function (4.1.1).

1 E .z/ D C zE .z/; (4.2.3) ˛;ˇ .ˇ/ ˛;ˇC˛ d E .z/ D ˇE .z/ C ˛z E .z/: (4.2.4) ˛;ˇ ˛;ˇC1 dz ˛;ˇC1

Now we present some other relations of the two-parametric Mittag-Leffler function to certain special functions introduced by different authors. The hyperbolic functions of order n, denoted by hr .z;n/,aredefined,e.g.,in [Bat-3, 18.2 (2)]. Their series representations relate these functions to the two- parametric Mittag-Leffler function (see [Bat-3, 18.2 (16)]):

X1 znkCr1 h .z;n/D D zr1E .zn/; r D 1;2;::: (4.2.5) r .nk C r 1/Š n;r kD0

The trigonometric functions of order˚ n«, denoted by kr .z;n/,aredeﬁned,e.g., i in [Bat-3, 18.2 (18)]. With D exp n the functions kr .z;n/ and hr .z;n/ are related by:

1r kr .z;n/D hr .z;n/; from which it follows

X1 .1/j znjCr1 k .z;n/D D zr1E .zn/; r D 1;2;::: (4.2.6) r .nj C r 1/Š n;r j D0

The relation to the complementary error function is also well-known (see [MatHau08, pp. 80–81], [Dzh66, p. 297]): 58 4 The Two-Parametric Mittag-Lefﬂer Function

X1 k z z2 E 1 .z/ D D e erfc.z/; (4.2.7) 2 ;1 k kD0 2 C 1 where erfc is complementary to the error function erf: Z 1 2 2 erfc.z/ WD p eu du D 1 erf.z/; z 2 C: z

The Miller–Ross function is deﬁned as follows (see [MilRos93]):

X1 .at/k E .; a/ D t D t E .at/: (4.2.8) t .C k C 1/ 1;C1 kD0

The Rabotnov function is represented as follows (see [RaPaZv69]):

X1 ˇk t k.˛C1/ R .ˇ; a/ D t ˛ D t ˛E .ˇt ˛C1/: (4.2.9) ˛ ..1 C ˛/.k C 1// ˛C1;˛C1 kD0

4.3 Differential and Recurrence Relations

The following differentiation formula is an immediate consequence of the deﬁnition of the two-parametric Mittag-Lefﬂer function (4.1.1) Â Ã m d ˇ1 ˛ ˇm1 ˛ Œz E .z / D z E .z /.m 1/: (4.3.1) dz ˛;ˇ ˛;ˇ m

We consider now some corollaries of formula (4.3.1). Let ˛ D m=n; .m; n D 1; 2; :::/in (4.3.1). Then Â Ã d m Œzˇ1E .zm=n/ dz m=n;ˇ n m X n k ˇ1 m=n ˇ1 z D z Em=n;ˇ.z / C z .m; n 1/: (4.3.2) ˇ m k kD1 n

Since 1 D 0.sD 0; 1; 2; : : : /; .s/ it follows from (4.3.2) with n D 1 and any ˇ D 0; 1;:::; mthat 4.3 Differential and Recurrence Relations 59 Â Ã d m Œzˇ1E .zm/ D zˇ1E .zm/.m 1/: (4.3.3) dz m;ˇ m;ˇ

Substituting zn=m in place of z in (4.3.2)weget Â Ã m h i m 1 n d .ˇ1/ n z m z m E .z/ n dz m;ˇ

Xn k .ˇ1/ n .ˇ1/ n z D z m Em;ˇ.z/ C z m .m; n D 1; 2;:::/: (4.3.4) ˇ m k kD1 n

Let m D 1 in this formula. We then obtain the ﬁrst-order differential equation .ˇ1/n for the function z E1=n;ˇ.z/:

1 d z.ˇ1/nE .z/ zn1Œz.ˇ1/nE .z/ n dz 1=n;ˇ 1=n;ˇ Xn zk ˇn1 D z k ;.nD 1; 2;:::/: (4.3.5) kD1 ˇ n

Solving this equation we obtain for any z0 6D 0; n n .ˇ1/n n .1ˇ/n z z0 E1=n;ˇ.z/ D z e z0 e E1=n;ˇ.z0/ Z ! ) z Xn k n ˇn1 Cn e k d .n D 1; 2;:::/: (4.3.6) z0 kD1 ˇ n

Formula (4.3.6) is true with z0 D 0 if ˇ D 1. In this case we have ( Z ! ) z Xn1 k1 zn n E1=n;1.z/ D e 1 C n e d .n 2/: (4.3.7) k 0 kD1 n

In particular, Z z z2 2 2 z2 z2 E1=2;1.z/ D e 1 C p e d D e f1 C erf zg D e erfc .z/; 0 (4.3.8) and, consequently,

2 E .z/ 2 ez ; jargzj < ; jzj!1: 1=2;1 4 In the following lemma we collect together a number of known recurrence relations for the two-parametric Mittag-Lefﬂer function. 60 4 The Two-Parametric Mittag-Lefﬂer Function

Lemma 4.1 ([GupDeb07]). For all ˛>0, ˇ>0the following relations hold

1 z z2E .z/ D E .z/ ; (4.3.9) ˛;ˇC2˛ ˛;ˇ .ˇ/ .ˇC ˛/ 1 z z2 z3E .z/ D E .z/ ; (4.3.10) ˛;ˇC3˛ ˛;ˇ .ˇ/ .ˇC ˛/ .ˇC 2˛/ 1 z z2 z3 z4E .z/ D E .z/ : ˛;ˇC3˛ ˛;ˇ .ˇ/ .ˇC ˛/ .ˇC 2˛/ .ˇC 3˛/ (4.3.11)

4.4 Integral Relations and Asymptotics

Using the well-known discrete orthogonality relation ( m1 X m; if k Á 0.mod m/ ; ei2hk=m D hD0 0; if k 6Á 0.mod m/ and deﬁnition (4.1.1) of the function E˛;ˇ.z/ we have

mX1 i2h=m m E˛;ˇ.z e / D mE˛=m;ˇ.z /.m 1/: (4.4.1) hD0

Substituting here m˛ for ˛ and z1=m for z we obtain

1 mX1 E .z/ D E .z1=mei2h=m/.m 1/: (4.4.2) ˛;ˇ m m˛;ˇ hD0

Similarly, the formula

1 Xm E .z/ D E .z1=.2mC1/ei2h=.2mC1//.m 0/ (4.4.3) ˛;ˇ 2m C 1 .2mC1/˛;ˇ hDm can be obtained via the relation

Xm 2m C 1; if k Á 0.mod 2m C 1/ ; eei2hk=.2mC1/ D 0; if k 6Á 0.mod 2m C 1/ : hDm

Using (4.1.1) and term-by-term integration we arrive at 4.4 Integral Relations and Asymptotics 61 Z z ˛ ˇ1 ˇ ˛ E˛;ˇ.t /t dt D z E˛;ˇC1.z /.ˇ>0/; (4.4.4) 0 and furthermore, at the more general relation Z z 1 1 ˛ ˇ1 .z t/ E˛;ˇ.t /t dt .˛/ 0 Cˇ1 ˛ D z E˛;Cˇ.z / . > 0; ˇ > 0/; (4.4.5) where the integration is performed along the straight line connecting the points 0 and z. It follows from formulas (4.4.5), (4.4.2)and(4.4.3)that Z 1 z .z t/ˇ1et dt D zˇE .z/.ˇ>0/; (4.4.6) .ˇ/ 1;ˇC1 Z 0 z p 1 ˇ1 ˇ 2 .z t/ cosh t dt D z E2;ˇC1.z /.ˇ>0/; (4.4.7) .ˇ/ 0 Z p z 1 ˇ1 sinh t ˇC1 2 .z t/ p dt D z E2;ˇC2.z /.ˇ>0/: (4.4.8) .ˇ/ 0

Let us prove the relation

ˇ1 ˛ ˇ1 2˛ z E˛;ˇ.z / D z E2˛;ˇ.z / Z z 1 ˛1 2˛ ˇ1 C .z t/ E2˛;ˇ.t /t dt.ˇ>0/: (4.4.9) .˛/ 0

First of all, we have by direct evaluations Z z ˛ 2˛ ˇ1 .z t/ E2˛;ˇ.t /t 1 C dt 0 .˛C 1/ Z X1 1 z .z t/˛ D t 2k˛Cˇ1 1 C dt .2k˛C ˇ/ .˛C 1/ kD0 0 X1 z2k˛ X1 z.2kC1/˛ D zˇ C zˇ .2k˛C ˇ C 1/ ..2k C 1/˛ C ˇ C 1/ kD0 kD0 X1 zk˛ D zˇ D zˇE .z˛/: .k˛C ˇ C 1/ ˛;ˇC1 kD0

This relation and formula (4.4.4)imply 62 4 The Two-Parametric Mittag-Lefﬂer Function Z z .z t/˛ E .t 2˛/t ˇ1 1 C dt 2˛;ˇ .˛C 1/ 0 Z z ˛ ˇ1 D E˛;ˇ.t /t dt.ˇ>0/: 0

Differentiation of this formula with respect to z gives us formula (4.4.9). Let us prove the formula Z l ˇ1 ˛ 1 ˛ x E˛;ˇ.x /.l x/ E˛;. .l x/ / dx 0 ˛ ˛ E C .l / E C .l / D ˛;ˇ ˛;ˇ lˇC1 .ˇ > 0 ; > 0/ ; (4.4.10) where and . 6D / are any complex parameters. Indeed, using (4.1.1)forany and . 6D / and ˇ>0;>0we ﬁnd Z l ˇ1 ˛ 1 ˛ x E˛;ˇ.x /.l x/ E˛;. .l x/ / dx 0 Z X1 X1 n./m l D xn˛Cˇ1.l x/m˛C1 dx .n˛C ˇ/ .m˛ C / nD0 mD0 0 X1 X1 n./ml.nCm/˛CˇC1 X1 X1 n./kn lk˛ D D lˇC1 ..m C n/˛/ C ˇ C / .k˛C ˇ C / nD0 mD0 nD0 kDn Â Ã X1 ./k lk˛ Xk n lˇC1 X1 lk˛.kC1 ./kC1/ D lˇC1 D : .k˛C ˇ C / .k˛C ˇ C / kD0 nD0 kD0

Using formula (4.1.1) once more, we arrive at (4.4.10). Finally, we obtain two integral relations: Z C1 t ˛ ˇ1 1 e E˛;ˇ.zt /t dt D .ˇ > 0; jzj <1/; (4.4.11) 0 1 z Z C1 t 2=.4x/ ˛ ˇ1 p ˇ=2 2˛ C e E˛;ˇ.t /t dt D x E2˛; 1 ˇ x .ˇ > 0; x > 0/: 0 2 (4.4.12)

First of all, since the Mittag-Lefﬂer type function (4.1.1) is an entire function of order D 1=.Re ˛/ and type 1 (see Sect. 4.1), we have for any >1the estimate:

jE˛;ˇ.z/jÄC expf jzj g:

Consequently, the integrals in the formulae (4.4.11)and(4.4.12) are convergent. 4.4 Integral Relations and Asymptotics 63

It is easy to check that term-by-term integration of the expansion

X1 et t k˛Cˇ1 et E .zt˛/t ˇ1 D zk ˛;ˇ/ .k˛C ˇ/ kD0 can be performed with respect to t along the interval .0; C1/ if jzj <1. As a result, we arrive at formula (4.4.11). Similarly, we have using term-by-term integration of the following expansion with respect to t:

X1 k˛Cˇ1 2 t 2 et =.4x/ E .t ˛/tˇ1 D et =.4x/ .ˇ > 0/ ˛;ˇ .k˛C ˇ/ kD0 along the same interval .0; C1/ (integration is performable with any ﬁxed x>0!), Á Z k˛Cˇ C1 X1 2 2 p et =.4x/ E .t ˛/tˇ1 dt D .2 x/k˛Cˇ : ˛;ˇ 2.k˛C ˇ/ 0 kD0

Rewriting the right-hand side of the last relation by using the Lagrange formula p .s/.sC 1=2/ D 212s .2s/ we obtain formula (4.4.12). The Mittag-Lefﬂer type function E˛;˛ .z/ plays an essential role in the linear Abel integral equation of the second kind (see, e.g., [GorVes91, GorMai97]).

Theorem 4.2. Let a function f.t/be in the function space L1.0; l/.Let˛>0and be an arbitrary complex parameter. Then the integral equation Z t u.t/ D f.t/C .t /˛1 u./ d; t 2 .0; l/; (4.4.13) .˛/ 0 has a unique solution Z t ˛1 ˛ u.t/ D f.t/C .t / E˛;˛ Œ .t / /f./d; t 2 .0; l/; (4.4.14) 0 in the space L1.0; l/ : This result was discovered in the pioneering work of Hille and Tamarkin [HilTam30]. We give the proof of this theorem in Chap. 7. We consider now a simple application of Theorem 4.2. 64 4 The Two-Parametric Mittag-Lefﬂer Function

Let z D t>0and t D be the integration variable in (4.4.9). Then this representation can be considered as an integral equation of type (4.4.13) with

ˇ1 ˛ f.t/ D t E˛;ˇ.t /; D 1 I and solution

ˇ1 2˛ u.t/ D t E2˛;ˇ.x /:

Using formula (4.4.14) we obtain for the solution u.t/ the representation

ˇ1 ˛ u.t/ D t E˛;ˇ.t / Z t ˛1 ˛ ˇ1 ˛ .t / E˛;ˇ. / E˛;˛ Œ.t / d: (4.4.15) 0

We also mention asymptotic results for the Mittag-Lefﬂer function E˛;ˇ.z/ which are essentially a reﬁnement of results of Dzhrbashyan (see [Dzh66]). Theorem 4.3. For all 0<˛<2, ˇ 2 C, m 2 N, the following asymptotic formulas hold: If jarg zj < minf; ˛g,then

Xm k 1 1=˛ z E .z/ D z.1ˇ/=˛ez C O jzjm1 ; jzj!1: (4.4.16) ˛;ˇ ˛ .ˇ k˛/ kD1

If 0<˛<1, ˛ < jarg zj <,then

Xm zk E .z/ D C O jzjm1 ; jzj!1: (4.4.17) ˛;ˇ .ˇ k˛/ kD1

Theorem 4.4. For all ˛ 2, ˇ 2 C, m 2 N, the following asymptotic formula holds: X 1 1ˇ 1=˛ 2in=˛ E .z/ D z1=˛ e2in=˛ ez e ˛;ˇ ˛ 3˛ jarg zC2nj< 4 Xm zk C O jzjm1 ; jzj!1: (4.4.18) .ˇ k˛/ kD1

A more exact description of the remainders in formulas (4.4.16)–(4.4.18)is presented, e.g., in [PopSed11]. 4.5 The Two-Parametric Mittag-Lefﬂer Function as an Entire Function 65

4.5 The Two-Parametric Mittag-Lefﬂer Function as an Entire Function

One more immediate consequence of the results in [Sed94, Sed00, Sed04, Sed07] (see also [Pop02, Pop06, PopSed03]) is the fact that E˛;ˇ with ˛>0;ˇ2 R,isa function of completely regular growth in the sense of Levin–Pfluger. For an entire function F.z/ of finite order this means that the following limit exists ˇ ˇ log ˇF.rei /ˇ hF . / D lim ; (4.5.1) r!1 r when r !1takes all positive values avoiding an exceptional set C 0 of relatively small measure common for all rays arg z D (see the formal definition in Sect. B.4). Sometimes the limit in (4.5.1) is called the weak limit and is denoted by

lim : r!1

The above property is known to be equivalent to the regularity of the distribution of zeros of an entire function (see [Lev56, Ron92]). The corresponding result is presented in Sect. B.4. To prove that E˛;ˇ possesses this property let us consider the Weierstrass product representation (see Theorem B.3.1). For the two-parametric Mittag-Lefﬂer function this representation has the form

1 Â Ã 2 Œ1=˛ Y z C z C:::C z z zn 2 Œ1=˛ E D 1 e 2zn Œ1=˛zn ˛;ˇ z nD1 n 1 Â Ã 2 Œ1=˛ Y z C z C:::C z z zn 2 Œ1=˛ q Cq zC:::Cq zŒ1=˛ 1 e 2zn Œ1=˛zn e 0 1 Œ1=˛ ; (4.5.2) z nD1 n where zn; zn are the zeros of E˛;ˇ in the neighbourhoods of the rays arg z D ˛ ˛ 2 ; arg z D 2 , respectively. This formula can be written in terms of simple Weierstrass factors:

2 p C C:::C G. ; p/ D .1 /e 2 p ;p2 N: (4.5.3)

Then (4.5.2) becomes Â Ã Â Ã Y1 Y1 z z Œ1=˛ q0Cq1zC:::CqŒ1=˛z E˛;ˇ.z/ D G ; Œ1=˛ G ; Œ1=˛ e : z z nD1 n nD1 n (4.5.4) 66 4 The Two-Parametric Mittag-Lefﬂer Function

Formula (4.5.4)(or(4.5.2)) is not too useful for obtaining asymptotic results. By using the technique developed for the entire functions of completely regular growth (see, e.g., [GoLeOs91,Lev56,Ron92]), we can compare the asymptotic behaviour of E˛;ˇ with that of more simple functions. In order to formulate it in a more rigorous form let us introduce two pairs of sequences:

i ˛ i ˛ wn Djznje 2 ;nD 1;2;;:::; wn Djznje 2 ;nD 1;2;;:::I (4.5.5)

˛ i ˛ ˛ i ˛ !n D .2n/ e 2 ;nD 1;2;;:::; !n D .2n/ e 2 ;nD 1;2;;:::I (4.5.6) and construct the corresponding Weierstrass products for these sequences Â Ã Â Ã Y1 Y1 z z Œ1=˛ W.z/ D G ; Œ1=˛ G ; Œ1=˛ eq0Cq1zC:::CqŒ1=˛z ; w w nD1 n nD1 n (4.5.7) Â Ã Â Ã Y1 Y1 z z Œ1=˛ ˝.z/ D G ; Œ1=˛ G ; Œ1=˛ eq0Cq1zC:::CqŒ1=˛z : ! ! nD1 n nD1 n (4.5.8)

Lemma 4.5. The functions E˛;ˇ.z/, W.z/, ˝.z/ are functions of completely regular growth with the same characteristics, i.e. with the same angular density (see formula (B.4.1a)) and in the case of integer D Œ1=˛ also with the same coefﬁcients of angular symmetry (see formula (B.4.1b)). To prove this result it sufﬁces to examine the corresponding properties of the sequences .z˙n/; .w˙n/; .!˙n/.

Lemma 4.6. The functions E˛;ˇ.z/, W.z/, ˝.z/ have the same asymptotic behaviour, i.e. the following weak limits exist1: ˇ ˇ ˇ ˇ ˇ i ˇ ˇ i ˇ E˛;ˇ.re / W.re / lim ˇ ˇ D lim ˇ ˇ D 1: (4.5.9) r!1 ˇW.rei /ˇ r!1 ˇ˝.rei /ˇ

G Comparing representations (4.5.4)and(4.5.7) we arrive at the conclusion that the quotient under this limit is in fact a Blaschke type product for two rays. Such products were studied in [Gov94]. Repeating calculations of [Gov94]wegetthe existence of the ﬁrst limit (4.5.9). The second result of the lemma also follows from quite general considerations. Comparing sequences .w˙n/ and .!˙n/ we can see that there exists a proximate

1 The meaning of the weak limit limr!1 is the same as in the deﬁnition of entire functions of completely regular growth, see (4.5.1). 4.6 Distribution of Zeros 67 order (cf. [Lev56, GoLeOs91]) (i.e. a non-negative, non-decreasing function such 0 that limr!1 .r/ D , limr!1 .r/r log r D 0)forwhich

.r/ jw˙nj D j!˙nj : (4.5.10)

A possible way to see this is to use an interpolating C1-function .r/ obeying the interpolation conditions

log j! j .n/ D ˙n ;nD 1;2;::: (4.5.11) log jw˙nj

Since we need to obtain only an asymptotic result, the interpolation problem can be simpliﬁed and even solved in closed form. After constructing the proximate order it remains to apply the Valiron–Levin theory of entire functions of a proximate order (see, e.g., [Lev56]). This completes the proof.F We are now in a position to discuss the global behaviour of the Mittag-Lefﬂer type function E˛;ˇ.z/. This behaviour is described in terms of the indicator function for E˛;ˇ.z/ in the case 0<˛<2(we assume additionally that ˇ 6D 1; 0; 1;:::, when ˛ D 1,see(B.2.5)):

˛ cos ˛ ;0Äj j < 2 ; hE˛;ˇ . / D ˛ (4.5.12) 0; 2 Äj jÄ:

4.6 Distribution of Zeros

In this section we mainly follow the article [Sed94](seealso[PopSed11]). First of all we recall some relations of the Mittag-Lefﬂer function to elementary functions for special values of the parameters ˛; ˇ. In these cases the zeros of the Mittag-Lefﬂer function can be found explicitly.

mC1 E1;1.z/ D expfzg;E1;m.z/ D z expfzg;m2 ZCI (4.6.1) p p p sinh z cosh z 1 E .z/ D cosh z;E .z/ D p ;E .z/ D p ; (4.6.2) 2;1 2;2 z 2;3 z

mC1=2 p m p E2;2m.z/ D z sinh z;m2 ZC;E2;.2m1/.z/ D z cosh z;m2 N: (4.6.3)

Thus the function E1;1.z/ has no zero, while the function E1;m.z/ has its only zero at z D 0 of order m C 1. Below we can see that for all other values of parameters ˛; ˇ (including those described in (4.6.2)and(4.6.3)) the Mittag-Leffler function E˛;ˇ.z/ has an infinite number of zeros. We should also mention that the function 68 4 The Two-Parametric Mittag-Leffler Function

2 E2;3.z/ has double zeros at the points zn D.2n/ ;n2 N. This is the only case when the Mittag-Leffler function E˛;ˇ.z/ has an infinite number of multiple zeros. In order to describe the distribution of zeros of the Mittag-Leffler function E˛;ˇ.z/ we introduce the following constants:

˛ ˛ 1 ˇ c D ;d D ; D 1 C ;ˇ6D ˛ l; l 2 Z ; ˇ .ˇ ˛/ ˇ .ˇ 2˛/ ˇ ˛ C (4.6.4) ˛ ˛ 1 ˇ c D ;d D ; D 2C ;ˇD ˛l; l 2 Z ;˛ 62 N: ˇ .ˇ 2˛/ ˇ .ˇ 3˛/ ˇ ˛ C (4.6.5)

By construction cˇ 6D 0 for all considered ˛ and ˇ. The values of parameters ˛ and ˇ, which are not mentioned in (4.6.4)and(4.6.5), are called exceptional values. Theorem 4.7. Let the values of parameters ˛; ˇ satisfy one of the following relations: (1) 0<˛<2;ˇ2 C, and ˇ 6D 0; 1; 2;:::when ˛ D 1; (2) ˛ D 2, Re ˇ>3.

Then all zeros zn of the Mittag-Lefﬂer function E˛;ˇ.z/ with sufﬁciently large modulus are simple and the asymptotic relation holds for n !˙1

d =c log 2in log c .z /1=˛ D 2in ˛ log 2inC ˇ ˇ C. ˛/2 . ˛/2 ˇ C˛ ; n ˇ .2in/˛ ˇ 2in ˇ 2in n (4.6.6) where for ˛<2 Â Ã Â Ã ! log jnj 1 log2 jnj ˛ D O C O C O ; (4.6.7) n jnj1C˛ jnj2˛ jnj2 but for ˛ D 2 Â Ã Â Ã ! e˙iˇ 1 log jnj log2 jnj ˛n D C O C O C O : (4.6.8) 2 4ˇ 8ˇ 14ˇ 2 cˇ.2n/ jnj jnj jnj

The proof of the theorem is based on the integral representation of the Mittag- Leffler function E˛;ˇ.z/ and on the following lemma. Lemma 4.8. Let A 2 C and ı 2 .0; =2/ be fixed numbers and let the sets Z, W be defined for all R>0by the relations

Z D Zı;R D fz Wjarg zj < ı; jzj >Rg ;

W D Wı;R D fz Wjarg zj < 2ı; jzj >2Rg : 4.6 Distribution of Zeros 69

Then for sufﬁciently large R>0the equation

z A log z D w; w 2 W; has a unique zero z 2 Z. This zero is simple and we have the asymptotics ! log w log2 w z D w C A log w C A2 C O ; w !1: (4.6.9) w w2

G Proof of Theorem 4.7 ([PopSed11, pp. 35–37]). Let us put in the formulas (4.4.16)and(4.4.17) m D 1 if ˇ is defined as in (4.6.4), and m D 2 if ˇ is defined as in (4.6.5). Then cˇ 6D 0. Hence one can conclude from (4.4.16)and(4.4.17)that for any ">0all zeros zn of the Mittag-Leffler function E˛;ˇ.z/ with sufficiently ˛ large modulus are situated in the angle jarg zj < 2 C". In this angle E˛;ˇ.z/ satisfies the asymptotic relation Â Ã dˇ 1 ˛ ˛zmE .z/ D zˇ expfz1=˛gc C O ; jarg zj < C ": (4.6.10) ˛;ˇ ˇ z z2 2

Therefore, there exists a sufﬁciently large r0 such that all zeros zn; jznj >r0, can be found from the equation Â Ã dˇ 1 expfz1=˛ C log zgDc C C O : (4.6.11) ˇ ˇ z z2

Let us put

1=˛ 1=˛ w D z C ˛ˇ log z : (4.6.12)

Then, by Lemma 4.8 we obtain the solution to (4.6.12) with respect to z1=˛ in the form

z1=˛ D w C O.log w/:

Hence Â Â ÃÃ Â Ã 1 1 log w 1 log w D 1 C O D C O : z w˛ w w˛ w1C˛

Substituting this relation into (4.6.11) we obtain the equation Â Ã Â Ã dˇ log w 1 expfwgDc C C O C O : (4.6.13) ˇ w˛ w1C˛ w2˛ 70 4 The Two-Parametric Mittag-Lefﬂer Function

In particular,

expfwgDcˇ C o.1/; w !1: (4.6.14)

Since all zeros of the function expfwgcˇ aresimpleandaregivenbytheformula 2in C log cˇ;n 2 Z, by Rouché’s theorem all zeros wn of Eq. (4.6.14) with sufﬁciently large modulus are simple too and can be described by the formula

wn D 2in C log cˇ C n; n ! 0; n !˙1: (4.6.15)

Thus Â Â ÃÃ 1 1 1 D C 1 C O ; log wn D log jnjCO.1/; n!˙1: wn 2in n (4.6.16)

Therefore, if w D wn in (4.6.14), then Â Ã Â Ã d log jnj 1 c expf gDc C ˇ C O C O : ˇ n ˇ .2in/˛ jnj1C˛ jnj2˛ 2 Since theÂ left-handÃ side of this relation is equal to cˇ C cˇ n C O n ,wehave 1 D O and hence n jnj˛ Â Ã Â Ã d =c log jnj 1 D ˇ ˇ C O C O : (4.6.17) n .2in/˛ jnj1C˛ jnj2˛

Substituting this relation into (4.6.15)weget Â Ã Â Ã dˇ=cˇ log jnj 1 w D 2in C log c C C O C O ;n!˙1; n ˇ .2in/˛ jnj1C˛ jnj2˛ (4.6.18) Â Ã log c 1 log w D log 2in C ˇ C O ;n!˙1: (4.6.19) n 2in jnj1C˛

˛ Now we note that the pre-images zn of wn satisfy jarg znj < 2 C " and 1 ˛ ˛ 2 C " <. Thus the conditions of Lemma 4.8 are satisﬁed and we obtain from this lemma and from (4.6.12)and(4.6.18) ! log w log2 w z1=˛ D w ˛ w C ˛ 2 n C O n : n n ˇ log n ˇ 2 (4.6.20) wn wn

Then the proof of the theorem in case (1) follows from (4.6.18)and(4.6.19). 4.6 Distribution of Zeros 71

In case (2) one can use a similar argument (see [PopSed11, pp. 36–37]) based on the following asymptotic formula

Á m Â Ã 1 p p X 1 1 E .z/ D z.1ˇ/=2 e z C ei.1ˇ/e z C O ; 2;ˇ 2 zk.ˇ 2k/ zmC1 kD1 (4.6.21) which is valid for jzj!1in the angles 0 Ä arg z Ä and Ä arg z Ä 0, respectively. F The most attractive result (see, e.g., [PopSed11, p. 37]) concerning the distribution of zeros of the Mittag-Lefﬂer function is the following:

Theorem 4.9. Let Re ˇ<3;ˇ6D 2 l;l 2 ZC.Thenallzeroszn of the Mittag- Leffler function E2;ˇ with sufficiently large modulus are simple and the following asymptotic formula is valid (n !˙1) Â Ã Â Ã iˇ=2 p n cˇe 1 1 zn D i .n 1 C ˇ=2/ C .1/ C O C O ; 2.in/2ˇ n6Re ˇ n1C2Re ˇ (4.6.22) p where the single-valued branch of the function z is chosen by the relation 0 Ä arg z <2. If ˇ is real then all zeros zn of E2;ˇ with sufficiently large modulus are real.

Now we present a result on the distribution of zeros of the function E2;3Ci.z/, 6D 0, 2 R. For the proof of the following theorem we refer to [PopSed11, pp. 39–44].

Theorem 4.10. (1) The set of multiple zeros of the function E2;3Ci.z/ is at most ﬁnite. C C (2) The sequence of all zeros zn consists of two subsequences zn ;n > n , and zn ;n

where the sequence ın is deﬁned by Â q Ã 1 ı D ı ./ D log ei log 2n C .ei log 2n/2 1 ;D ; n n .1C i/ (4.6.24)

where the principal branch is used for the values of the logarithmic function. p (3) The sequence n D zn asymptotically belongs to the semi-strips ˇ ˇ ˇ ˇ log < ˇRe C ˇ < log ; Im >0; (4.6.25) 1 2 2 72 4 The Two-Parametric Mittag-Lefﬂer Function

where p p 2 2 1 DjjC jj 1; 2 DjjC jj C 1:

(4) Every point of the interval Œ0; is a limit point of the sequence Im ın, and every point of the intervals Œlog 1; log 2, Œlog 1= 2; log 1= 1 is a limit point of the sequence Re ın. (5) There exist R D R./ > 0 and N D N./ 2 N such that there is no point of p the sequence k D zk in the disks

j inj N: (4.6.26)

It remains to consider the distribution of zeros of the function E˛;ˇ in the case ˛>2.

Theorem 4.11 ([PopSed11, p. 45]). Let ˛>2.Thenallzeroszn of the Mittag- Leffler function E˛;ˇ with sufficiently large modulus are simple and the following asymptotic formula holds: Â Â Ã Ã 1 ˇ 1 ˛ z D n C ˛ ; (4.6.27) n sin =˛ 2 ˛ n where the sequence ˛n is defined as described below: (1) If the pair .˛; ˇ/ is not mentioned in (4.6.4) and (4.6.5), then Á n.cos cos 3 /= sin ˛n D O e ˛ ˛ ˛ :

(2) If 2<˛<4,then Á ˛Re ˇ ncot ˛n D O n e ˛ :

(3) If ˛ 4,then Á Á 3 ncot ncos = sin ˛Re ˇ ˛n D e ˛ O e ˛ ˛ C O n :

If ˇ is real then all zeros zn with sufficiently large modulus are real too. In a series of articles by different authors the following question, which is very important for applications, was discussed: “Are all zeros of the Mittag-Leffler function E˛;ˇ with ˛>2simple and negative?” This question goes back to an article by Wiman [Wim05b]. Several attempts to answer this question have shown its non-triviality. In [OstPer97] this question was reformulated as the following problem: “For any ˛ 2,findasetW˛ consisting of those values of the positive parameter ˇ such that all zeros of the Mittag-Leffler function E˛;ˇ are simple and negative.” 4.6 Distribution of Zeros 73

Let us give some answers to the above question, following [PopSed11].

Theorem 4.12. For any ˛>2, ˇ 2 .0; 2˛ 1, all complex zeros .zn.˛; ˇ//n2N of the Mittag-Lefﬂer function E˛;ˇ are simple and negative and satisfy the following inequalities

.˛C ˇ/ ˛.˛; ˇ/ < z .˛; ˇ/ < ; (4.6.28) 1 1 .ˇ/ ˛ ˛ n .˛; ˇ/ < zn.˛; ˇ/ < n1.˛; ˇ/; n 2; (4.6.29) where Á n C ˇ1 ˛ ˛ n .˛; ˇ/ D : sin ˛

If ˛ 4, then all zeros are simple and negative for any ˇ 2 .0; 2˛.

˛ Theorem 4.13. Let ˛ 6, 0<ˇÄ 2˛. Then for all n; 1 Ä n Ä 3 1,thezeros zn.˛; ˇ/ of the Mittag-Lefﬂer function satisfy the following inequalities

p .˛nC ˇ/ .˛nC ˇ/ 2 < z .˛; ˇ/ < : (4.6.30) .˛.n 1/ C ˇ/ n .˛.n 1/ C ˇ/

Theorem 4.14. For any N 2 N, N 3,thezeroszn.N; N C 1/ of the Mittag- Lefﬂer function EN;N C1.z/ satisfy the relation Ä nC =2 C ˛ .N / ˛ z .N; N C 1/ D n ;n2 N;n ŒN=3 ; (4.6.31) n sin =˛ where ˛n.N / 2 R; j˛n.N /jÄxn.N /, 8 ˆ expfncot =˛g;3Ä N Ä 6; ˆ <ˆ

xn.N / D expf2n sin 2=˛g;7Ä N Ä 1;400; ˆ ˆ :ˆ 1:01expf2n sin 2=˛g; N > 1;400:

If N 6; 1 Ä n Ä ŒN=3 1,then Â Ã ..n C 1/N /Š 3=2Œ..n C 1/N /Š2 1 C minf1; N n2g .nN /Š .nN /Š..n C 2/N /Š ..n C 1/N /Š < z .N; N C 1/ < : (4.6.32) n .nN /Š 74 4 The Two-Parametric Mittag-Lefﬂer Function

Next we present a few non-asymptotic results on the distribution of zeros of the Mittag-Lefﬂer function. Theorem 4.15. Let 0<˛<1.Then Â Ã S1 S (1) For ˇ 2 Œn C ˛; n C 1 Œ1; C1/ the function E˛;ˇ.z/ has no nD0 negative zero; S1 (2) For ˇ 2 .n; n C ˛/ the function E˛;ˇ.z/ has one negative zero and it is nD0 simple. Theorem 4.16. (I) Let 0<˛<1, ˇ<0.Then

(1) For ˇ 2 Œ2n 1; 2n/;n 2 ZC, the function E˛;ˇ.z/ has one positive zero and it is simple; (2) For ˇ 2 Œ2n;2n C 1/; n 2 N, the set of zeros of the function E˛;ˇ.z/ is either empty, or consists of two simple points, or consists of one double point.

(II) The function E1;ˇ.z/ has a unique simple positive zero, whenever ˇ 2 .2n 1; 2n/; n 2 ZC, and has no positive zero, whenever ˇ 2 .2n;2n C 1/; n 2 N. Theorem 4.17. (I) Let one of the following conditions be satisﬁed: (1) 0<˛<1; ˇ2 Œ1; 1 C ˛, (2) 1<˛<2; ˇ2 Œ˛ 1; 1 [ Œ˛; 2.

Then all zeros of the function E˛;ˇ.z/ are located outside of the angle ˛ jarg zjÄ 2 . (II) Let 1<˛<2;ˇD 0.Thenallzeros(6D 0) are located outside of the angle ˛ jarg zjÄ 2 .

4.7 Computations with the Two-Parametric Mittag-Lefﬂer Function

Mittag-Leffler type functions play a basic role in the solution of fractional differential equations and integral equations of Abel type. Therefore, it seems important as a first step to develop their theory and stable methods for their numerical computation. Integral representations play a prominent role in the analysis of entire functions. For the two-parametric Mittag-Leffler function (4.1.1) such representations in the form of an improper integral along the Hankel loop have been treated in the case ˇ D 1 and in the general case with arbitrary ˇ by Erdélyi et al. [Bat-3]and Dzherbashyan [Dzh54a, Dzh66]. They considered the representations 4.7 Computations with the Two-Parametric Mittag-Leffler Function 75

Z 1=˛ .1ˇ/=˛ 1 e ./ E˛;ˇ.z/ D d ; z 2 G . I ı/; (4.7.1) 2i˛ . Iı/ z Z 1=˛ .1ˇ/=˛ 1 .1ˇ/=˛ z1=˛ 1 e .C/ E˛;ˇ.z/ D z e C d ; z 2 G . I ı/; ˛ 2i˛ . Iı/ z (4.7.2) under the conditions

0<˛<2;˛=2<ı

The contour . I ı/ consists of two rays Sı (arg Dı;j j )andSı (arg D ı;j j ) and a circular arc Cı.0I / (j jD ; ı Ä arg Ä ı). On its left side there is a region G./. ; ı/, on its right side a region G.C/. ; ı/. Using the integral representations in (4.7.1)and(4.7.2) it is not difﬁcult to obtain asymptotic expansions for the Mittag-Lefﬂer function in the complex plane (see Theorems 4.3 and 4.4). Let 0<˛<2, ˇ be an arbitrary number, and ı be chosen to satisfy the condition (4.7.3). Then we have, for any p 2 N (and for p D 0 if the “empty sum convention” is adopted) and jzj!1

Xp k 1 1=˛ z E .z/ D z.1ˇ/=˛ez C O jzj1p ; 8z; jarg zjÄı: ˛;ˇ ˛ .ˇ ˛k/ kD1 (4.7.4)

Analogously, for all z;ı Äjarg zjÄ,wehave

p X zk E .z/ D C O jzj1p : (4.7.5) ˛;ˇ .ˇ ˛k/ kD1

These formulas are used in the numerical algorithm presented in this section (proposed in [GoLoLu02]). In what follows attention is restricted to the case ˇ 2 R, the most important one in the applications. For the purpose of numerical computation we look for integral representations better suited than (4.7.1)and (4.7.2). Denoting

e 1=˛ .1ˇ/=˛ . ;z/ D z we represent the integral in formulas (4.7.1)and(4.7.2) in the form Z Z 1 1 I D . ;z/d D . ;z/d 2i˛ . Iı/ 2i˛ S Z Z ı 1 1 C . ;z/d C . ;z/d D I1 C I2 C I3: (4.7.6) 2i˛ Cı.0I / 2i˛ Sı 76 4 The Two-Parametric Mittag-Lefﬂer Function

iı The integrals I1, I2 and I3 have to be transformed. For I1 we take D re , Ä r<1, and get

Z Z iı 1=˛ 1 1 e.re / .reiı/.1ˇ/=˛ I D . ;z/ D iı r: 1 d iı e d 2i˛ Sı 2i˛ C1 .re / z (4.7.7) Analogously, by using D reiı, Ä r<1,

Z Z 1=˛ 1 1 C1 e.reiı/ .reiı/.1ˇ/=˛ I D . ;z/ D iı r: 3 d iı e d (4.7.8) 2i˛ Sı 2i˛ .re / z

i' For I2 with D e ; ı Ä ' Ä ı

Z Z 1=˛ 1 1 ı e. ei'/ . ei'/.1ˇ/=˛ I D . ;z/ D i i' ' 2 d i' e d 2i˛ Cı.0I / 2i˛ ı . e / z Z Z 1C.1ˇ/=˛ ı e 1=˛ .ei'=˛/e.i'.1ˇ/=˛C1/ ı D ' D PŒ˛;ˇ; ;';z '; i' d d 2˛ ı e z ı (4.7.9) where

1C.1ˇ/=˛ e 1=˛ cos .'=˛/.cos .!/ C i sin .!// PŒ˛;ˇ; ;';z D ; 2˛ ei' z (4.7.10) ! D 1=˛ sin .'=˛/ C '.1 C .1 ˇ/=˛/:

The sum I1 and I3 can be rewritten as Z C1 I1 C I3 D KŒ˛;ˇ;ı;r;zdr; (4.7.11)

where

1 1=˛ r sin . ı/ z sin . / KŒ˛;ˇ;ı;r;z D r.1ˇ/=˛er cos .ı=˛/ ; 2˛ r2 2rz cos .ı/ C z2 (4.7.12) D r1=˛ sin .ı=˛/ C '.1 C .1 ˇ/=˛/:

Using the above notation formulas (4.7.1)and(4.7.2) can be rewritten in the form Z Z C1 ı ./ E˛;ˇ.z/ D KŒ˛;ˇ; ;';zdr C PŒ˛;ˇ; ;';zd'; z 2 G . I ı/; ı (4.7.13) 4.7 Computations with the Two-Parametric Mittag-Lefﬂer Function 77 Z Z C1 ı E˛;ˇ.z/ D KŒ˛;ˇ; ;';zdr C PŒ˛;ˇ; ;';zd' ı

1 1=˛ C z.1ˇ/=˛ez ; z 2 G.C/. I ı/: (4.7.14) ˛

Let us now consider the case 0<˛Ä 1, z 6D 0. By condition (4.7.3) we can choose ı D min f; ˛gD˛. Then the kernel function (4.7.12) looks simpler:

KŒ˛;ˇ;˛;r;z D KŒ˛;ˇ;r;Q z (4.7.15)

1 1=˛ r sin ..1 ˇ// z sin ..1 ˇ C ˛/ D r.1ˇ/=˛er : 2˛ r2 2rz cos .˛/ C z2

We distinguish three possibilities for arg z in the formulas (4.7.13)–(4.7.15)for the computation of the function E˛;ˇ.z/ at an arbitrary point z 2 C; z 6D 0, namely (A) jarg zj >˛; (B) jarg zjD˛; (C) jarg zj <˛. The following theorems give representation formulas suitable for further numerical calculations. Theorem 4.18. Under the conditions

0<˛Ä 1; ˇ 2 R; jarg zj >˛; z 6D 0; the function E˛;ˇ.z/ has the representations Z Z C1 ˛ E˛;ˇ.z/ D KŒ˛;ˇ;r;Q zdr C PŒ˛;ˇ; ;';zd'; > 0; ˇ 2 R; ˛ (4.7.16) Z C1 E˛;ˇ.z/ D KŒ˛;ˇ;r;Q zdr; if ˇ<1C ˛; (4.7.17) 0 Z 1=˛ sin .˛/ C1 er 1 E .z/ D r ; ˇ D 1 C ˛: ˛;ˇ 2 2 d if ˛ 0 r 2rz cos .˛/ C z z (4.7.18)

Theorem 4.19. Under the conditions

0<˛Ä 1; ˇ 2 R; jarg zjD˛; z 6D 0; the function E˛;ˇ.z/ has the representations 78 4 The Two-Parametric Mittag-Lefﬂer Function Z Z C1 ˛ E˛;ˇ.z/ D KŒ˛;ˇ;r;Q zdr C PŒ˛;ˇ; ;';zd'; > jzj; (4.7.19) ˛ where the kernel functions KŒ˛;ˇ;r;Q z and PŒ˛;ˇ; ;';z are given by the formulas (4.7.15) and (4.7.10), respectively. Theorem 4.20. Under the conditions

0<˛Ä 1; ˇ 2 R; jarg zj <˛; z 6D 0; the function E˛;ˇ.z/ has the representations Z Z C1 ˛ E˛;ˇ.z/ D KŒ˛;ˇ;r;Q zdr C PŒ˛;ˇ; ;';zd' ˛

1 1=˛ C z.1ˇ/=˛ez ;0<

Z 1=˛ sin .˛/ C1 er E .z/ D r ˛;ˇ 2 2 d ˛ 0 r 2rz cos .˛/ C z

1 1 1=˛ C ez ; if ˇ D 1 C ˛; (4.7.22) z ˛z where the kernel functions KŒ˛;ˇ;r;Q z and PŒ˛;ˇ; ;';z are given by formulas (4.7.15) and (4.7.10), respectively.

Therefore, for arbitrary z 6D 0 and 0<˛Ä 1 the Mittag-Leffler function E˛;ˇ.z/ can be represented by one of the formulas (4.7.16)–(4.7.22). These formulas are used for numerical computation if q0by using series representation (4.1.1). The case ˛>1is reduced to the case 0<˛Ä 1 by using recursion formulas. To compute the function E˛;ˇ.z/ for arbitrary z 2 C with arbitrary indices ˛>0;ˇ2 R, three possibilities are distinguished: (A) jzjÄq, 00; (B) jzj >q, 0<˛Ä 1; (C) jzj >q, ˛>1. In each case the Mittag-Leffler function can be computed with the prescribed accuracy >0. In the case (A) the computations are based on the following result: 4.7 Computations with the Two-Parametric Mittag-Leffler Function 79

Theorem 4.21. In the case .A/ the Mittag-Lefﬂer function can be computed with the prescribed accuracy >0by use of the formula

Xk0 zk E .z/ D C .z/; j.z/j < ; (4.7.23) ˛;ˇ .˛kC ˇ/ kD0 where

k0 D max fŒ.1 ˇ/=˛ C 1I Œln . .1 jzj//= ln .jzj/g:

In the case (B) one can use the integral representations (4.7.16)–(4.7.22). For this it is necessary to compute numerically either the improper integral

Z1 I D KŒ˛;ˇ;r;Q zdr; a 2f0I g; a and/or the integral

Z˛ J D PŒ˛;ˇ; ;';zd'; > 0: ˛

To calculate the ﬁrst (improper) integral I of the bounded function KŒ˛;ˇ;r;Q z the following theorem is used: Theorem 4.22. The representation Z Z 1 r0 I D KŒ˛;ˇ;r;Q zdr D KŒ˛;ˇ;r;Q zdr C .r/; j.r/jÄ ; a 2f0I g; a a (4.7.24) is valid under the conditions

0<˛Ä 1; jzj >q>0; 8 < max f1; 2jzj;. ln . =6//˛g; if ˇ 0I r0 D : max f.1 Cjˇj/˛;2jzj;.2 ln . =.6.jˇjC2/.2jˇj/jˇj///˛g; if ˇ<0:

The second integral J (the integrand PŒ˛;ˇ; ;';z being bounded and the limits of integration being ﬁnite) can be calculated with prescribed accuracy >0by one of many product quadrature methods. In the case (C) the following recursion formula is used (see [Dzh66]) 80 4 The Two-Parametric Mittag-Lefﬂer Function

1 mX1 E .z/ D E .z1=me2il=m/; m 1: (4.7.25) ˛;ˇ m ˛=m;ˇ lD0

In order to reduce case (C) to the cases (B) and (A) one can take m D Œ˛ C1 in for- 1=m 2il=m mula (4.7.25). Then 0<˛=m:1, and we calculate the functions E˛=m;ˇ.z e / as in case (A) if jzj1=m Ä q<1, and as in case (B) if jzj1=m >q. Remark 4.23. The ideas and techniques employed for the Mittag-Leffler function can be used for numerical calculation of other functions of hypergeometric type. In particular, the same method with some small modifications can be applied to the Wright function, which plays a very important role in the theory of partial differential equations of fractional order (see, e.g., [BucLuc98, GoLuMa00, Luc00, LucGor98, MaLuPa01]). To this end, the following representations of the Wright function (see [GoLuMa99]) can be used in place of the corresponding representations of the Mittag-Leffler function:

X1 zk . ;ˇI z/ D ; >1; ˇ 2 C; . kC ˇ/ kD0 Z 1 . ;ˇI z/ D e Cz ˇd ; > 1; ˇ 2 C; 2i Ha where Ha denotes the Hankel path in the -plane with a cut along the negative real semi-axis arg D .

4.8 Extension for Negative Values of the First Parameter

The two-parametric Mittag-Leffler function (4.1.1), defined in the form of a series, exists only for the values of parameters Re ˛>0and ˇ 2 C.However,byusing an existing integral representation formula for the two-parametric Mittag-Leffler function (see, e.g., [Dzh66, KiSrTr06]) it is possible to determine an extension of the two-parametric Mittag-Leffler function to other values of the first parameter. In this section we present an analytic continuation of the Mittag-Leffler function depending on real parameters ˛; ˇ 2 R by extending its domain to negative ˛<0. Here we follow the results of [Han-et-al09]. The following integral representation of the Mittag-Leffler function is known (see, e.g., [Dzh66, KiSrTr06]) Z 1 t ˛ˇ et E C ˛;ˇ.z/ D ˛ dt; z 2 ; (4.8.1) 2 Ha t z 4.8 Extension for Negative Values of the First Parameter 81 where the contour of integration Ha is the so-called Hankel path, a loop starting and ending at 1, and encircling the disk jtjÄjzj1=˛ counterclockwise. To find an equation which can determine E˛;ˇ.z/, we rewrite the integral representation of the Mittag-Leffler function (4.8.1)as Z 1 et E .z/ D t ˛;ˇ ˇ ˛Cˇ d (4.8.2) 2 Ha t zt and expand part of the integrand in (4.8.2) in partial fractions as follows:

1 1 1 D : (4.8.3) t ˇ zt ˛Cˇ t ˇ t ˇ z1t ˛Cˇ

Substituting Eq. (4.8.3)into(4.8.2) yields Z Z 1 et 1 et E .z/ D t t; z 2 C nf0g: ˛;ˇ ˇ d ˇ 1 ˛Cˇ d (4.8.4) 2 Ha t 2 Ha t z t

This gives the following definition of the Mittag-Leffler function with negative value of the first parameter: Â Ã 1 1 E .z/ D E ;˛>0;ˇ2 RI z 2 C nf0g: (4.8.5) ˛;ˇ .ˇ/ ˛;ˇ z

In particular, Â Ã 1 E .z/ WD E .z/ D 1 E ;˛>0I z 2 C nf0g: ˛ ˛;1 ˛ z

By using the known recurrence formula

1 E .z/ D C zE .z/ ˛;ˇ .ˇ/ ˛;˛Cˇ we obtain another variant of the deﬁnition (4.8.5) Â Ã 1 1 E .z/ D E ;˛>0;ˇ2 RI z 2 C nf0g: (4.8.6) ˛;ˇ z ˛;˛Cˇ z

Direct calculations show that deﬁnitions (4.8.5)and(4.8.6) determine the same function, analytic in C nf0g. By taking the limit in (4.8.5)as˛ !C0 we get the deﬁnition of E0;ˇ.z/

1 E .z/ D ;ˇ2 RIjzj <1: (4.8.7) 0;ˇ .ˇ/.1 z/

Obviously, this function can be analytically continued in the domain C nf1g. 82 4 The Two-Parametric Mittag-Lefﬂer Function

From the definition of the two-parametric Mittag-Leffler function (4.1.1)we obtain the following series representation of the extended Mittag-Leffler function (i.e. the function corresponding to negative values of the first parameter): Â Ã X1 1 1 k E .z/ D ; z 2 C nf0g: (4.8.8) ˛;ˇ .˛z C ˇ/ z kD1

By using this representation and the above definitions of the extended Mittag- Leffler function (4.8.5)(or(4.8.6)) one can obtain functional, differential and recurrence relations which are analogous to corresponding relations for the two- parametric function with positive first parameter. Proposition 4.24. Let ˛>0, ˇ 2 R. Then the following formulas are valid for all values of parameters for which all items are defined. A. Recurrence relations.

2 E˛;ˇ.z/ C E˛;ˇ.z/ D 2E2˛;ˇ.z /I (4.8.9)

1 Xn1 E .z/ D E .ze2ik=n/I (4.8.10) n˛;ˇ n ˛;ˇ kD0

Xn1 zk E .z/ D znE .z/ C I (4.8.11) ˛;ˇ ˛;ˇ˛n .ˇ ˛k/ kD0 2 2 2 E˛.z/ D E2˛.z / C E2˛.z / zE2˛;˛C1.z /: (4.8.12)

B. Differential relations.

d 1ˇ ˛ ˇ ˛ z E .z / Dz E .z /I (4.8.13) dz ˛;ˇ ˛;ˇ 1 d 1 1 ŒE .z/ D C E .z/I (4.8.14) dz ˛ .˛C 1/ ˛ ˛;˛ n d n n ŒE .z / D E .z /: (4.8.15) dzn n n

C. Functional relations.

Zz ˛ ˇ1 ˇ ˛ E˛;ˇ.t /t dt Dz E˛;ˇC1.z /I (4.8.16) 0 Ä Â Ã 1 a L zˇ1E D : (4.8.17) ˛;ˇ ˙az˛ sˇ.s˛ a/ 4.9 Further Analytic Properties 83

4.9 Further Analytic Properties

Here we present a number of integral and differential formulas for the Mittag-Lefﬂer function. The integral relations below can easily be established by applying classical formulas for Gamma and Beta functions (see Appendix A) and other techniques.

Z1 1 exxˇ1E .x˛z/ dx D .jzj <1;˛;ˇ2 C; Re ˛>0;Re ˇ>0/: ˛;ˇ 1 z 0 (4.9.1) Zx ˇ1 ˛ ˇ ˛ .x / E˛ . / d D.ˇ/x E˛;ˇC1.x /.˛;ˇ2 C; Re ˛>0; Re ˇ>0/: 0 (4.9.2) Z1 mŠS ˛ˇ esxxm˛Cˇ1E.m/ .˙x˛/ dx D ; (4.9.3) ˛;ˇ s˛ 0 where js˛j <1;˛;ˇ2 C; Re ˛>0;Re ˇ>0.

ˇ11 ˛ ˇ21 ˛ E˛;ˇ1 . /.x / E˛;ˇ2 ..x / / d 0 xˇ1Cˇ21 ˚ « D E .x˛/ E .x˛ / : (4.9.4) ˛;ˇ1Cˇ2 ˛;ˇ1Cˇ2

The differential relations below follow by direct calculation: Â Ã @ n zˇ1E .z˛/ D zˇn1E .z˛/: (4.9.5) @z ˛;ˇ ˛;ˇn Â Ã @ n zˇ1E .z˛/ D nŠz˛nCˇ1EnC1 .z˛/; (4.9.6) @ ˛;ˇ ˛;˛nCˇ

nC1 where E˛;˛nCˇ.z/ is the Prabhakar three-parametric function (see Sect. 5.1 below). A special case of the two-parametric Mittag-Lefﬂer function (the so-called ˛- exponential function) is of interest for many applications. It is deﬁned in the following way:

z ˛1 ˛ C C e˛ WD z E˛;˛.z /.z 2 nf0g;2 /: (4.9.7)

For all ˛ 2 C; Re ˛>0, it can be represented in the form of the series 84 4 The Two-Parametric Mittag-Lefﬂer Function

X1 z˛k ez D z˛1 k ; (4.9.8) ˛ ..k C 1/˛/ kD0 which converges in C nf0g and determines in this domain an analytic function. The simple properties of this function:

1˛ z 1 .1/ lim z e˛ D .Re ˛>0/; (4.9.9) z!0 .˛/ z z .2/ e1 D e ; (4.9.10) justify its name. However, the ˛-exponential function does not satisfy the main property of the exponential function, i.e.,

z z .C/z e˛ e˛ 6D e˛ : (4.9.11)

For 0<˛<2,the˛-exponential function satisﬁes a simple asymptotic relation Â Ã .1˛/=˛ NX1 k1 1 1 ez D expf1=˛zg C O ; (4.9.12) ˛ ˛ .˛k/ z˛kC1 z˛N C1 kD1

N ˛ ˛ where z !1, N 2 nf1g, jarg.z /jÄ, 2 <

4.10 The Two-Parametric Mittag-Lefﬂer Function of a Real Variable

4.10.1 Integral Transforms of the Two-Parametric Mittag-Lefﬂer Function

The following form of the Laplace transform of the two-parametric Mittag-Lefﬂer function is most often used in applications:

s˛ˇ L t ˇ1E .t ˛/ .s/ D .Re s>0;2 C; js˛j <1/: (4.10.1) ˛;ˇ s˛ 4.10 The Two-Parametric Mittag-Lefﬂer Function of a Real Variable 85

It can be shown directly that the Laplace transform of the two-parametric Mittag-Lefﬂer function E˛;ˇ.t/ is given in terms of the Wright function (see, e.g., [KiSrTr06,p.44]) Ä ˇ ˇ 1 .1; 1/; .1; 1/ ˇ1 L E˛;ˇ.t/ .s/ D 21 ˇ .Re s>0/: (4.10.2) s .˛; ˇ/ s

From the Mellin–Barnes integral representation of the two-parametric Mittag- Lefﬂer function we arrive at the following formula for the Mellin transform of this function

Z1 .s/.1 s/ M E .t/ .s/ D E .t/ts1dt D .0 < Re <1/: ˛;ˇ ˛;ˇ .ˇ ˛s/ 0 (4.10.3)

To conclude this subsection, we consider the Fourier transform of the two- parametric Mittag-Lefﬂer function E˛;ˇ.jtj/ with ˛>1. Performing a term-by-term integration of the series we get the formula (˛>1):

ZC1 Ä ˇ ˇ ixt ı.x/ 2 .2; 2/; .1; 1/ ˇ 1 F E˛;ˇ.jtj/ .x/ WD e E˛;ˇ.jtj/dt D 21 ˇ ; .ˇ/ x2 .˛ C ˇ; 2˛/ x2 1 (4.10.4) where ı./ is the Dirac delta function. Since for all t 1 E .jtj/ DjtjE .jtj/; (4.10.5) ˛;ˇ .ˇ/ ˛;˛Cˇ formula (4.10.4) can be simpliﬁed Ä ˇ ˇ 2 .2; 2/; .1; 1/ ˇ 1 F jtjE˛;˛Cˇ.jtj/ .x/ D 21 ˇ .˛ > 1; ˇ 2 C/: x2 .˛ C ˇ; 2˛/ x2 (4.10.6)

4.10.2 The Complete Monotonicity Property

Let us show that the generalized Mittag-Leffler function E˛;ˇ.x/ possesses the complete monotonicity property for 0 Ä ˛ Ä 1; ˇ ˛. In fact, this result follows from the complete monotonicity of the classical Mittag-Leffler function E˛.x/ due to the following technical lemmas. 86 4 The Two-Parametric Mittag-Leffler Function

Lemma 4.25. For all ˛ 0

d E .x/ D˛ E .x/: ˛;˛ dx ˛ G This follows from the standard properties of the integral depending on a parameter. F Lemma 4.26. Let ˇ>˛>0. Then the following identity holds:

Z1 1 ˇ˛1 E .x/ D 1 t 1=˛ E .tx/dt: (4.10.7) ˛;ˇ ˛ .ˇ ˛/ ˛;˛ 0

G Let us take E˛;˛.tx/ in the form of a series and substitute it into the right-hand side of (4.10.7). By interchanging the order of integration and summation (which can be easily justiﬁed) we obtain that the right-hand side is equal to

Z1 X1 k 1 .x/ ˇ˛1 t k 1 t 1=˛ dt: ˛ .ˇ ˛/ .˛kC ˛/ kD0 0

Calculating these integrals we arrive at the series representation for E˛;ˇ.x/. F Observe that 1 1 E .x/ D ;ˇ>0; 0;ˇ ˛ .ˇ/ 1 C x

E0;ˇ.x/ D 0; ˇ D 0:

In both cases E0;ˇ.x/ is completely monotonic. The complete monotonicity of E˛;ˇ.x/ then follows immediately from Pollard’s result [Poll48], see Sect. 3.7.2 of this book.

4.10.3 Relations to the Fractional Calculus

Here we present a few formulas related to the values of the fractional integrals and derivatives of the two-parametric Mittag-Lefﬂer function (see, e.g., [HaMaSa11, pp. 15–16]). Let us start with the left-sided Riemann–Liouville integral. Suppose that Re ˛>0;Re ˇ>0;Re >0;a2 R. Then by using the series representation and the left-sided Riemann–Liouville integral of the power function we get ˛ 1 ˇ ˛C1 ˇ I0C t Eˇ; .at / .x/ D x Eˇ;˛C .ax / ; (4.10.8) 4.10 The Two-Parametric Mittag-Lefﬂer Function of a Real Variable 87 and, in particular, if a 6D 0, then (for ˇ D ˛) Â Ã x1 1 I ˛ t 1E .at˛/ .x/ D E .ax˛/ : (4.10.9) 0C ˛; a ˛; ./

In the same manner one can obtain the formula Â Ã x˛1 1 I ˛ t ˛1E .at˛/ .x/ D E .ax˛/ : (4.10.10) 0C ˛;ˇ a ˛;ˇ .ˇ/

Analogously, one can calculate the right-sided fractional Riemann–Liouville integral of the two-parametric Mittag-Lefﬂer function in the case Re ˛>0; Re ˇ>0;a2 R;a6D 0 ˛ ˛ ˇ ˇ I t Eˇ; .at / .x/ D x Eˇ;˛C .ax / : (4.10.11)

If we suppose additionally that Re .˛ C / > Re ˇ, then the last formula can be rewritten as Â Ã xˇ 1 I ˛ t ˛ E .atˇ/ .x/ D E .axˇ/ ; ˇ; a ˇ;˛Cˇ .˛C ˇ/ (4.10.12) and, in particular, Â Ã x˛ˇ 1 I ˛ t ˛ˇ E .at˛/ .x/ D E .ax˛/ : (4.10.13) ˛;ˇ a ˛;ˇ .ˇ/

In the case of the fractional differentiation of the two-parametric Mittag-Lefﬂer function we have ˛ 1 ˇ ˛1 ˇ R D0C t Eˇ; .at / .x/ D x Eˇ;˛.ax / ; Re ˛>0; Re ˇ>0;a2 : (4.10.14)

If we assume extra conditions on the parameters, namely Re >Re ˇ,Re> Re .˛ C ˇ/, a 6D 0, then the following relations hold: Â Ã x˛ˇ1 1 D˛ t 1E .atˇ/ .x/ D E .axˇ/ : 0C ˇ; a ˇ;˛ˇ . ˛ ˇ/ (4.10.15)

In particular (see [KilSai95b]), for Re ˛>0;Re ˇ>Re ˛ C 1, one can prove

xˇ˛1 D˛ t ˇ1E .at˛/ .x/ D C axˇ1 E .ax˛/ : (4.10.16) 0C ˛;ˇ .ˇ ˛/ ˛;ˇ

Finally, the right-sided (Liouville) fractional derivative of the two-parametric Mittag-Leffler function satisfies the relation (see, e.g., [KiSrTr06,p.86]) 88 4 The Two-Parametric Mittag-Leffler Function

xˇ D˛ t ˛ˇE .at˛/ .x/ D C ax˛ˇ E .ax˛/ ; (4.10.17) ˛;ˇ .ˇ ˛/ ˛;ˇ valid for all Re ˛>0;Re ˇ>Re ˛ C 1. We also mention two extra integral relations for the two-parametric Mittag- Lefﬂer function which are useful for applications. Lemma 4.27. Let ˛>0and ˇ>0. Then the following formula is valid

Zx 1 t ˇ1E .t 2˛/ 2˛;ˇ dt D xˇ1 E .x˛/ E .x2˛ / : (4.10.18) .˛/ .x t/1˛ ˛;ˇ 2˛;ˇ 0

Corollary 4.28. Formula (4.10.18) means ˛ ˇ1 2˛ ˇ1 ˛ 2˛ I0C t E2˛;ˇ.t / .x/ D x E˛;ˇ.x / E2˛;ˇ.x / : (4.10.19)

4.11 Historical and Bibliographical Notes

The two-parametric Mittag-Leffler function first appeared in the paper by Wiman 1905 [Wim05a], but he did not pay too much attention to it. Much later this function was rediscovered by Humbert and Agarval, who studied it in detail in 1953 [Aga53](seealso[Hum53, HumAga53]). A new function was obtained by replacing the additive constant 1 in the argument of the Gamma function in (3.1.1) by an arbitrary complex parameter ˇ. Later, when we deal with Laplace transform pairs, the parameter ˇ will be required to be positive like ˛. Using the integral representations for E˛; ˇ.z/ Dzherbashian [Dzh54a], [Dzh54b], [Dzh66, Ch. III, §2] proved formulas for the asymptotic representation of E˛; ˇ.z/ at infinity, and in [Dzh66, Ch. III, §4] he gave applications of these to the construction of Fourier type integrals and to the proof of theorems on pointwise convergence of these integrals on functions defined and summable with exponential- power weight on a finite system of rays. Note that the developed technique is based on the representation of entire functions in the form of sums of integral transforms with kernels of the form E˛; ˇ.z/. By using asymptotic properties of the function E˛; ˇ.z/, Dzherbashian (see[Dzh66, Ch. III, §2]) found its Mellin transform, established certain functional identities and proved the inversion formula for the following integral transform with the function E˛; ˇ.z/ in the kernel Z1 i' ˛ ˛ ˇ1 E˛;ˇ.e x t /t f.t/dt (4.11.1) 0 in the space L2.RC/. 4.11 Historical and Bibliographical Notes 89

In [Bon-et-al02] the properties of the integral transforms with Mittag-Lefﬂer function in the kernel

E˛; ˇ.xt/f .t/dt.x>0/ (4.11.2) 0 are studied in weighted spaces of r-summable functions 8 0 1 9 <ˆ Z1 1=r => @ dt A L;r D f Wkf k;r Á jt f.t/j ;1Ä r<1;2 R : :ˆ t ;> 0 (4.11.3)

The conditions for the boundedness of such an operator as a mapping from one space to another were found, the images of these spaces under such a mapping were described, and inversion formulas were established. These results are based on the representation of (4.11.2) as a special case of the general H-transform (see Sect. F.3). In recent years mathematicians’ attention towards the Mittag-Leffler type functions has increased, both from the analytical and numerical point of view, overall because of their relation to the fractional calculus. In addition to the books and papers already quoted in the text, here we would like to draw the reader’s attention to some recent papers on the Mittag-Leffler type functions, e.g., Al Saqabi and Tuan [Al-STua96], Kilbas and Saigo [KilSai96], Gorenflo, Luchko and Rogosin [GoLuRo97] and Mainardi and Gorenflo [MaiGor00]. Since the fractional calculus has now received wide interest for its applications in different areas of physics and engineering, we expect that the Mittag-Leffler function will soon occupy its place as the Queen Function of Fractional Calculus. The remarkable asymptotic properties of the Mittag-Leffler function have pro- voked an interest in the investigation of the distribution of the zeros of E˛;ˇ.z/. Several articles have been devoted to this problem (see [Dzh84,DzhNer68,OstPer97, Poly21,Pop02,Psk05,Sed94,Sed00,Wim05b]). An extended survey of the results is presented in [PopSed11]. Also studied is the related question of the distribution of zeros of sections and tails of the Mittag-Leffler function (see [Ost01,Zhe02]) and of some associated special functions (see [GraCso06, Luc00]). The obtained results have found an application in the study of certain problems in spectral theory (see, e.g. [Dzh70,Djr93,Nak03]), approximation theory (see, e.g. [Sed98]), and in treating inverse problems (see, e.g., [TikEid02]). Except in the case when ˛ D 1; ˇ Dm; m 2f1g[ZC, the function E˛;ˇ.z/ has an infinite set of zeros (see [Sed94]). In [Wim05b] it was shown that for ˛ 2 all zeros of the classical Mittag-Leffler function E˛;1.z/ are negative and simple (see also [Poly21], where the case ˛ D N 2 N;N > 1, is considered). In [Dzh84]it was proved that same result is valid for E2;ˇ.z/; 1 < ˇ < 3. Note that all zeros 90 4 The Two-Parametric Mittag-Leffler Function

p z1 of the function E2;3.z/ D cosh z are twofold and negative, but the function E2;ˇ.z/; ˇ > 3, has no real zero. Ostrovski and Pereselkova [OstPer97] formulated the problem to describe the set W of pairs .˛; ˇ/ such that all zeros of E˛;ˇ.z/ are negative and simple. The authors conjectured that

W Df.˛; ˇ/j˛ 2;0 < ˇ < 1 C ˛g:

It was shown, in particular, that .˛; 1/; .˛; 2/ 2 W for all real ˛ 2,and f.2m;ˇ/jm 2 N;0<ˇ<1C 2mg W. The asymptotic behaviour of the zeros of the function E˛;ˇ.z/ is the subject of several investigations. In [Sed94] asymptotic formulas for the zeros zn.˛; ˇ/ of E˛;ˇ.z/ were found for all ˛>0and ˇ 2 C.For0<˛<2this asymptotic representation as n !˙1is more exact and has the form Â Ã i .z .˛; ˇ//1=˛ D 2in C a.˛; ˇ/ log jnjC sign n n 2 Â Ã 1 Cb.˛;ˇ/ C O n˛ C log jnj : n

The values of a.˛;ˇ/;b.˛;ˇ/ are given in [Sed94]. A way of evaluating the zeros compatible with this asymptotical formula is proposed in [Sed00]. Schneider [Sch96] has proved that the generalized Mittag-Leffler function E˛;ˇ.x/ is completely monotonic for positive values of parameters ˛; ˇ if and only if 0<˛Ä 1; ˇ ˛. The proof was based on the use of the corresponding probability measures and the Hankel integration path. An analytic proof presented in Sect. 4.1.5 is due to Miller and Samko [MilSam97]. Note that the main formula (4.10.7) used in this proof is a special case of a more general relation due to Dzherbashian [Dzh66, p. 120] which ˇC1 ˛ states that x E˛;ˇC .x / is the fractional integral of order of the ˇ1 ˛ function x E˛;ˇ.x /. However, the above presented result is more simple and straightforward. Later (see, [MilSam01]), the proof of the complete monotonicity of some other special functions was given by Miller and Samko. As a challenging open problem related to the Special Functions of Fractional Cal- culus (such as the multi-index Mittag-Leffler functions), we mention the possibility of their numerical computation and graphical interpretation, plots and tables, and implementations in software packages such as Mathematica, Maple, Matlab, etc. As mentioned earlier, the Classical Special Functions are already implemented there. For their Fractional Calculus analogues, numerical algorithms and software packages have been developed only for the classical Mittag-Leffler function E˛Iˇ.z/ and the Wright function .˛;ˇI z/! Numerical results and plots for the Mittag-Leffler functions for basic values of indices can be found in Caputo–Mainardi [CapMai71b] (one of the first attempts!) and Gorenflo–Mainardi [GorMai97]. Among the very 4.12 Exercises 91 recent achievements, we mention the following results: Podlubny [Pod11] (a Matlab routine that calculates the Mittag-Leffler function with desired accuracy), Gorenflo et al. [GoLoLu02] and Diethelm et al. [Die-et-al05] (algorithms for the numerical evaluation of the Mittag-Leffler function and a package for computation with Mathematica), Hilfer–Seybold [HilSey06] (an algorithm for extensive numerical calculations for the Mittag-Leffler function in the whole complex plane, based on its integral representations and exponential asymptotics), Luchko [Luc08] (algorithms for computation of the Wright function with prescribed accuracy), etc. The results concerning calculation of the Mittag-Leffler function presented in Sect. 4.7 are based on the paper [GoLoLu02]. In [GoLoLu02] a numerical scheme for computation of the Mittag-Leffler function is given in pseudocode using a specially developed algorithm based on the above formulated results (see also the MatLab routine by Podlubny [Pod06, Pod11], and numerical computations of the Mittag-Leffler function performed by Hilfer and Seybold [SeyHil05, HilSey06, SeyHil08]). The increasing interest in Mittag-Leffler functions and their wide application has given rise to the need for effective strategies for their numerical computation (see [HilSey06,Pod11,SeyHil08]). Thus the analysis of efficient and accurate numerical methods for generalized Mittag-Leffler functions with two parameters

ˇ1 ˛ e˛;ˇ.tI / D t E˛;ˇ.t /; with scalar or matrix arguments, is actually a compelling subject. In [GarPop12]the problem of the numerical computation of generalized Mittag-Leffler functions with two parameters e˛;ˇ.tI / is studied with applications to fractional calculus. The inversion of their Laplace transform is an effective tool in this direction; however, the choice of the integration contour is crucial. Here parabolic contours are used and the quadrature rules for numerical integration are applied. An in-depth error analysis is carried out to select suitable contour parameters, depending on the parameters of the Mittag-Leffler function, in order to achieve any fixed accuracy. Numerical experiments to validate theoretical results are performed and some computational issues are discussed.

4.12 Exercises

4.12.1. Prove the following relations ([Ber-S05b]):

2 2 E˛.x/ D E2˛.x / xE2˛;1C˛.x /; x 2 R; Re ˛>0; (4.12.1a) 2 2 E˛.ix/ D E2˛.x / ixE2˛;1C˛.x /; x 2 R; Re ˛>0: (4.12.1b) 92 4 The Two-Parametric Mittag-Lefﬂer Function

4.12.2. Prove the following recurrence relations ([SaKaKi03])

mX1 zn zmE .z/DE .z/ ; Re ˛>0; Re ˇ>0; m 2 N: ˛;ˇCm˛ ˛;ˇ .ˇC n˛/ nD0 (4.12.2)

4.12.3. Let the family of functions H˛ be given by the formula ([Ber-S05b, p. 432])

Z1 2 H .k/ D E .t 2/ cos .kt/dt; k > 0; 0 Ä ˛ Ä 1; ˛ 2˛ 0

where its power series in k has the form

1 X1 H .k/ D b .˛/kn;0Ä ˛<1: ˛ n nD0

Deduce the following asymptotic formula for E˛.x/:

1 X1 b .˛/ E .x/ D n ;0Ä ˛<1: ˛ xnC1 nD0

Hint. Use the relation (4.12.1a). 4.12.4. Using the series representation of the two-parametric Mittag-Lefﬂer function (4.1.1) prove the following recurrence relations 4.12.4.1.

2 z z E1;1.z/ C E1;1.z/ D 2E2;1.z / ” e C e D 2 cosh.z/; 2 z z E1;1.z/ E1;1.z/ D 2z E2;2.z / ” e e D 2 sinh.z/;

or in more general form: 4.12.4.2.

2 E˛;ˇ.z/ C E˛;ˇ.z/ D 2E2˛;ˇ.z /; 2 E˛;ˇ.z/ E˛;ˇ.z/ D 2z E2˛;˛Cˇ.z / I

4.12.4.3. ez 1 z E .z/ D 1;3 z2 4.12 Exercises 93

or in more general form (for any m 2 N): 4.12.4.4. ( ) 1 mX2 zk E .z/ D ez : 1;m zm1 kŠ kD0

4.12.5. With ˛>0show that d t ˛1 E .t ˛/ D E .t ˛/: (2par 1par) ˛;˛ dt ˛ 4.12.6. Prove the following differential relations for the two-parametric Mittag- Lefﬂer function ([GupDeb07]) 4.12.6.1.

0 2 00 3E1;4.z/ C 5zE1;4 C z E1;4 D E1;2 E1;3:

4.12.6.2.

0 2 00 n.n C 2/E˛;nC3.z/ C z˛Œ2n C ˛ C 2E˛;nC3 C z E˛;nC3

D E˛;nC1 E˛;nC2;

which hold for any ˛>0and any n D 1;2;:::. 4.12.7. Prove the following Laplace transform pair for the auxiliary functions of Mittag-Lefﬂer type deﬁned below

s˛ˇ sˇ e .tI / WD t ˇ1 E .t˛/ D : (LT E2) ˛;ˇ ˛;ˇ s˛ C 1 C s˛

4.12.8. Evaluate the following integrals ([HaMaSa11]): 4.12.8.1.

Zx E .t ˛/ ˛ dt .x t/1ˇ 0

for Re ˛>0,Reˇ>0. 4.12.8.2.

Z1 st m˛Cˇ1 .m/ ˛ e t E˛;ˇ .˙at /dt 0 94 4 The Two-Parametric Mittag-Lefﬂer Function

for Re s>0,Re˛>0,Reˇ>0,where

dm E.m/.z/ D E .z/: ˛;ˇ dzm ˛;ˇ

Answers. 4.12.8.1.

ˇ ˛ .ˇ/x E˛;ˇC1.x /:

4.12.8.2.

mŠs˛ˇ : .s˛ a/mC1

4.12.9. Prove the following formulas for half-integer values of parameters [Han-et-al09]: 4.12.9.1.

1 x2 E1=2;1=2.˙x/ D p ˙ xe Œ1 ˙ erf.x/: x

4.12.9.2.

x2 E1=2;1.˙x/ D e Œ1 ˙ erf.x/:

4.12.9.3.

1 p Cx p E1;1=2.Cx/ D p C xe erf. x/: x

4.12.9.4.

1 p x p E1;1=2.x/ D p C i xe erf.i x/: x

4.12.9.5. p Cx erf. x/ E1;3=2.Cx/ D e p : x

4.12.9.6. p x erf.i x/ E1;3=2.x/ Die p : x 4.12 Exercises 95

4.12.10. Prove the following formulas for negative integer values of parameters [Han-et-al09]: 4.12.10.1. Â Ã 1 1 e˙x E ˙ D 1 ˙ : 1;2 x x

4.12.10.2. Â Ã 1 E C D 1 cosh x: 2;1 x2

4.12.10.3. Â Ã 1 E D 1 cos x: 2;1 x2

4.12.10.4. Â Ã 1 sinh x E C D 1 : 2;2 x2 x

4.12.10.5. Â Ã 1 sin x E D 1 : 2;2 x2 x

4.12.11. Prove the following formulas for negative semi-integer values of parameters [Han-et-al09]: 4.12.11.1. Â Ã 1 2 E ˙ Dxex Œ1 ˙ erf.x/: 1=2;1=2 x

4.12.11.2. Â Ã 1 2 E ˙ D 1 ex Œ1 ˙ erf.x/: 1=2;1 x

4.12.11.3. Â Ã 1 p p E C D xeCxerf. x/: 1;1=2 x 96 4 The Two-Parametric Mittag-Lefﬂer Function

4.12.11.4. Â Ã 1 p p E Di xexerf.i x/: 1;1=2 x

4.12.11.5. Â Ã p 1 2 Cx erf. x/ E1;3=2 C D p e p : x x x

4.12.11.6. Â Ã p 1 2 x erf.i x/ E1;3=2 D p C ie p : x x x

4.12.12. Prove the following formula for the Laplace transform of the derivatives of the Mittag-Lefﬂer function ([KiSrTr06, p. 50]): Â Â Ã Ã n ˛ˇ ˛nCˇ1 @ ˛ nŠs ˛ L t E˛;ˇ.t / .s/ D .js j <1/: @ .s˛ /nC1

4.12.12. Prove the following relations for the Mittag-Lefﬂer functions with positive integer values of parameters (Capelas relations) ([Cap13]).

Xm k1 m z z Em;k.z / D e ;m2 N; (4.12.12a) kD1 ! 1 mX2 zk E .z/ D ez ;m2 N: (4.12.12b) 1;m zm1 kŠ kD0 Chapter 5 Mittag-Lefﬂer Functions with Three Parameters

5.1 The Prabhakar (Three-Parametric Mittag-Lefﬂer) Function

5.1.1 Deﬁnition and Basic Properties

The Prabhakar generalized Mittag-Lefﬂer function [Pra71]isdeﬁnedas

X1 ./ E .z/ WD n zn ; Re.˛/>0;Re.ˇ/>0;>0; (5.1.1) ˛;ˇ nŠ .˛n C ˇ/ nD0 where ./n D . C 1/:::. C n 1/ (see formula (A.1.17)). For D 1 we recover the two-parametric Mittag-Lefﬂer function

X1 zn E .z/ WD ; (5.1.2) ˛;ˇ .˛nC ˇ/ nD0 and for D ˇ D 1 we recover the classical Mittag-Lefﬂer function

X1 zn E .z/ WD : (5.1.3) ˛ .˛nC 1/ nD0

Let ˛; ˇ > 0. Then termwise Laplace transformation of series (5.1.1) yields

Z 1 Á 1 X . C n/ a n est t ˇ1 E .at˛/dt D sˇ : (5.1.4) ˛;ˇ ./ s 0 nD0

© Springer-Verlag Berlin Heidelberg 2014 97 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__5 98 5 Mittag-Leffler Functions with Three Parameters

On the other hand (binomial series!)

X1 .1 / X1 . C n/ .1 C z/ D zn D .1/n zn : (5.1.5) .1 n/nŠ ./nŠ nD0 nD0

Comparison of (5.1.4)and(5.1.5) yields the Laplace transform pair

sˇ t ˇ1 E .at˛/ : (5.1.6) ˛;ˇ .1 as˛/

Equation (5.1.6) holds (by analytic continuation) for Re ˛>0;Re ˇ>0. In particular we get the known Laplace transform pairs

s˛ˇ t ˇ1 E .at˛/ ; (5.1.7) ˛;ˇ s˛ a s˛1 E .at˛/ : (5.1.8) ˛ s˛ a

Note that the pre-factor t ˇ1 is essential for the above Laplace transform pairs. From the above Laplace transform pair one can obtain the complete monotonicity (CM) of the function

1 E .t ˛/ : (5.1.9) ˛;1 s.1C s˛/

The proof is based on the following theorem (see [GrLoSt90, Thm. 2.6]): if the real-valued function F.x/; x > 0I limx!C1 F.x/ D 0; possesses an analytic continuation in C n R and satisﬁes the inequalities Im zF.z/ 0 for Im z >0and Im F.x/ 0 for 00(see, ˇ1 ˛ e.g. [CapMai11]) for the function e˛;ˇ.tI / D t E˛;ˇ .t /:

0<˛;ˇÄ 1; t ˇ1 E .t˛/ CM iff (5.1.10) ˛;ˇ 0<Ä ˇ=˛:

In the same manner we recover the known result

˛ E˛.t / CM if 0<˛Ä 1: (5.1.11) 5.1 The Prabhakar (Three-Parametric Mittag-Lefﬂer) Function 99

Cases of Reducibility

Here we present some formulas connecting the values of three-parametric (Prabhakar) Mittag-Lefﬂer functions with different values of parameters (see, e.g. [MatHau08]). (i) If ˛; ˇ; 2 C are such that Re ˛>0,Reˇ>0,Re.ˇ ˛/ > 0,then

1 zE˛;ˇ D E˛;ˇ˛ E˛;ˇ˛ : (5.1.12)

(ii) If ˛; ˇ 2 C are such that Re ˛>0,Reˇ>0, .˛ ˇ/ 62 N0,then

1 zE1 D E : (5.1.13) ˛;ˇ ˛;ˇ˛ .ˇ ˛/

(iii) If ˛; ˇ 2 C are such that Re ˛>0,Reˇ>1,then

2 ˛E˛;ˇ D E˛;ˇ1 .1 C ˛ ˇ/E˛;ˇ: (5.1.14)

Differentiation of the Three-Parametric Mittag-Lefﬂer Function

If ˛; ˇ; ; z; w 2 C, then for any n D 1;2;:::,andanyˇ; Re ˇ>n, the following formula holds: Â Ã h i d n zˇ1E .wz˛/ D zˇn1E .wz˛ /: (5.1.15) dz ˛;ˇ ˛;ˇn

In particular, for any n D 1;2;:::,andanyˇ; Re ˇ>n, Â Ã n d ˇ1 ˛ ˇn1 ˛ z E .wz / D z E .wz / (5.1.16) dz ˛;ˇ ˛;ˇ n and for any n D 1;2;:::,andanyˇ; Re ˇ>n, Â Ã d n .ˇ/ zˇ1.;ˇI wz/ D zˇn1.;ˇ nI wz/; (5.1.17) dz .ˇ n/ where

.;ˇI z/ WD 1F1.; ˇI z/ D .ˇ/E1;ˇ: (5.1.18)

C To prove formula (5.1.15) one can use term-by-term differentiation of the power series representation of the three-parametric Mittag-Lefﬂer function. Thus we get 100 5 Mittag-Lefﬂer Functions with Three Parameters

Â Ã h i 1 Â Ã Ä d n X .˛/ d n wkzˇkC1 zˇ1E .wz˛ / D k dz ˛;ˇ .˛kC ˇ/ dz kŠ kD0

ˇn1 ˛ D z E˛;ˇn.wz /; Re ˇ>n; and the result follows. B

Integrals of the Three-Parametric Mittag-Lefﬂer Function

By integration of series (5.1.1) we get the following. If ˛; ˇ; ; z; w 2 C,Re˛>0,Reˇ>0,Re>0,then

Zz ˇ1 ˛ ˇ ˛ t E˛;ˇ.wt /dt D z E˛;ˇC1.wz /: (5.1.19) 0

In particular,

Zz ˇ1 ˛ ˇ ˛ t E˛;ˇ.wt /dt D z E˛;ˇC1.wz /; (5.1.20) 0 and

Zz 1 t ˇ1.;ˇI wz/dt D zˇ.;ˇ C 1I wz/: (5.1.21) ˇ 0

5.1.2 Integral Representations and Asymptotics

As for any function of the Mittag-Lefﬂer type, the three parametric Mittag-Lefﬂer function can be represented via the Mellin–Barnes integral. Let ˛ 2 RC, ˇ; 2 C, ˇ 6D 0,Re>0. Then we have the representation Z 1 1 .s/. s/ s E˛;ˇ.z/ D .z/ ds; (5.1.22) ./2i L .ˇ ˛s/ where jarg zj <, the contour of integration begins at c i1, ends at c C i1, 0

C The integral in the RHS of (5.1.22) is equal to the sum of residues at the poles s D 0; 1; 2;:::. Hence Z .s/. s/ .z/s ds L .ˇ ˛s/ Ä X1 .s C k/.s/. s/.z/s D lim s!k .ˇ ˛s/ kD0 X1 .1/k . C k/ D .z/k kŠ .ˇC ˛k/ kD0 X1 ./ zk D ./ k D ./E .z/; .ˇC ˛k/ kŠ ˛;ˇ kD0 and thus (5.1.22) follows. B If ˇ is a sufﬁciently large real number, one can use Strirling’s formula, valid for any ﬁxed a p .z C a/ 2zzCa1=2ez; as jzj!1; (5.1.23) in order to get the following asymptotic formula (a > 0; ˛ > 0; ˇ > 0; > 0) p X1 .ˇ/ akxk 2˛˛1=2e˛ .˛/E .a.˛x/ / k p ˛;ˇ kŠ ˛1=2Ck ˛ kD0 2˛ e (5.1.24) 1 X Á Ák .ˇ/k x 1 D a D ; as x !C1: kŠ ˛ x ˇ kD0 1 C a ˛

As in the case of the Mittag-Lefﬂer function with two parameters, the asymptotic behaviour of the three parametric function critically depends on the values of the parameters ˛; ˇ; and cannot easily be described. In principle, an asymptotic expansion of the Prabhakar function can be found from its representation via a generalized Wright function or H-function (see Sect. 5.1.5 below) by using an approach of Braaksma [Bra62](cf.[KiSrTr06]). To the best of the authors’ knowledge, the asymptotic behaviour in different domains of the complex plane (similar to that of Proposition 3.6, and of Theorems 4.3 and 4.4) for the Prabhakar function has not yet been described in an explicit form.

5.1.3 Integral Transforms of the Prabhakar Function

In a similar way as for the two-parametric Mittag-Leffler function one can calculate the Laplace transform of the Prabhakar function (the three-parametric Mittag-Leffler function) 102 5 Mittag-Leffler Functions with Three Parameters

Z1 Ä ˇ Á ˇ st 1 .; 1/; .1; 1/ ˇ1 L E .t/ .s/ D e E .t/dt D 21 ˇ .Res>0/: ˛;ˇ ˛;ˇ s .ˇ; ˛/ s 0 (5.1.25)

The most useful variant of the Laplace transform of the Prabhakar function is the following formula (see, e.g., [KiSrTr06, p. 47]): Á s˛ˇ L t ˇ1E .t ˛/ .s/ D ; (5.1.26) ˛;ˇ .s˛ / which is valid for all Res>0,Reˇ>0, 2 C such that js˛j <1. Applying the Mellin inversion formula to (5.1.22) we obtain the Mellin transform of three-parametric Mittag-Lefﬂer function Á Z 1 .s/. s/ M s1 s E˛;ˇ.wt/ .s/ D t E˛;ˇ.wt/dt D w : 0 ./.ˇ ˛s/ (5.1.27)

Further we take into account the integral relation for the Whittaker function

Z 1 1 t=2 .1=2C C / .1=2 C / t e W;.t/dt D ; Re . ˙ / > 1=2; 0 .1 C / (5.1.28) where

x=2 c=2 W;.x/ D e x U.a;c;x/; a D 1=2 C ; c D 2 C 1; and U.a;c;x/ is the Tricomi function (or conﬂuent hypergeometric function,or degenerate hypergeometric function) deﬁned, e.g., by the integral Z 1 1 t a1 U.a;c;x/ D extdt; b D 1 C a c: .a/ 0 t C b

By using (5.1.28) we obtain the so-called Whittaker integral transform of the three- parametric Mittag-Leffler function Z Ä ˇ 1 ˇ 1 1 pt ı p w ˇ .; 1/; . ˙ C ; ı/ t 2 W .pt/E .wt / t D 2 ; e ; ˛;ˇ d 3 2 ı ˇ 0 ./ p .ˇ; ˛/; .1 C ; ı/ (5.1.29) 5.1 The Prabhakar (Three-Parametric Mittag-Leffler) Function 103 ˇ ˇ ˇ ˇ ˇ w ˇ where 32 is the generalized Wright function, and jRe j1=2,Re >0, ˇ ˇ <1. pı As a particular case of this formula we can obtain the Laplace transform of the three- parametric Mittag-Leffler function. Indeed, since

t=2 W˙1=2;0.t/ D e ;

the Laplace transform of E˛;ˇ can be represented by the relation Z Ä ˇ 1 p w ˇ .; 1/; . ; ı/ t 1 ptE .wt ı / t D ˇ ; (5.1.30) e ˛;ˇ d 2 1 ı ˇ 0 ./ p .ˇ; ˛/

1 where Re ˛>0,Reˇ>0,Re >0,Rep>0, p>jwj Re ˛ . In particular, for D ˇ and ı D ˛ this result coincides with that obtained in [Pra71, Eq. 2.5] Z 1 ˇ1 pt ˛ ˇ ˛ t e E˛;ˇ.wt /dt D p .1 wp / ; (5.1.31) 0

1 where Re ˛>0,Reˇ>0,Rep>0, p>jwj Re ˛ .

5.1.4 Fractional Integrals and Derivatives of the Prabhakar Function

Theorem 5.1. Let ; ˛; ˇ > 0, a 2 R. Then the following formulas for the Riemann–Liouville and the Liouville fractional integration and differentiation of the Prabhakar function are valid: (i) n h io ˇ1 ˛ ˇC1 ˛ I0C t E˛;ˇ.at / .x/ D x E˛;ˇC.ax /; (5.1.32)

where

Zx 1 '.t/ I '.t/ .x/ D dt; Re >0; 0C ./ .x t/1 0

is the left-sided Riemann–Liouville fractional integral (see, e.g., [SaKiMa93, p. 33]). (ii) n h io ˇ ˛ ˇ ˛ I t E˛;ˇ.at / .x/ D x E˛;ˇC.ax /; (5.1.33) 104 5 Mittag-Lefﬂer Functions with Three Parameters

where

Z1 1 '.t/ I '.t/ .x/ D dt; Re >0; ./ .t x/1 x

is the right-sided Liouville fractional integral (see, e.g., [SaKiMa93, p. 94]). (iii) n h io ˇ1 ˛ ˇ1 ˛ D0C t E˛;ˇ.at / .x/ D x E˛;ˇ.ax /; (5.1.34)

where

Â Ã Zx 1 d n '.t/ D '.t/ .x/ D dt; Re >0;nD Œ C 1; 0C .n / dx .x t/nC1 0

is the left-sided Riemann–Liouville fractional derivative (see, e.g., [SaKiMa93, p. 37]). (iv) If ˇ Cfg >1,then n h io ˇ ˛ ˇ ˛ D t E˛;ˇ.at / .x/ D x E˛;ˇ.ax /; (5.1.35)

where

Â Ã Z1 n n .1/ d '.t/ D'.t/ .x/ D dt; Re >0;nD ŒC1; .n / dx .t x/nC1 x

is the right-sided Liouville fractional derivative (see, e.g., [SaKiMa93,p.95]). C The proof follows from the deﬁnitions of the corresponding fractional integrals and derivatives. Thus, to prove relation (5.1.32) we put

n h io Zx 1 1 X ./ ant n˛Cˇ1 K Á I t ˇ1E .at˛/ .x/ D .x t/1 n dt: 0C ˛;ˇ ./ .n˛C ˇ/nŠ 0 nD0

Since the series converges for any t>0, interchanging the order of integration and summation and evaluating the inner integral by means of the Beta function yields

X1 ./ .ax˛/n K Á xCˇ1 n D xˇC1E .ax˛/: .C n˛ C ˇ/nŠ ˛;ˇC nD0

The proof is complete. B 5.1 The Prabhakar (Three-Parametric Mittag-Lefﬂer) Function 105

5.1.5 Relations to the Wright Function, H -Function and Other Special Functions

Due to the integral representation (5.1.22) the three-parametric Mittag-Lefﬂer function can be considered as a special case of the H-function (see, e.g., [KilSai04]) Ä ˇ ˇ 1 1;1 ˇ .1 ; 1/ E .z/ D H z ˇ (5.1.36) ˛;ˇ ./ 1;2 .0; 1/; .1 ˇ; ˛/ as well as a special case of the Wright generalized hypergeometric function pq (see, e.g. Slater [Sla66], Mathai and Saxena [MatSax73]) Ä ˇ ˇ 1 ˇ .; 1/ E .z/ D 11 z ˇ : (5.1.37) ˛;ˇ ./ .ˇ; ˛/

In particular, when ˛ D 1 the Prabhakar function E1;ˇ.z/ coincides with the Kummer conﬂuent hypergeometric function ˚.I ˇI z/, apart from the constant factor . .ˇ//1

1 E .z/ D ˚.I ˇI z/; (5.1.38) ˛;ˇ .ˇ/

N and when ˛ D m 2 is a positive integer then Em;ˇ.z/ is related to the generalized hypergeometric function Â Ã 1 ˇ ˇ C 1 ˇ C m 1 z E .z/ D F I ; ;:::; I : (5.1.39) m;ˇ .ˇ/1 m m m m mm

Here we present other special functions, which are connected with the three- parametric Mittag-Lefﬂer function (5.1.1)(see[KiSaSa04]). There holds the relation

.kC 1/ Ek .z/ D Z.ˇ/.zI m/ .k; m 2 NI ˇ 2 C/; (5.1.40) m;ˇC1 .kmC ˇ C 1/ k

.ˇ/ m where Zk .zI m/ is a polynomial of degree k in z studied in [Kon67]. In particular,

.ˇ/ ˇ N C Zk .zI 1/ D Lk ;.k2 I ˇ 2 /; (5.1.41)

ˇ where Lk is the Laguerre polynomial (see, e.g., [Bat-2, Sec. 10.12]), and hence

.kC 1/ Ek .z/ D Lˇ;.k2 NI ˇ 2 C/: (5.1.42) 1;ˇC1 .kC ˇ C 1/ k 106 5 Mittag-Lefﬂer Functions with Three Parameters

ˇ The Laguerre function L (see, e.g., [Bat-1, 6.9(37)]) is also a special case of the three-parametric Mittag-Lefﬂer function (5.1.1):

.C 1/ E .z/ D Lˇ;.;ˇ2 C/: (5.1.43) 1;ˇC1 .ˇC 1/

The following relation in terms of the Kummer confluent hypergeometric function .m 2 NI ˇ; 2 C/ Â Ã .2/.m1/=2 mY1 1 ˇ C k z E .z/ D ˚ ; I (5.1.44) m;ˇ .m/ˇ1=2 ..ˇ C k/=m/ m mm kD0 is deduced from the definition of the three-parametric Mittag-Leffler function and from the multiplication formula of Gauss and Legendre for the Gamma function [Bat-1, 1.2(11)] (see Sect. A.1.4).

5.2 Generalized (Kilbas–Saigo) Mittag-Lefﬂer Type Functions

5.2.1 Deﬁnition and Basic Properties

Consider a function deﬁned for real ˛; m 2 R and complex l 2 C by Kilbas and Saigo in the following form

X1 k E˛;m;l .z/ D ckz .z 2 C/; (5.2.1) kD0 where

kY1 .˛Œjm C l C 1/ c D 1; c D .k D 1; 2; /: (5.2.2) 0 k .˛Œjm C l C 1 C 1/ j D0

In (5.2.2) an empty product is assumed to be equal to one,1 ˛; m are real numbers and l 2 C such that

˛ > 0; m > 0; ˛.jm C l/ C 1 ¤1; 2;3; .j D 0; 1; 2; /: (5.2.3)

The function (5.2.1) was introduced in [KilSai95b](seealso[KilSai95a]).

1In what follows we will call this assumption the “Empty Product Convention”. 5.2 Generalized (Kilbas–Saigo) Mittag-Lefﬂer Type Functions 107

In particular, if m D 1, the conditions in (5.2.3) take the form

˛>0;˛.jC l/C 1 ¤1; 2;3; .j D 0; 1; 2; / (5.2.4) and (5.2.1) is reduced to the Mittag-Lefﬂer type function given in Chap. 4,formula (4.1.1):

E˛;1;l .z/ D .˛l C 1/E˛;˛lC1.z/: (5.2.5)

Therefore we call E˛;m;l .z/ a generalized Mittag-Lefﬂer type function (or three- parametric Mittag-Lefﬂer function, or Kilbas–Saigo function). As we shall see later, this function is used to solve in closed form new classes of integral and differential equations of fractional order. When ˛ D n 2 N Df1; 2; g, En;m;l .z/ takes the form Â Ã X1 kY1 1 E .z/ D 1 C zk; (5.2.6) n;m;l Œn.jm C l/ C 1 Œn.jm C l/C n kD1 j D0 where n; m and l are real numbers such that

n 2 N;m>0;n.jm C l/ ¤1; 2; ; n.j D 0; 1; 2; /: (5.2.7)

5.2.2 The Order and Type of the Entire Function E˛;m;l .z/

In this subsection we give a few characteristics of E˛;m;l .z/. First of all we show that the generalized Mittag-Lefﬂer type function (5.2.1) is an entire function. Lemma 5.2. If ˛; m and l are real numbers such that the conditions (5.2.3)are satisﬁed, then E˛;m;l .z/ is an entire function of the variable z. Proof. According to (5.2.2) and the relation (Appendix A,formula(A.1.27)) with z D ˛nm, a D ˛l C ˛ C 1 and b D ˛l C 1, we have the asymptotic estimate

c Œ˛.km C l C 1/ C 1 k D .˛mk/˛ !1 .k !1/: ckC1 Œ˛.km C l/ C 1

Therefore the radius of convergence R of the series (5.2.1) is equal to C1,i.e. E˛;m;l .z/ is an entire function.

Corollary 5.3. For ˛>0;m>0and Re l>1=˛ the function E˛;m;l .z/ is an entire function of z. 108 5 Mittag-Lefﬂer Functions with Three Parameters

Corollary 5.4. If ˛ D n 2 N;m>0and l are real numbers such that the conditions (5.2.7) are satisfied, then the generalized Mittag-Leffler type function En;m;l .z/ given by (5.2.6) is an entire function of z. The order and type of the generalized Mittag-Leffler type function (5.2.1)isgiven by the following statement. Theorem 5.5. If ˛; m and l are real numbers such that the conditions (5.2.3)are satisfied, then E˛;m;l .z/ is an entire function of order D 1=˛ and type D 1=m. Moreover we have the asymptotic estimate ÂÄ Ã 1 jE .z/j < exp C jzj1=˛ ; jzjr >0; (5.2.8) ˛;m;l m 0 whenever >0is sufficiently small.

G Applying (Appendix B,formula(B.2.3)) we ﬁrst ﬁnd the order of E˛;m;l .z/. According to (5.2.2)wehave

.˛l C 1/ .˛l C ˛m C 1/ .˛l C ˛mŒk 1 C 1/ c D : k .˛l C ˛ C 1/ .˛l C ˛ C ˛m C 1/ .˛l C ˛ C ˛mŒk 1 C 1/

Let

zn D ˛l C ˛mn C 1.n2 N Df1; 2; g/: (5.2.9)

Using (5.2.8) with z D zn;aD 0 and b D ˛ we obtain that for any d>0there exists an n0 2 N such that ˇ ˇ ˇ ˇ ˛ ˇ.zn C ˛/ˇ ˛ .1 d/zn Ä ˇ ˇ Ä .1 C d/zn 8n>n0: (5.2.10) .zn/

Therefore for k>n0 we have Â Ã ˇ ˇ Xk1 ˇ ˇ 1 ˇ.zn C ˛/ˇ log D log ˇ ˇ jc j .z / k nD0 n ˇ ˇ ˇ ˇ Xn0 ˇ ˇ Xk1 ˇ ˇ ˇ.zn C ˛/ˇ ˇ.zn C ˛/ˇ D log ˇ ˇ C log ˇ ˇ .zn/ .zn/ nD0 nDn0C1 Xk1 Xk1 ˛ Ä d1 C log.1 C d/C log.zn/ nDn0C1 nDn0C1

D d1 C .k n0 1/ log.1 C d/ Ä Â Ã Xk1 ˛l C 1 C˛ log.n/ C log.˛m/ C log 1 C n˛m nDn0C1 5.2 Generalized (Kilbas–Saigo) Mittag-Lefﬂer Type Functions 109 and hence Â Ã 1 log Ä d3 C kd4 C ˛k log.k/; (5.2.11) jckj where d3 and d4 are certain positive constants. Similarly Â Ã 1 log d4 C .k n0 1/ log.1 d/ jckj Ä Â Ã Xk1 ˛l C 1 C˛ log.n/ C log.˛m/ C log 1 C nlm nDn0C1 and Â Ã 1 log d5 C kd6 C ˛k log.k/ (5.2.12) jckj for some real constants d5, d6. It follows from (5.2.11)and(5.2.12) that the usual limit

k log.k/ 1 lim D (5.2.13) k!1 log.1=jckj/ ˛ exists and hence in accordance with (Appendix B,formula(B.2.3)) the order of E˛;m;l .z/ is given by

1 D : (5.2.14) ˛ Next we use (Appendix B,formula(B.2.4)) to ﬁnd the type of the function E˛;m;l .z/. Applying (5.2.10), (5.2.9)and(5.2.2)wehave ˇ ˇ Â Ã Yn0 ˇ ˇ kn01 kY1 ˇ.zn C ˛/ˇ 1 ˛ ˇ ˇ zn Äjckj .zn/ 1 C d nD0 nDn0C1 ˇ ˇ Â Ã Yn0 ˇ ˇ kn01 kY1 ˇ.zn C ˛/ˇ 1 ˛ Ä ˇ ˇ zn: (5.2.15) .zn/ 1 d nD0 nDn0C1

Using this formula and the asymptotic relation

k1 Â Ã Â Ã Â Ã Á Y 1 ˛ 1 ˛ 1 ˛k e ˛k .2k/˛ .k !1/ ˛nm kŠ ˛m ˛km nDn0 110 5 Mittag-Lefﬂer Functions with Three Parameters from (Appendix B,formula(B.2.4)), we obtain ˛ D m˛ and hence the type of E˛;m;l .z/ is given by

1 D : (5.2.16) m The asymptotic estimate (5.2.8) follows from (5.2.13)–(5.2.14) and the definitions of the order and type of an entire function given in Appendix B. This completes the proof of the theorem. F Corollary 5.6. If ˛ D n 2 N;m>0and l are real numbers such that the conditions (5.2.7) are satisfied, then the generalized Mittag-Leffler type function En;m;l .z/ given by (5.2.6) is an entire function of z with order D 1=n and type D 1=m.

Corollary 5.7. The classical Mittag-Lefﬂer function E˛.z/ and the two-parametric Mittag-Lefﬂer function E˛;ˇ.z/, given respectively in (Chap. 3,formula(3.1.1)) and in (Chap. 4,formula(4.1.1)), have the same order and type:

1 D ; D 1: (5.2.17) ˛ Remark 5.8. The assertions of Corollary 5.7 coincide with those in (Chap. 3, Proposition 3.1) and (Chap. 4, Sect. 4.1). Remark 5.9. Theorem 5.5 shows that the three-parametric Mittag-Lefﬂer type function (5.2.1) has the same order as the Mittag-Lefﬂer functions (Chap. 3,formula (3.1.1)) and (Chap. 4,formula(4.1.1)). But the type of E˛;m;l .z/ depends on m.

5.2.3 Recurrence Relations for E˛;m;l .z/

In this subsection we give recurrence relations for E˛;m;l .z/. Theorem 5.10. Let ˛; m and l be real numbers such that the condition (5.2.3)is satisﬁed and let n 2 N. Then the following recurrence relation holds

nY1 Œ˛.jm C l C 1/ C 1 zn ŒE .z/ 1 D ˛;m;lCnm Œ˛.jm C l/ C 1 j D0 2 0 1 3 Xn kY1 Œ˛.jm C l/C 1 4E .z/ 1 @ A zk 5 : (5.2.18) ˛;m;l Œ˛.jm C l C 1/ C 1 kD1 j D0 5.2 Generalized (Kilbas–Saigo) Mittag-Lefﬂer Type Functions 111

G By (5.2.1)wehave 0 1 X1 kY1 Œ˛.jm C nm C l/ C 1 E .z/ D 1 C @ A zk: ˛;m;lCnm Œ˛.jm C nm C l C 1/ C 1 kD1 j D0

Changing the summation indices s D j C n and p D k C n, we obtain ! X1 nCYk1 Œ˛.sm C l/ C 1 E .z/ D 1 C zk ˛;m;lCnm Œ˛.sm C l C 1/ C 1 kD1 sDn ! p1 X1 Y Œ˛.sm C l/C 1 D 1 C zpn Œ˛.sm C l C 1/ C 1 pDnC1 sDn ! p1 nY1 Œ˛.sm C l C 1/ C 1 X1 Y Œ˛.sm C l/ C 1 D 1 C zpn Œ˛.sm C l/ C 1 Œ˛.sm C l C 1/ C 1 sD0 pDnC1 sD0

1 nY1 Œ˛.sm C l C 1/ C 1 D 1 C zn Œ˛.sm C l/ C 1 sD0 2 ! p1 X1 Y Œ˛.sm C l/ C 1 41 C zp Œ˛.sm C l C 1/ C 1 pD1 sD0 ! 3 p1 Xn Y Œ˛.sm C l/ C 1 zp 15 Œ˛.sm C l C 1/ C 1 pD1 sD0

1 nY1 Œ˛.sm C l/ C 1 D 1 C zn Œ˛.sm C l C 1/ C 1 sD0 2 ! 3 p1 Xn Y Œ˛.sm C l/ C 1 4E .z/ zp 15 ˛;m;l Œ˛.sm C l C 1/ C 1 pD1 sD0 and (5.2.18)isproved.F Corollary 5.11. If the conditions of Theorem 5.10 are satisﬁed, then

.˛l C ˛ C 1/ zE .z/ D ŒE .z/ 1 (5.2.19) ˛;m;lCm .˛l C 1/ ˛;m;l 112 5 Mittag-Lefﬂer Functions with Three Parameters and

nY1 Œ˛.jm C l C 1/ C 1 znE .z/ D ˛;m;lCnm Œ˛.jm C l/ C 1 j D0 2 0 1 3 Xn1 kY1 Œ˛.jm C l/ C 1 4E .z/ 1 @ A zk5 (5.2.20) ˛;m;l Œ˛.jm C l C 1/ C 1 kD1 j D0 for n D 2;3;. The following two Corollaries show how the above properties look like in special cases. Corollary 5.12. If ˛>0;ˇ>0and n 2 N, then for the Mittag-Lefﬂer type function E˛;˛nCˇ.z/ we have the recurrence relations 1 zE .z/ D E .z/ (5.2.21) ˛;˛Cˇ ˛;ˇ .ˇ/ and

n z .˛nC ˇ/E˛;˛nCˇ.z/ 2 0 1 3 nY1 .˛j C ˛ C ˇ/ Xn1 kY1 .˛j C ˇ/ D 4.ˇ/E .z/ 1 @ A zk5 .˛j C ˇ/ ˛;ˇ .˛j C ˛ C ˇ/ j D0 kD1 j D0 (5.2.22) for n D 2;3;.

Corollary 5.13. If ˛>0and n 2 N, then the Mittag-Lefﬂer function E˛;˛C1.z/ is expressed via the Mittag-Lefﬂer function (3.1.1)by

zE˛;˛C1.z/ D E˛.z/ 1 (5.2.23) and Xn zj znE .z/ D E .z/ (5.2.24) ˛;˛nC1 ˛ .˛j C 1/ j D1 for n D 2;3;.

5.2.4 Connection of En;m;l .z/ with Functions of Hypergeometric Type

As we have mentioned in Sect. 5.2.1, E˛;m;l .z/ is a generalization of the Mittag- Lefﬂer type function E˛;ˇ.z/ in Chap. 4,formula(4.1.1), in particular, of the 5.2 Generalized (Kilbas–Saigo) Mittag-Lefﬂer Type Functions 113

Mittag-Lefﬂer function E˛.z/ in Chap. 3,formula(3.1.1). When ˛ D n 2 N, En;m;l .z/ in (5.2.6) becomes a function of hypergeometric type.S Such a function pFq a1;a2: ;apI b1;b2: ;bqI z for p 2 N0 D N f0g, q 2 N0, a1;a2: ;ap 2 C, b1;b2; ;bq 2 C and z 2 C; jzj <1;is deﬁned by the hypergeometric series [Bat-1]

X1 k .a1/k.a2/k .ap/k z pFq a1;a2: ;ap I b1;b2: ;bqI z D ; (5.2.25) .b1/k.b2/k .aq /k kŠ kD0 where ./k is the Pochhammer symbol deﬁned in (Appendix A,formula(A.1.17)).

Theorem 5.14. Let the conditions (5.2.7) be satisﬁed. Then the function En;m;l .z/ is given via the hypergeometric function by Â Ã nl C 1 nl C 2 nl C n z E .z/ D F 1I ; I (5.2.26) n;m;l 1 n nm nm nm .nm/n and is an entire function of z with order D 1=n and type D 1=m. G According to (5.2.1), (5.2.25)and(5.2.26)wehave 0 1 1 X kY1 .nŒjm C l C 1/ E .z/ D 1 C @ A zk n;m;l .nŒjm C l C 1 C 1/ kD1 j D0 1 X kŠ zk D 1 C .nl C 1/ .nl C 1 C .k 1/nm/ .nl C n/ .nl C n C .k 1/nm/ kŠ kD1 1 X .1/ 1 zk D 1 C k nk .Œnl C 1/=Œnm/k .Œnl C n=Œnm/k .nm/ kŠ kD1 Â Ã nl C 1 nl C 2 nl C n z D F 1I ; I 1 n nm nm nm .nm/n and (5.2.26) is proved. The last assertion follows from Theorem 5.5. This completes the proof. F

Corollary 5.15. If n 2 N and ˇ>0, then the Mittag-Lefﬂer function En;ˇ.z/ is given by

En;1;.ˇ1/=n.z/ D .ˇ/En;ˇ.z/ Â Ã ˇ ˇ C 1 ˇ C n 1 z D F 1I ; I (5.2.27) 1 n n n n .n/n and is an entire function of order D 1=n and type D 1. Corollary 5.16. If

m>0;l2 R; jm C l ¤1; 2;3; .j D 0; 1; 2; /; (5.2.28) 114 5 Mittag-Lefﬂer Functions with Three Parameters then E1;m;l .z/ is given by Â Ã Â Ã Á l C 1 z l C 1 z E .z/ D F 1I I D E (5.2.29) 1;m;l 1 1 m m m 1;.lC1/=m m and is an entire function of z with order D 1 and type D 1=m.

Corollary 5.17. If l ¤1; 2;3; ,thenE1;1;l .z/ is given by

E1;1;l .z/ D 1F1.1I l C 1I z/ D .l C 1/E1;lC1.z/ (5.2.30) and is an entire function of z with order D 1 and type D 1. In particular, if l 2 N0, " # lŠ Xl1 zk E .z/ D lŠE .z/ D ez (5.2.31) 1;1;l 1;lC1 zl kŠ kD0 and

z E1;1;0.z/ D E1.z/ D e : (5.2.32)

5.2.5 Differentiation Properties of En;m;l .z/

In this subsection we give two differentiation formulas for the three-parametric Mittag-Leffler type function (5.2.1). The first of these is given by the following statement. Theorem 5.18. Let n 2 N;m>0and l 2 R be such that the conditions in (5.2.7) are satisfied and let 2 C. Then the following differentiation formula Â Ã d n Yn zn.lmC1/E .znm/ D Œn.l m/ C jzn.lm/ C znlE .znm/ dz n;m;l n;m;l j D1 (5.2.33) holds. In particular, if

n.l m/ Dj for some j D 1; 2; ;n; (5.2.34) then Â Ã d n zn.lmC1/E .znm/ D znlE .znm/: (5.2.35) dz n;m;l n;m;l 5.2 Generalized (Kilbas–Saigo) Mittag-Lefﬂer Type Functions 115

G If D 0,thenby(5.2.1) En;m;l .z/ D 1, and formula (5.2.33) takes the well- known form Â Ã d n Yn zn.lmC1/ D Œn.l m/ C jzn.lm/: dz j D1

If ¤ 0,thenby(5.2.1)wehave Â Ã d n zn.lmC1/E .znm/ dz n;m;l " # Â Ã 1 d n X D zn.lmC1/ C c kzn.lmC1/Cnmk dz k kD1 D Œn.l m C 1/Œn.l m C 1/ 1 Œn.l m C 1/ n C 1zn.lm/ 0 1 X1 kY1 .nŒjm C l C 1/ Œnf.k 1/m C l C 1gC1 C @ A kzn.lm/Cnmk .nŒjm C l C 1 C 1/ Œn.f.k 1/m C lgC1 kD1 j D0 Yn D Œn.l m/ C kzn.lm/ C znl kD1 0 1 X1 kY2 .nŒjm C l C 1/ C @ A kzn.lm/Cnmk: .nŒjm C l C 1 C 1/ kD2 j D0

By index substitution s D k 1 we obtain Â Ã d n zn.lmC1/E .znm/ dz n;m;l Yn D Œn.l m/ C jzn.lm/ C znl kD1 0 1 X1 Ys1 .nŒjm C l C 1/ C @ A sC1znlCnms .nŒjm C l C 1 C 1/ sD1 j D0 Yn D Œn.l m/ C kzn.lm/ kD1 2 0 1 3 X1 Ys1 .nŒjm C l C 1/ Cznl 41 C @ A .znm/s5 .nŒjm C l C 1 C 1/ sD1 j D0 116 5 Mittag-Lefﬂer Functions with Three Parameters and (5.2.33) is proved in accordance with (5.2.1); (5.2.35) follows from (5.2.33). The theorem is proved. F Corollary 5.19. If ˛ D n D 1; 2; , ˇ>0and 2 C, then for the Mittag-Lefﬂer n type function En;ˇ.az / in (Chap. 4,formula(4.1.1)) we have Â Ã d n 1 zˇ1E .zn/ D zˇn1 C zˇ1E .zn/: (5.2.36) dz n;ˇ .ˇ n/ n;ˇ

Special cases of the above property have the form. Corollary 5.20. If ˛ D n D 1; 2; , ˇ D k 2 N .1 Ä k Ä n/ and 2 C, then for n the Mittag-Lefﬂer type function En;k.z / we have Â Ã d n zk1E .zn/ D zk1E .zn/: (5.2.37) dz n;k n;k

n In particular, when k D 1, for the Mittag-Lefﬂer function En.z / in (Chap. 3, formula (3.1.1)) we have Â Ã d n ŒE .zn/ D E .zn/: (5.2.38) dz n n

Remark 5.21. By (Chap. 4,formula(4.1.1)), the relation (5.2.36) can be represented in the form Â Ã n d ˇ1 n ˇn1 n z E .z / D z E .z /; (5.2.39) dz n;ˇ n;ˇ n which coincides with (Chap. 4,formula(4.3.1)). Remark 5.22. When a D 1, the relation (5.2.38) coincides with (Chap. 3,formula (3.1.1)). Another differentiation relation for the generalized Mittag-Leffler type function (5.2.1) is given by the following: Theorem 5.23. Let ˛ D n 2 N;m>0and l 2 R be such that the conditions nm in (5.2.7) are satisfied and let 2 C. Then for En;m;l .az / the differentiation formula Â Ã d n zn.ml/1E .znm/ dz n;m;l Yn n.ml1/1 n n.lC1/1 nm D Œn.m l/ jz C .1/ z En;m;l .z / (5.2.40) j D1 holds. In particular, if the conditions 5.2 Generalized (Kilbas–Saigo) Mittag-Leffler Type Functions 117

n.l m/ D j for some j D 1; 2; ;n; (5.2.41) are satisﬁed, then Â Ã d n zn.ml/1E .znm/ D .1/nzn.lC1/1E .znm/: (5.2.42) dz n;m;l n;m;l

G If D 0,thenby(5.2.1) En;m;l .z/ D 1, and formula (5.2.40) takes the well- known form Â Ã d n Yn zn.ml/1 D Œn.m l/ jzn.ml1/1: dz j D1

When ¤ 0, according to (5.2.1)wehave Â Ã d n zn.ml/1/E .znm/ dz n;m;l " # Â Ã 1 d n X D zn.ml/1 1 C c kzn.ml/nmk1 dz k kD1 D Œn.m l/ 1 Œn.m l/ nzn.ml/1n 0 1 X1 kY1 .nŒjm C l C 1/ C k @ A .nŒjm C l C 1 C 1/ kD1 j D0

Œn.m l/ nmk 1 Œn.m l/ nmk nzn.ml/nmkn1 Yn D Œn.m l/ kzn.ml1/1 kD1 0 1 X1 kY1 .˛Œjm C l C 1/ C.1/n @ A .˛Œjm C l C 1 C 1/ kD1 j D0 .nŒfk 1gm C l C 1 C 1/ j zn.ml/nmkn1 .nŒfk 1gm C l C 1/ Yn D Œn.m l/ kzn.ml1/1 kD1 2 0 1 3 X1 kY2 .nŒjm C l C 1/ C.1/nazn.lC1/1 41 C k @ A kznm.k1/5 : .nŒjm C l C 1 C 1/ kD2 j D0 118 5 Mittag-Lefﬂer Functions with Three Parameters

By changing the index s D k 1 we obtain Â Ã d n zn.ml/1E .znm/ dz n;m;l Yn D Œn.m l/ kzn.ml1/1 kD1 2 0 1 3 X1 Ys1 .nŒjm C l C 1/ C.1/nzn.lC1/1 41 C @ A .znm/s5 .nŒjm C l C 1 C 1/ sD1 j D0 and (5.2.40) is proved in accordance with (5.2.1); (5.2.42) follows from (5.2.40). The theorem is proved. F Special cases of the above property have the form. Corollary 5.24. If ˛ D n D 1; 2; , ˇ>0and 2 C, then for the two- n parametric Mittag-Lefﬂer function En;ˇ.az / in (Chap. 4,formula(4.1.1)) we have Â Ã d n .1/n znˇE .zn/ D zˇ C .1/nznˇE .zn/: dz n;ˇ .ˇ n/ n;ˇ (5.2.43) Corollary 5.25. If ˛ D n D 1; 2; , ˇ D k 2 N .1 Ä k Ä n/ and 2 C, then for n the two-parametric Mittag-Lefﬂer function En;k.az / we have Â Ã d n znkE .zn/ D .1/nznkE .zn/: (5.2.44) dz n;k n;k

n In particular, when k D 1, for the Mittag-Lefﬂer function En.z / we have Â Ã d n zn1E .zn/ D .1/nzn1E .zn/: (5.2.45) dz n n

Remark 5.26. The relations (5.2.33), (5.2.40)and(5.2.35), (5.2.42) can be considered as inhomogeneous and homogeneous differential equations of order n for n.lmC1/ nm n.ml/1 nm the functions z En;m;l .z / and z En;m;l .z /, respectively. In this way explicit solutions of new classes of ordinary differential equations were obtained in [KilSai95b, SaiKil98, SaiKil00]. In particular, (5.2.36), (5.2.43)and (5.2.37), (5.2.44) are inhomogeneous and homogeneous differential equations for ˇ1 n nj n j 1 n the functions z En;ˇ.z / and z En;ˇ.z /. The function z En;ˇ.z / as an explicit solution to a differential equation was found earlier by reduction of the differential equation to the corresponding Volterra integral equation [SaKiMa93, Section 42.1]. 5.2 Generalized (Kilbas–Saigo) Mittag-Lefﬂer Type Functions 119

5.2.6 Fractional Integration of the Generalized Mittag-Lefﬂer Function

In this subsection we present applications of the Riemann–Liouville and Liouville ˛ ˛ fractional integrals .I0C'/.x/ and .I'/.x/ of order ˛>0, defined by the respective formulas [SaKiMa93, formulas (5.1) and (5.3)] Z 1 x '.t/ .I ˛ '/.x/ D t.x>0/ 0C 1˛ d (5.2.46) .˛/ 0 .x t/ and Z 1 1 '.t/ .I ˛'/.x/ D t.x>0/; 1˛ d (5.2.47) .˛/ x .t x/ to the generalized Mittag-Leffler function (5.2.1). ˛ The first statement shows the effect of I0C on E˛;m;l .z/. Theorem 5.27. Let ˛>0, m>0, l>1=˛ and 2 C.Then ˛ ˛l ˛m ˛.lmC1/ ˛m I0C t E˛;m;l .t / .x/ D x ŒE˛;m;l .x / 1 : (5.2.48)

G If D 0,then(5.2.48) takes the form 0 D 0.Let ¤ 0. In accordance with (5.2.46)and(5.2.1)wehave ˛ ˛l ˛m J Á I0C t E˛;m;l .t / .x/ 2 3 Z x X1 kY1 .˛Œjm C l C 1/ D .x t/˛1 4t ˛l C k t ˛.mkCl/5 dt: .˛/ .˛Œjm C l C 1 C 1/ 0 kD1 j D0

Interchanging integration and summation and evaluating the inner integrals by using the well-known formula [SaKiMa93, formula (2.44)]

.ˇ/ I ˛ t ˇ1 .x/ D x˛Cˇ1 .˛ > 0; ˇ > 0/; (5.2.49) 0C .˛C ˇ/ we ﬁnd 2 0 1 X1 kY1 .˛Œjm C l C 1/ J D 4 I ˛ t ˛l .x/ C kC1 @ A 0C .˛Œjm C l C 1 C 1/ kD1 j D0 3 ˛ ˛.mkCl/ 5 I0C t .x/ 120 5 Mittag-Lefﬂer Functions with Three Parameters 0 1 X1 Yk .˛Œjm C l C 1/ D kC1 @ A x˛.mkClC1/; .˛Œjm C l C 1 C 1/ kD0 j D0 where the interchange is possible since all integrals converge under the conditions of the theorem. Shifting the summation index k C 1 to k we obtain 0 1 X1 kY1 .˛Œjm C l C 1/ J D x˛.lmC1/ k @ A x˛mk .˛Œjm C l C 1 C 1/ kD1 j D0

˛.lmC1/ ˛m D x ŒE˛;m;l .x / 1 :

This completes the proof of the theorem. F The following Corollary shows how the above properties look like in special cases. Corollary 5.28. For ˛>0, ˇ>0and 2 C the following formulas hold: ˛ ˇ1 ˛ ˇ1 ˛ I0C t E˛;ˇ .t / .x/ D x E˛;ˇ .x / 1 ; (5.2.50) ˛ ˛ ˛ I0C ŒE˛ .t / .x/ D E˛ .x / 1: (5.2.51)

Remark 5.29. In view of (Chap. 4,formula(4.1.1)) (5.2.50) can be written as ˛ ˇ1 ˛ ˛Cˇ1 ˛ I0C t E˛;ˇ .t / .x/ D x E˛;˛Cˇ .x /; (5.2.52) which coincides with the formula [SaKiMa93, Table 9.1, formula 23], if D 1.In particular, (5.2.51) takes the form ˛ ˛ ˛ ˛ I0C ŒE˛ .t / .x/ D x E˛;˛C1 .x /; (5.2.53) by putting ˇ D 1 in (5.2.52). ˛ The next statement is valid for the right-sided Liouville fractional operator I in (5.2.47). Theorem 5.30. Let ˛>0, m>0, l>1=˛ and 2 C. Then the following formula holds: ˛ ˛.lC1/1 ˛m ˛.lm/1/ ˛m I t E˛;m;l .t / .x/ D x ŒE˛;m;l .x / 1 : (5.2.54) G If D 0,then(5.2.54) takes the form 0 D 0.If ¤ 0 we have in accordance with (5.2.47)and(5.2.1) h iÁ J Á I ˛ t˛.lC1/1E .t˛m/ .x/ D ˛;m;l .˛/ 5.2 Generalized (Kilbas–Saigo) Mittag-Lefﬂer Type Functions 121

2 3 Z 1 k1 1 X Y .˛Œjm C l C 1/ .t x/˛1 4t˛.lC1/1 C k t˛.mkClC1/15 dt: .˛Œjm C l C 1 C 1/ x kD1 j D0

Interchanging integration and summation and evaluating the inner integrals be using the formula [SaKiMa93, Table 9.3, formula 1]

. ˛/ I ˛ Œt .x/ D x˛ . > ˛ > 0/; (5.2.55) ./ and using the same arguments as in the proof of Theorem 5.27 we have ˛ ˛.lC1/1 J D I t .x/ 0 1 3 X1 kY1 .˛Œjm C l C 1/ C kC1 @ A I ˛ t ˛.mkClC1/1 .x/5 .˛Œjm C l C 1 C 1/ kD1 j D0 0 1 X1 Yk .˛Œjm C l C 1/ D kC1 @ A x˛.mkCl/1 .˛Œjm C l C 1 C 1/ kD0 j D0 0 1 X1 kY1 .˛Œjm C l C 1/ D x˛.lm/1 k @ A x˛mk .˛Œjm C l C 1 C 1/ kD1 j D0

˛.lm/1 ˛m D x ŒE˛;m;l .x / 1 ; and the theorem is proved. F The following Corollary shows how the above properties look like in special cases. Corollary 5.31. For ˛>0, ˇ>0and 2 C the following formulas hold: Ä 1 I ˛ t ˛ˇ E .t ˛/ .x/ D x˛ˇ E .x˛/ ; (5.2.56) ˛;ˇ ˛;ˇ .ˇ/ ˛ ˛1 ˛ ˛1 ˛ I t E˛ .t / .x/ D x ŒE˛ .x / 1 : (5.2.57)

5.2.7 Fractional Differentiation of the Generalized Mittag-Lefﬂer Function

In this subsection we present applications of the Riemann–Liouville and Liouville ˛ ˛ fractional derivatives D0C and D, defined by the respective formulas [SaKiMa93, formulas (5.8)] 122 5 Mittag-Leffler Functions with Three Parameters Â Ã d n .D˛ f /.x/ D .I n˛f /.x/ 0C dx 0 Â Ã Z d n 1 x f.t/ D t.x>0I n D Œ˛ C 1/ 1nC˛ d (5.2.58) dx .n ˛/ 0 .x t/ and Â Ã d n .D˛ f /.x/ D .I n˛f /.x/ dx Â Ã Z d n 1 1 f.t/ D t.x>0I n D Œ˛ C 1//; 1nC˛ d (5.2.59) dx .n ˛/ x .t x/ to the generalized Mittag-Leffler function (5.2.1). ˛ The application of D0C to E˛;m;l .z/ is given by the following statement. Theorem 5.32. Let ˛>0and m>0be such that 1 l>m 1 ;˛.jm C l/ ¤ 0; 1; 2; .j 2¤ N /; ˛ 0 and let 2 C. Then the following formula holds: ˛ ˛.lmC1/ ˛m D0C t E˛;m;l .t / .x/ Œ˛.l m C 1/ C 1 D x˛.lm/ C x˛l E .x˛m/: (5.2.60) Œ˛.l m/ C 1 ˛;m;l

In particular, if ˛.l m/ Dj for some j D 1; ; Œ˛,then ˛ ˛.lmC1/ ˛m ˛l ˛m D0C t E˛;m;l .t / .x/ D x E˛;m;l .x /: (5.2.61)

G If D 0,thenE˛;m;l .z/ D 1, and applying the formula [SaKiMa93,formula (2.44)]

.ˇ/ D˛ t ˇ1 .x/ D xˇ˛1 .˛ > 0; ˇ > 0/; (5.2.62) 0C .ˇ ˛/ with ˇ D ˛.l m C 1/ C 1,wehave

Œ˛.l m C 1/ C 1 D˛ t ˛.lmC1/ .x/ D x˛.lm/; 0C Œ˛.l m/ C 1 which proves (5.2.60)for D 0. If ¤ 0, then setting n D Œ˛ C 1,using(5.2.1)and(5.2.58), interchanging the order of summation and integration and applying (5.2.62), we have 5.2 Generalized (Kilbas–Saigo) Mittag-Lefﬂer Type Functions 123 ˛ ˛.lmC1/ ˛m J Á D0C t E˛;m;l .t / .x/ ˛ ˛.lmC1/ D D0C t .x/ 0 2 0 1 31 Â Ã d n X1 kY1 .˛Œjm C l C 1/ C @I n˛ 4 k @ A t ˛.kmClmC1/5A .x/ dx 0C .˛Œjm C l C 1 C 1/ kD1 j D0 Œ˛.l m C 1/ C 1 D x˛.lm/ Œ˛.l m/ C 1 0 1 X1 kY1 .˛Œjm C l C 1/ C k @ A D˛ t ˛.kmClmC1/ .x/ .˛Œjm C l C 1 C 1/ 0C kD1 j D0 Œ˛.l m C 1/ C 1 D x˛.lm/ Œ˛.l m/ C 1 0 1 X1 kY1 .˛Œjm C l C 1/ Œ˛.km C l m C 1/ C 1 C k @ A x˛.kmClm/ .˛Œjm C l C 1 C 1/ Œ˛.km C l m/ C 1 kD1 j D0 Œ˛.l m C 1/ C 1 D x˛.lm/ Œ˛.l m/ C 1 2 0 1 3 X1 kY2 .˛Œjm C l C 1/ Cx˛l 41 C k1 @ A x˛.kmm/5 .˛Œjm C l C 1 C 1/ kD2 j D0 Œ˛.l m C 1/ C 1 D x˛.lm/ C x˛l E .x˛m/: Œ˛.l m/ C 1 ˛;m;l

This yields (5.2.60). Formula (5.2.61) then follows from the fact that the Gamma function has poles at every non-positive integer. This completes the proof of the theorem. F Corollary 5.33. For ˛>0, ˇ>0and 2 C the following formula holds:

xˇ˛1 D˛ t ˇ1E .t ˛/ .x/ D C xˇ1E .x˛/: (5.2.63) 0C ˛;ˇ .ˇ ˛/ ˛;ˇ

If further ˇ ˛ D 0; 1; 2;,then ˛ ˇ1 ˛ ˇ1 ˛ D0C t E˛;ˇ .t / .x/ D x E˛;ˇ .x /: (5.2.64)

In particular, for ˇ D 1 we have

x˛ D˛ ŒE .t ˛/ .x/ D C E .x˛/: (5.2.65) 0C ˛ .1 ˛/ ˛ 124 5 Mittag-Lefﬂer Functions with Three Parameters

Remark 5.34. If ˛ D n 2 N,then n .n/ N D0Cf .x/ D y .x/ .n 2 /; (5.2.66) and therefore formula (5.2.60) is reduced to relation (5.2.33) proved in Theorem 5.18 under weaker conditions. Similarly (5.2.65) is reduced to (5.2.38). ˛ The next statement holds for the right-hand sided operator D givenby(5.2.59). Theorem 5.35. Let 2 C and let ˛>0and m>0be such that l>mf˛g=˛, where f˛g is the fractional part of ˛.Then ˛ ˛.ml/1 ˛m D t E˛;m;l .t / .x/ Œ˛.l m C 1/ C 1 D x˛.ml1/1 C x˛.lC1/1E .x˛m/: (5.2.67) Œ˛.l m/ C 1 ˛;m;l

G If D 1,thenE˛;m;l .z/ D 1 and applying the formula [KiSrTr06,formula (2.2.13)]

. C ˛/ D˛ Œt .x/ D x˛ .˛>0;>1f˛g/ (5.2.68) ./ with D ˛.l m/ we get

Œ˛.l m C 1/ C 1 D˛ t ˛.ml/1 .x/ D x˛.ml1/; Œ˛.l m/ C 1 which proves (5.2.67)for D 0. If ¤ 0, then we note that, in accordance with the condition l>mf˛g=˛, l>f˛g=˛ > 1=˛. Therefore condition (5.2.3) is satisfied and the three- parametric Mittag-Leffler function E˛;m;l .z/ is properly defined. In view of (5.2.59) and (5.2.1)wehave ˛ ˛.ml/1 ˛m ˛ ˛.lm/1 J Á D t E˛;m;l .t / .x/ D D t .x/ 0 1 X1 kY1 .˛Œjm C l C 1/ C k @ A D˛ t ˛.kmClm/1 .x/: .˛Œjm C l C 1 C 1/ kD1 j D0

Applying (5.2.68) with D ˛.km C l m/ C 1.k2 N0/, we obtain

Œ˛.l m C 1/ C 1 J D x˛.ml1/1 Œ˛.l m/ C 1 5.3 Historical and Bibliographical Notes 125 0 1 X1 kY1 .˛Œjm C l C 1/ C k @ A .˛Œjm C l C 1 C 1/ kD1 j D0 Œ˛.km C l m C 1/ C 1 x˛.kmClm1/1 Œ˛.km C l m/ C 1 Œ˛.l m C 1/ C 1 D x˛.ml1/1 C x˛.l1/1 Œ˛.l m/ C 1 0 1 X1 kY2 .˛Œjm C l C 1/ C k @ A x˛.kmClm1/1 .˛Œjm C l C 1 C 1/ kD2 j D0 Œ˛.l m C 1/ C 1 D x˛.ml1/1 C x˛.l1/1C Œ˛.l m/ C 1 2 0 1 3 X1 kY1 .˛Œjm C l C 1/ x˛.l1/1 41 C k @ A x˛km5 : .˛Œjm C l C 1 C 1/ kD1 j D0

This, in accordance with (5.2.1), yields (5.2.67), and the theorem is proved. F Corollary 5.36. For ˛>0, ˇ>Œ˛C 1 and 2 C the following formula holds:

xˇ D˛ t ˛ˇ E .t ˛/ .x/ D C x˛ˇ E .x˛ /: (5.2.69) ˛;ˇ .ˇ ˛/ ˛;ˇ

Remark 5.37. If ˛ D n 2 N, then according to (5.2.66), formula (5.2.67) is reduced to relation (5.2.40), proved in Theorem 5.23 under weaker conditions. Similarly (5.2.69) is reduced to (5.2.43).

5.3 Historical and Bibliographical Notes

By means of the series representation, a generalization of the Mittag-Leffler function E˛;ˇ.z/ was introduced by Prabhakar in [Pra71](seealso[MatHau08]). This function is a special case of the Wright generalized hypergeometric function [Wri34, Wri35b]aswellastheH-function [MaSaHa10]. For various properties of this function with applications, see [Pra71]. Like any function of the Mittag-Leffler type, the three parametric Mittag-Leffler function E˛;ˇ.z/ can be represented via the Mellin–Barnes integral. Differentiation and integration formulas for the three-parametric Mittag-Leffler function (Prabhakar function) were obtained in [KiSaSa04]. Some of these formulas were generalized and given in the form of the Laplace transform (see [Sax02]). 126 5 Mittag-Leffler Functions with Three Parameters

Relations connecting the function E˛;ˇ.z/ and the Riemann–Liouville fractional integrals and derivatives are given in [SaxSai05]. Another three-parametric generalization of the Mittag-Lefﬂer function E˛;m;l .z/ was proposed in the form of a power series by Kilbas and Saigo as the solution of a certain Abel–Volterra type integral equation (see [KilSai95b]). In particular, in [KilSai95b], a number of differential relations involving the function E˛;m;l .z/ were obtained. The corresponding formulas were considered as inhomogeneous and homogeneous differential equations of order n for the func- n.lmC1/ nm n.ml/1 nm tions z En;m;l .z / and z En;m;l .z /, respectively. In this way explicit solutions of new classes of ordinary differential equations were obtained in [KilSai95a, SaiKil98, SaiKil00](seealso[KiSrTr06]). Analytic properties of this function are studied in [GoKiRo98]. In [HanSer12] the solution of the Bloch–Torrey equation for space-time fractional anisotropic diffusion is expressed in terms of the three-parametric Mittag-Lefﬂer function E˛;m;l .z/.

5.4 Exercises

5.4.1. Prove the following reducibility formulas for the three-parametric Mittag- Lefﬂer function E˛;ˇ.z/: (a) If ˇ; 2 C are such that Re ˇ>0,Re. ˇ/ > 2,then

3 1 zEˇ; D 2ˇ2 Eˇ;ˇ2/ .2 3ˇ 3/Eˇ;ˇ1

2 2 C .2ˇ C 3ˇ C 3ˇ 2 C 1/Eˇ;ˇ :

(b) If ˇ; 2 C are such that Re ˇ>0,Re>2,then

3 1 Eˇ; D 2ˇ2 Eˇ;2/ .2 3ˇ 3/Eˇ;1

2 2 C .2ˇ C 3ˇ C 3ˇ 2 C 1/Eˇ;ˇ :

5.4.2. Prove the following integral relations for the three-parametric Mittag- Lefﬂer function E˛;ˇ.z/ valid for all Re ˛>0;Re ˇ > 0; > 0; Re ı>0 [MatHau08, p. 96]: R1 1 ˇ1 ı1 ˛ (a) .ı/ u .1 u/ E˛;ˇ.zu /du D E˛;ˇCı.z/I 0 Rx 1 ı1 ˇ1 ˛ (b) .ı/ .x u/ .u t/ E˛;ˇ..u t/ /du t ˇCı1 ˛ D .x t/ E˛;ˇCı..x t/ /: 5.4 Exercises 127

5.4.3. Let [KiSaSa02, p. 383–384]

Á Zx 1 E ;;!IaC' .x/ D .x t/ E ;Œ!.x t/ '.t/dt; x > a; a

be the integral transform with the Prabhakar function in the kernel. (a) Find the value of this transform of the power-type function .t a/ˇ1. (b) Calculate the composition of the operator E ;;!IaC and the left-sided ˛ Riemann–Liouville fractional integration operator IaC. (c) Prove the semigroup property of the integral transform with the Prabhakar function in the kernel:

C E ;;!IaCE ;;!IaC' D E ;;!IaC':

5.4.4. Show that the linear homogeneous differential equation [KilSai00, p. 194]

y.n/.x/ D axˇy.x/ .0 < x Ä d

.a 6D 0; ˇR;ˇ > n; .n C ˇ/.i C 1/ 6D 1;2;:::;n 1; i 2 N/ has n solutions of the form

j 1 ˇCn yj .x/ D x En;1Cˇ=n;.ˇCj /=n.ax /.jD 1;2;:::;n/:

Prove that if ˇ 0, then these solutions are linearly independent. 5.4.5. Let n 2 N, ˇ 2 R;ˇ > n, fk;k 2 R;k D 0;1;:::;p,andi.n C ˇ/ C k 6Dj (i 2 N0;k D 0;1;:::;p;j D 1;2;:::;n)[KilSai00, p. 197]. Show that the inhomogeneous differential equation

Xp .n/ ˇ k y .x/ D ax y.x/ C fkx .0 < x Ä d

has a particular solution of the form 2 3 p X Yn 1 4 5 k Cn nCˇ y0.x/ D fkx En;1Cˇ=n;1C.ˇCk /=n.ax /: k C j kD0 j D1

5.4.6. Let ˛>0, m>0, k > 1, fk 2 R (k0;1;:::;l). Prove that the Abel– Volterra equation with quasi-polynomial free term [KilSai96, p. 365]

Zx ax˛.m1/ '.t/ Xl '.x/ D dt C f xk .0 < x < d ÄC1/ .˛/ .x t/1˛ k 0 kD0 128 5 Mittag-Lefﬂer Functions with Three Parameters

has a unique solution of the form

Xl k ˛m '.x/ D fkx E˛;m;k =˛ .ax /: kD0 Chapter 6 Multi-index Mittag-Lefﬂer Functions

6.1 The Four-Parametric Mittag-Lefﬂer Function: Luchko–Kilbas–Kiryakova’s Approach

6.1.1 Deﬁnition and Special Cases

R 2 2 C Consider the function deﬁned for ˛1;˛2 2 .˛1 C ˛2 ¤ 0/ and ˇ1;ˇ2 2 by the series

X1 k z C E˛1;ˇ1I˛2;ˇ2 .z/ Á .z 2 /: (6.1.1) .˛1k C ˇ1/ .˛2k C ˇ2/ kD0

Such a function with positive ˛1 >0, ˛2 >0and real ˇ1;ˇ2 2 R was introduced by Dzherbashian [Dzh60]. When ˛1 D ˛; ˇ1 D ˇ and ˛2 D 0, ˇ2 D 1, this function coincides with the Mittag-Lefﬂer function (4.1.1):

X1 zk E .z/ D E .z/ Á .z 2 C/: (6.1.2) ˛;ˇI0;1 ˛;ˇ .˛kC ˇ/ kD0

Therefore (6.1.1) is sometimes called the generalized Mittag-Lefﬂer function or four-parametric Mittag-Lefﬂer function.

Certain special functions of Bessel type are expressed in terms of E˛1;ˇ1I˛2;ˇ2 .z/: The Bessel function of the ﬁrst kind (see, e.g., [Bat-2, n. 7.2.1-2], [NIST, p. 217, 219]) Á Â Ã z z2 J .z/ D E : (6.1.3) 2 1;C1I1;1 4

The Struve function (see, e.g., [Bat-2, n. 7.5.4], [NIST, p. 288])

© Springer-Verlag Berlin Heidelberg 2014 129 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__6 130 6 Multi-index Mittag-Leffler Functions

Á Â Ã z C1 z2 H .z/ D E : (6.1.4) 2 1;C3=2I1;3=2 4

The Lommel function (see, e.g., [Bat-2, n. 7.5.5]) Â Ã Â Ã Â Ã zC1 C 1 C C 1 z2 C C C S;.z/ D E1; 1 I1; 1 : 4 2 2 2 2 4 (6.1.5)

The Bessel–Maitland function (see, e.g., [Kir94, App. E. ii])

J .z/ D E;C1I1;1.z/: (6.1.6)

The generalized Bessel–Maitland function (see, e.g., [Kir94, App. E. ii]) Á z C2 J .z/ D E .z/: (6.1.7) ; 2 ;CC1I1;C1

6.1.2 Basic Properties

First of all we prove that (6.1.1) is an entire function if ˛1 C ˛2 >0. R C 2 2 Theorem 6.1. Let ˛1;˛2 2 and ˇ1;ˇ2 2 be such that ˛1 C ˛2 ¤ 0 and C ˛1 C ˛2 >0.ThenE˛1;ˇ1I˛2;ˇ2 .z/ is an entire function of z 2 of order

1 D (6.1.8) ˛1 C ˛2 and type

Â Ã ˛1 Â Ã ˛2 ˛ C ˛ ˛1C˛2 ˛ C ˛ ˛1C˛2 D 1 2 1 2 : (6.1.9) j˛1j j˛2j

G Rewrite (6.1.1) as the power series

X1 k 1 E˛1;ˇ1I˛2;ˇ2 .z/ D ck z ;ck D : (6.1.10) .˛1k C ˇ1/ .˛2k C ˇ2/ kD0

Using Stirling’s formula for the Gamma function we obtain

jc j k ˛1 ˛2 ˛1C˛2 j˛1j j˛2j k .k !1/: jckC1j

Thus, E˛1;ˇ1I˛2;ˇ2 .z/ is an entire function of z when ˛1 C ˛2 >0. 6.1 The Four-Parametric Mittag-Lefﬂer Function: Luchko–Kilbas–. . . 131

We use (Appendix B,formulas(B.2.3)and(B.2.4)) to evaluate the order and the type of (6.2.1). For this we apply the asymptotic formula for the logarithm of the Gamma function .z/ at inﬁnity [Bat-1, 1.18(1)]: Â Ã Â Ã 1 1 1 log .z/ D z log z z C log.2z/ C O .jzj!1; jargzj </: 2 2 z (6.1.11)

Applying this formula and taking (6.1.10) into account, we deduce the asymptotic estimate Â Ã 1 log k log.k/.˛1 C ˛2/.k!1/ ck from which, in accordance with (Appendix B,(B.2.3)), we obtain (6.1.8). Further, according to (Appendix A,(A.1.24)), we have

.˛j k C ˇj / Ä Â Ã ˛ kCˇ 1 1 D .2/1=2 ˛ k C ˇ j j 2 e.˛j kCˇj / 1 C O .k !1/ j j k (6.1.12) for j D 1; 2, and we obtain the asymptotic estimate

Y2 1 ˛j kCˇj ˛j k .˛1k C ˇ1/ .˛2k C ˇ2/ 2 .˛j k/ 2 e .k !1/: (6.1.13) j D1

From (6.1.10)and(6.1.13)wehave

Y2 1= 1=k 1= ˛j ˛j lim sup k jckj D lim sup k .j˛j jk/ e k!1 k!1 j D1 Y2 Y2 ˛1C˛2 ˛j 1= ˛j D e j˛j j D e j˛j j : j D1 j D1

Substituting this relation into (Appendix B,(B.2.4)) we obtain 0 1

Y2 1 1 @ ˛ A ˛ ˛ C D j˛ j j D .˛ C ˛ /.j˛ j 1 j˛ j 2 / ˛1 ˛2 j 1 2 1 2 j D1

Â Ã ˛1 Â Ã ˛2 ˛ C ˛ ˛1C˛2 ˛ C ˛ ˛1C˛2 D 1 2 1 2 ; j˛1j j˛2j which proves (6.1.9).F 132 6 Multi-index Mittag-Lefﬂer Functions

Remark 6.2. For ˛1 >0and ˛2 >0, relations (6.1.8)and(6.1.9) were proved by Dzherbashian [Dzh60].

6.1.3 Integral Representations and Asymptotics

The four-parametric Mittag-Lefﬂer function has the Mellin–Barnes integral representation Z 1 .s/.1 s/ s E˛1;ˇ1I˛2;ˇ2 .z/ D .z/ ds; (6.1.14) 2i .ˇ1 ˛1s/ .ˇ2 ˛2s/ L where L D L1 is a left loop, i.e. the contour which is situated in a horizontal strip, starting at 1 C i'1 and ending at 1 C i'2, with 1 <'1 <0<'2 < C1. This contour separates poles of the Gamma functions .s/and .1 s/.

By using (6.1.14) the function E˛1;ˇ1I˛2;ˇ2 can be extended to non-real values of the parameters. If the parameters ˛1;ˇ1I ˛2;ˇ2 are such that Re .˛1 C ˛2/>0,then the integral (6.1.14)convergesforallz 6D 0. This is a consequence of the following asymptotic formulas for the function H.s/ D .s/.1s/ in the integrand .ˇ1˛1s/.ˇ2˛2s/ of (6.1.14), where s D t C i ;.t !1/, and the properties of the Mellin–Barnes integral:

–ForRe˛1 >0;Re ˛2 >0 Â Ã jtj Re.˛1C˛2/t ŒRe.˛ /Re.˛1/Re.˛ /Re.˛2/t jH.s/j M 1 2 I 1 P2 (6.1.15) e 1 ŒRe.ˇj /C Im.˛i / jtjj D1

–ForRe˛1 <0;Re ˛2 >0 Â Ã Re.˛1C˛2/t Re.˛1/ Re.˛2/ t jtj ŒjRe.˛1/j Re.˛2/ jH.s/j M Im.˛1/t I 2 P2 e e 1 ŒRe.ˇi /C Im.˛i / jtjiD1 (6.1.16)

–ForRe˛1 >0;Re ˛2 <0 Â Ã Re.˛1C˛2/t Re.˛1/ Re.˛2/ t jtj ŒRe.˛1/ jRe.˛2/j jH.s/j M Im.˛2/t : 3 P2 e e 1 ŒRe.ˇi /C Im.˛i / jtjiD1 (6.1.17)

We do not present here exact asymptotic formulas for E˛1;ˇ1I˛2;ˇ2 .z/ as z !1. They can be considered as formulas for a special case of the generalized Wright function and H-function (see Sect. 6.1.5 below). 6.1 The Four-Parametric Mittag-Lefﬂer Function: Luchko–Kilbas–. . . 133

From the series representation of the four-parametric Mittag-Lefﬂer function we derive a simple asymptotics at zero, valid in the case Re f˛1 C ˛2g >0for all N 2 N:

XN k z N C1 E˛1;ˇ1I˛2;ˇ2 .z/ D C O jzj ; z ! 0: .˛1k C ˇ1/ .˛2k C ˇ2/ kD0 (6.1.18)

The following integral representation of the four-parametric Mittag-Leffler function (see [RogKor10]) shows its tight connection to the generalized Wright function (see Appendix F). Let 0<˛j <2, ˇj 2 C, j D 1; 2. Then the following representation of the four- parametric generalized Mittag-Leffler function E˛1;ˇ1I ˛2;ˇ2 .z/ holds ([RogKor10]) 8 Î .z/; z G./. ; /; ˆ 0 2 2 ˆ ˆ ˆ ˇ C1 Â Ã ˆ 2 ˆ z ˛2 ˛ ˇ ˛ ˇ ˛ C 1 1 ˆ 1 1 2 2 1 ˛ ./ Î0.z/ C ; I z 2 ; z 2 G . ; 1;2/; ˆ 2i˛ ˛ ˛ <ˆ 2 2 2

C E˛1;ˇ1I ˛2;ˇ2 D ˇ2 1 Â Ã ˆ ˛ ˆ z 2 ˛1 ˇ1˛2 ˇ2˛1 C 1 1 ˆI .z/ C ; I z ˛2 ˆ 0 2i˛ ˛ ˛ ˆ 2 2 2 ˆ ˆ ˆ ˇ1C1 Â Ã ˆ z ˛1 ˛ ˇ ˛ ˇ ˛ C 1 1 : 2 2 1 1 2 ˛ .C/ C ; I z 1 ; z 2 G . ; 1/; 2i˛1 ˛1 ˛1 (6.1.19) with 8 9 ˆ Z Z .ˇ C1/ > < 1=˛2 2 = .ˇ C1/ 2 ˛2 1 1=˛1 1 e 2 d 2 I .z/ D 1 ˛1 ; 0 2 e 1 d 1 4 ˛1˛2 :ˆ 1 2 z ;> . I 1/ . I 2/ (6.1.20) X1 zk .˛;ˇI z/ WD ; (6.1.21) kŠ .˛k C ˇ/ kD0 where .˛;ˇI z/ is the classical Wright function (see Appendix F), j 2 ˛j 2 ; minf˛j ;g , 0<1 <2 <2,and >0is an arbitrary positive number. Here . I /. >0;0< Ä / is a contour with non-decreasing arg consisting of the following parts: 134 6 Multi-index Mittag-Lefﬂer Functions

1. The ray arg D ; j j ; 2. The arc Ä arg Ä of the circle j jD ; 3. The ray arg D ; j j . In the case 0< <the complex -plane is divided by the contour . I /into two unbounded parts: the domain G./. I / to the left of the contour and the domain G.C/. I / to the right. If D , the contour . I / consists of the circle j jD and of the cut 1 < Ä . In this case the domain G./. I / becomes the circle j j < and the domain G.C/. I / becomes the domain f Wjarg j <;j j > g. For two different values of 1; 2, 0< 1 < 2 <the union of the two unbounded ./ domains between the curves . I 1/ and . I 2/ is denoted by G . I 1; 2/.

6.1.4 Extended Four-Parametric Mittag-Lefﬂer Functions

Let the contour L in the Mellin–Barnes integral Z 1 .s/.1 s/ .z/sds; (6.1.22) 2i .ˇ1 ˛1s/ .ˇ2 ˛2s/ L now coincide with the right loop LC1, i.e. with a curve starting at C1 C i'1 and ending at C1 C i'2 (1 <'1 <'2 < C1), leaving the poles of .s/at the left and the poles of .1 s/ at the right. Then this integral exists for all z 6D 0 whenever Re f˛1 C ˛2g <0. Thus the integral (6.1.22) possesses an extension to another set of parameters. It deﬁnes a new function which is called the extended generalized Mittag-Lefﬂer E function and denoted ˛1;ˇ1I˛2;ˇ2 .z/ (see [KilKor05, KilKor06a]). Using the same approach as before, i.e. calculating the integral (6.1.22)bythe Residue Theorem, one can obtain the following Laurent series representation of E ˛1;ˇ1I˛2;ˇ2 .z/:

X1 d E .z/ D k ; (6.1.23) ˛1;ˇ1I˛2;ˇ2 zkC1 kD0 where 1 dk D : .˛1.k C 1/ ˇ1/ .˛2.k C 1/ ˇ2/

In the case Re f˛1 C ˛2g <0the series (6.1.23) is convergent for all z 2 C; z 6D E 0. The function ˛1;ˇ1I˛2;ˇ2 .z/ has an asymptotics at z ! 0 similar to that of the standard four-parametric Mittag-Lefﬂer function E˛1;ˇ1I˛2;ˇ2 .z/,Ref˛1 C ˛2g >0 E at z !1. The asymptotics of ˛1;ˇ1I˛2;ˇ2 .z/ at z !1can be displayed in the form 6.1 The Four-Parametric Mittag-Lefﬂer Function: Luchko–Kilbas–. . . 135

Â Ã XN d 1 E .z/ D k C O ; z !1: (6.1.24) ˛1;ˇ1I˛2;ˇ2 zkC1 jzjN C1 kD0

6.1.5 Relations to the Wright Function and the H-Function

For real values of the parameters ˛1;˛2 2 R and complex values of ˇ1;ˇ2 2 C the E four-parametric Mittag-Lefﬂer function ˛1;ˇ1I˛2;ˇ2 can be represented in terms of the generalized Wright function and the H–function. These representations follow immediately from the Mellin–Barnes integral E representation of the function ˛1;ˇ1I˛2;ˇ2 and the properties of the corresponding integrals. E Let us present some formulas relating ˛1;ˇ1I˛2;ˇ2 to the generalized Wright function pq:

1. If ˛1 C ˛2 >0and the contour of integration in (6.1.14) is chosen as L D L1, then Ä ˇ ˇ E .1; 1/ ˇ ˛1;ˇ1I˛2;ˇ2 .z/ D 12 ˇz : (6.1.25) .ˇ1;˛1/; .ˇ2;˛2/

2. If ˛1 C ˛2 <0and the contour of integration in (6.1.14) is chosen as L D LC1, then Ä ˇ ˇ E 1 .1; 1/ ˇ1 ˛1;ˇ1I˛2;ˇ2 .z/ D 12 ˇ : (6.1.26) z .ˇ1 ˛1; ˛1/; .ˇ2 ˛2; ˛2/ z

E Analogously, one can obtain the following representation of ˛1;ˇ1I˛2;ˇ2 in terms of the H-function:

1. If ˛1 >0;˛2 >0and the contour of integration in (6.1.14) is chosen as L D L1,then Ä ˇ ˇ E 1;1 .0; 1/ ˇ ˛1;ˇ1I˛2;ˇ2 .z/ D H1;3 ˇz : (6.1.27) .0; 1/; .1 ˇ1;˛1/; .1 ˇ2;˛2/

2. If ˛1 >0;˛2 <0and the contour of integration in (6.1.14) is chosen as L D L1 when ˛1 C ˛2 >0or L D LC1 when ˛1 C ˛2 <0,then Ä ˇ ˇ E 1;1 .0; 1/; .ˇ2; ˛2/ ˇ ˛1;ˇ1I˛2;ˇ2 .z/ D H2;2 ˇx : (6.1.28) .0; 1/; .1 ˇ1;˛1/

3. If ˛1 <0;˛2 >0and the contour of integration in (6.1.14) is chosen as L D L1 when ˛1 C ˛2 >0or L D LC1 when ˛1 C ˛2 <0,then 136 6 Multi-index Mittag-Lefﬂer Functions Ä ˇ ˇ E 1;1 .0; 1/; .ˇ1; ˛1/ ˇ ˛1;ˇ1I˛2;ˇ2 .z/ D H2;2 ˇx : (6.1.29) .0; 1/; .1 ˇ2;˛2/

4. If ˛1 <0;˛2 <0and the contour of integration in (6.1.14) is chosen as L D LC1,then Ä ˇ ˇ 1;1 .0; 1/; .ˇ1; ˛1/; .ˇ2; ˛2/ ˇ E˛ ;ˇ I˛ ;ˇ .z/ D H ˇx : (6.1.30) 1 1 2 2 3;1 .0; 1/

6.1.6 Integral Transforms of the Four-Parametric Mittag-Lefﬂer Function

In order to present elements of the theory of integral transforms of the extended generalized Mittag-Lefﬂer function we introduce a set of weighted Lebesgue spaces L;r.RC/. These spaces are suitable for the above mentioned integral transforms since the latter are connected with the classical Mellin transform (see, e.g., [Mari83, pp. 36–39]). Let us denote by L;r.RC/.1Ä r Ä1; 2 R/ the space of all Lebesgue measurable functions f such that kf k;r < 1,where 0 1 Z1 1=r @ r dt A kf k;r Á jt f.t/j < 1 .1 Ä r<1/Ikf k;1 Á ess sup kt f.t/k: t t>0 0 (6.1.31)

In particular, for D 1=r the spaces L;r coincide with the classical spaces of r-summable functions: L1=r;r D Lr .RC/ endowed with the norm 8 9

For any function f 2 L;r.RC/.1Ä r Ä 2/ its Mellin transform Mf is deﬁned (see, e.g., [KilSai04, (3.2.5)]) by the equality

ZC1 .Mf /.s/ D f.e / es d.sD C itI ; t 2 R/: (6.1.32) 1 T If f 2 L;r L;1, then the transform (6.1.32) can be written in the form of the classical Mellin transform with Re s D (see Appendix C): 6.1 The Four-Parametric Mittag-Lefﬂer Function: Luchko–Kilbas–. . . 137

ZC1 .Mf /.s/ D f.t/ts1 dt: (6.1.33)

An inverse Mellin transform in this case can be determined by the formula

ZCi1 1 f.t/D .Mf /.s/t s ds.D Re s/: 2i i1

We have for the Mellin transform of the generalized hypergeometric Wright function

Qp Ä Ä ˇ .s/ .a ˛ s/ ˇ i i M .ai ;˛i /1;p ˇ iD1 pq t .s/ D Qq ; (6.1.34) .bj ;ˇj /1;q .bj ˇj s/ j D1 Â Ä Ã Re .ai / ˛i >0;ˇj >0I i D 1;:::;pI j D 1;:::;qI 0

.s/ M Œ.˛; ˇI t/.s/ D .Re s>0/: (6.1.35) .ˇ ˛s/

The Mellin transform of the H-function under certain assumptions on its Hm;n parameters coincides with the function p;q .s/ in the Mellin–Barnes integral representation of the H-function (see [PrBrMa-V3, 8.4.51.11], [KilSai04, Theorem 2.2]). Let us introduce the following parameters characterizing the behaviour of the H-function (see Appendix F) Ä ˇ ˇ Hm;n Hm;n ˇ .ai ;˛i /1;p p;q .z/ D p;q zˇ .bj ;ˇj /1;q Pn Pp Pm Pq a D ˛i ˛i C ˇj ˇj ; iD1 iDnC1 j D1 j DmC1 Pq Pp Pq Pp pq (6.1.36) D ˇj ˛i C 2 ;D ˇj ˛i ; j D1 hiD1 i h j D1 i iD1 ˛ D min Re bj ;ˇ D min 1Re ai : 1Äj Äm ˇj 1ÄiÄn ˛i

Let a 0, s 2 C, be such that

Re s C Re <1: (6.1.38)

Then the Mellin transform of the H-function exists and satisﬁes the relation Â Ä ˇ Ã Ä ˇ ˇ ˇ M m;n ˇ .ai ;˛i /1;p Hm;n .ai ;˛i /1;p ˇ Hp;q z ˇ .s/ D p;q ˇ s : (6.1.39) .bj ;ˇj /1;q .bj ;ˇj /1;q

Since the four-parametric Mittag-Lefﬂer function is related to the generalized Wright function and to the H-function (see Sect. 6.1.5), then one can use (6.1.34) or (6.1.39) to deﬁne the Mellin transform of the function E˛1;ˇ1I˛2;ˇ2 .z/ and of its E extension ˛1;ˇ1I˛2;ˇ2 .z/.

6.1.7 Integral Transforms with the Four-Parametric Mittag-Lefﬂer Function in the Kernel

Integral transforms with the four-parametric Mittag-Lefﬂer function in the kernel can be considered as a special case of the more general H-transform. Let us recall a few facts from the theory of the H-transform following [KilSai04]. The H-transform is introduced as a Mellin-type convolution with the H-function in the kernel:

Z1 Ä ˇ ˇ m;n ˇ .ai ;˛i /1;p .Hf /.x/ D Hp;q xtˇ f.t/dt.x>0/: (6.1.40) .bj ;ˇj /1;q 0

Let us recall some results on the H-transform in L;2-type spaces following [KilSai04, Ch. 3] (elements of the so-called L;2-theory of H-transforms). Here we use the notation (6.1.36) for the parameters a;;;˛;ˇ. We also introduce a so-called exceptional set EH for the function H.s/:

EH D f 2 R W ˛<1 <ˇand H.s/ has zeros on Re s D 1 g : (6.1.41) Let (i) ˛<1 <ˇand suppose one of the following conditions holds: (ii) a >0,or (iii) a D 0, .1 / C Re Ä 0. Then the following statements are satisfied: (a) There exists an injective transform H 2 ŒL;2; L1;2 such that for any f 2 L;2 the Mellin transform satisfies the relation 6.1 The Four-Parametric Mittag-Leffler Function: Luchko–Kilbas–. . . 139 Ä ˇ ˇ M Hm;n .ai ;˛i /1;p ˇ M . H f /.s/ D p;q ˇs . f /.1 s/ .Re s D 1 /: .bj ;ˇj /1;q (6.1.42)

If a D 0, .1 / C Re D 0, 62 EH,thenH is bijective from L;2 onto L1;2. (b) For any f; g 2 L;2 the following equality holds:

Z1 Z1 f.x/.Hg/.x/ dx D .Hf /.x/g.x/ dx: (6.1.43)

0 0

d .H f /.x/ D hx1.C1/=h x.C1/=h dx Z1 Ä ˇ ˇ m;nC1 ˇ .;h/;.ai ;˛i /1;p HpC1;qC1 xtˇ f.t/dt: .bj ;ˇj /1;q ;. 1; h/ 0 (6.1.44)

If Re <.1 /h 1,then

d .H f /.x/ Dhx1.C1/=h x.C1/=h dx Z1 Ä ˇ ˇ mC1;n ˇ .ai ;˛i /1;p ;.;h/ HpC1;qC1 xtˇ f.t/dt: . 1; h/; .bj ;ˇj /1;q 0 (6.1.45)

(d) The H-transform does not depend on in the following sense: if two values of the parameter, say and Q, satisfy condition (i) and one of the conditions (ii) or (iii), and if the transforms H and HQ aredeﬁnedbytherelation(T 6.1.42) in L;2 and L;2Q , respectively, then H f D HQ f for any f 2 L;2 L;2Q . (e) If either a >0or a D 0,and.1 / C Re <0, then for any f 2 L;2 we have H f D H f ,i.e.H is deﬁned by the equality (6.1.40).

An extended L;r-theory (for any 1 Ä r ÄC1)oftheH-transform is presented in [KilSai04]. The integral transform with the four-parametric Mittag-Leffler function in the kernel is defined for ˛1;˛2 2 R, ˇ1;ˇ2 2 C by the formula: 140 6 Multi-index Mittag-Leffler Functions

Z1 E E˛1;ˇ1I˛2;ˇ2 f .x/ D ˛1;ˇ1I˛2;ˇ2 .xt/f .t/dt.x>0/; (6.1.46) 0 E where for ˛1 C ˛2 >0the kernel ˛1;ˇ1I˛2;ˇ2 D E˛1;ˇ1I˛2;ˇ2 (i.e. it is the four- parametric generalized Mittag-Leffler function defined by (6.1.1)), and for ˛1C˛2 < E 0 the kernel ˛1;ˇ1I˛2;ˇ2 is the extended four-parametric generalized Mittag-Leffler function defined by (6.1.22). The properties of this transform follow from its representation as a special case of the H-transform.

1. If ˛1 >0;˛2 >0,then 2 3 Z1 ˇ ˇ .0; 1/ 1;1 4 ˇ 5 E˛1;ˇ1I˛2;ˇ2 f .x/ D H1;3 xtˇ f.t/dt: 0 .0; 1/; .1 ˇ1;˛1/; .1 ˇ2;˛2/ (6.1.47)

2. If ˛1 >0;˛2 <0,then 2 3 Z1 ˇ ˇ .0; 1/; .ˇ2; ˛2/ 1;1 4 ˇ 5 E˛1;ˇ1I˛2;ˇ2 f .x/ D H2;2 xtˇ f.t/dt: (6.1.48) 0 .0; 1/; .1 ˇ1;˛1/

3. If ˛1 <0;˛2 >0,then 2 3 Z1 ˇ ˇ .0; 1/; .ˇ1; ˛1/ 1;1 4 ˇ 5 E˛1;ˇ1I˛2;ˇ2 f .x/ D H2;2 xtˇ f.t/dt: (6.1.49) 0 .0; 1/; .1 ˇ2;˛2/

4. If ˛1 <0;˛2 <0,then 2 3 Z1 ˇ ˇ .0; 1/; .ˇ1; ˛1/; .ˇ2; ˛2/ 1;1 4 ˇ 5 E˛1;ˇ1I˛2;ˇ2 f .x/ D H3;1 xtˇ f.t/dt: 0 .0; 1/ (6.1.50)

Based on (6.1.47)–(6.1.50) and on the above presented elements of the L;2- theory of the H-transform one can formulate the following results for the integral transforms with the four-parametric generalized Mittag-Lefﬂer function in the kernel. Let us present these only in the case (1) (i.e. when ˛1 >0;˛2 >0). All other cases can be considered analogously (see, e.g. [KilKor06a, KilKor06b]). Let ˛1 >0;˛2 >0. Then the parameters a ;;;˛;ˇ are related to the parameters of the four-parametric Mittag-Lefﬂer function as follows:

a D 2 ˛1 ˛2;D ˛1 C ˛2;D 1 ˇ1 ˇ2;˛D 0; ˇ D 1: 6.1 The Four-Parametric Mittag-Lefﬂer Function: Luchko–Kilbas–. . . 141

Let 0<<1, ˛1 >0;˛2 >0and ˇ1;ˇ2 2 C be such that ˛1 C ˛2 <2or ˛1 C ˛2 D 2 and 3 2 Ä Re .ˇ1 C ˇ2/. Then: (a) There exists an injective mapping E 2 ŒL ; L such that for any ˛1;ˇ1I˛2;ˇ2 ;2 1;2 f 2 L;2 the following relation holds: Á .s/.1 s/ ME f .s/ .Mf /.1 s/ . s 1 /: ˛1;ˇ1I˛2;ˇ2 D Re D .ˇ1 ˛1s/ .ˇ2 ˛2s/ (6.1.51)

If either ˛1 C ˛2 <2or ˛1 C ˛2 D 2 and 3 2 Ä Re .ˇ1 C ˇ2/ and the additional conditions

ˇ C k ˇ C l s ¤ 1 ;s¤ 2 .k; l D 0; 1; 2; /; for Re s D 1 ; (6.1.52) ˛1 ˛2

are satisﬁed, then the operator E is bijective from L;2 onto L1;2. (b) For any f; g 2 L;2 we have the integration by parts formula

Z1 Z1 f.x/E g.x/ x E f .x/g.x/ x: ˛1 ;ˇ1I˛2;ˇ2 d D ˛1;ˇ1I˛2;ˇ2 d (6.1.53) 0 0

Á d E f .x/ D hx1.C1/=h x.C1/=h ˛1;ˇ1I˛2;ˇ2 dx 2 3 Z1 ˇ ˇ .; h/; .0; 1/ 1;2 4 ˇ 5 H2;4 xtˇ f.t/dt 0 .0; 1/; .1 ˇ1;˛1/; .1 ˇ2;˛2/; . 1; h/ (6.1.54)

when Re >.1 /h 1, and in the form:

Á d E f .x/ Dhx1.C1/=h x.C1/=h ˛1;ˇ1I˛2;ˇ2 dx 2 3 Z1 ˇ ˇ .0; 1/; .;h/ 2;1 4 ˇ 5 H2;4 xtˇ f.t/dt 0 . 1; h/; .0; 1/; .1 ˇ1;˛1/; .1 ˇ2;˛2/ (6.1.55)

when Re <.1 /h 1. 142 6 Multi-index Mittag-Lefﬂer Functions

(d) The mapping E does not depend on in the following sense: if ˛1;ˇ1I˛2;ˇ2 0<; <1and the mappings E , E are deﬁned on 1 2 ˛1;ˇ1I˛2;ˇ2I1 ˛1;ˇ1I˛2;ˇ2I2 the spaces L , L respectively, then E f D E f for T1;2 2;2 ˛1;ˇ1I˛2;ˇ2I1 ˛1;ˇ1I˛2;ˇ2I2 L L all f 2 1;2 2;2. (e) If f 2 L;2 and either ˛1 C ˛2 <2or ˛1 C ˛2 D 2 and 3 2 Ä Re .ˇ1 C ˇ2/, then for all f 2 L we have E f D E f , i.e. the mapping ;2 ˛1;ˇ1I˛2;ˇ2 ˛1;ˇ1I˛2;ˇ2 E is deﬁned by the formula (6.1.47). ˛1;ˇ1I˛2;ˇ2

6.1.8 Relations to the Fractional Calculus

Let us present a number of the (left- and right-sided) Riemann–Liouville fractional integration and differentiation formulas for the four-parametric Mittag-Leffler function. Both cases (˛1 C ˛2 >0and ˛1 C ˛2 <0) will be considered E simultaneously (see [KiKoRo13]). For simplicity we use the notation ˛1;ˇ1I˛2;ˇ2 for the four-parametric Mittag-Leffler function in both cases. Let ˛1;˛2 2 R, ˛1 ¤ 0; ˛2 ¤ 0, ˇ1;ˇ2 2 C, and let the contour of integration in (6.1.14) be chosen as L D L1 when ˛1 C˛2 >0,andasL D LC1 when ˛1 C ˛2 <0. Let the additional parameters ; ; 2 C be such that Re >0;Re >0 and ! 2 R;.!¤ 0/. The left-sided Riemann–Liouville fractional integral of the four-parametric Mittag-Leffler function is given by the following formulas:

(a) If ˛1 <0and ˛2 >0, then for x>0 1E ! I0Ct ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ C1 1;2 ! ˇ .0; 1/; .1 ; !/; .ˇ1; ˛1/ ˆ x H3;3 x ˇ .! > 0/; < .0; 1/; .1 ; !/; .1 ˇ2;˛2/ ˆ Ä ˇ ˆ ˇ ˆ C1 2;1 ! ˇ .0; 1/; . C ; !/;.ˇ1; ˛1/ : x H3;3 x ˇ .! < 0/: .0; 1/; . ; !/; .1 ˇ2;˛2/

(b) If ˛1 <0and ˛2 <0, then for x>0 1E ! I0Ct ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ 1;2 ˇ.0; 1/; .1 ; !/; .ˇ1; ˛1/; .ˇ2; ˛2/ ˆx C1H x! ˇ .! > 0/; <ˆ 4;2 .0; 1/; .1 ; !/ ˆ Ä ˇ ˆ ˇ ˆ 2;1 ˇ.0; 1/; . C ; !/; .ˇ1; ˛1/; .ˇ2; ˛2/ :ˆx C1H x! ˇ .! < 0/: 4;2 .0; 1/; . ; !/ 6.1 The Four-Parametric Mittag-Lefﬂer Function: Luchko–Kilbas–. . . 143

Á 1E ! I0Ct ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 " ˇ # ˇ ˆ ˇ .0; 1/; .1 ; !/ ˆ C1 1;2 ! ˇ ˆ x H2;4 x ˇ .! > 0/; <ˆ .0; 1/; .1 ; !/; .1 ˇ1;˛1/; .1 ˇ2;˛2/ " ˇ # ˆ ˇ ˆ ˇ ˆ C1 2;1 ! ˇ .0; 1/; . C ; !/ : x H2;4 x ˇ .! < 0/: .0; 1/; . ; !/; .1 ˇ1;˛1/; .1 ˇ2;˛2/

The right-sided Riemann–Liouville fractional integral of the four-parametric Mittag-Lefﬂer function is given by the following formulas:

(a) If ˛1 <0and ˛2 >0, then for x>0 E ! It ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ 1;2 ! ˇ .0; 1/; .1 C ; !/; .ˇ1; ˛1/ ˆ x H3;3 x ˇ .! > 0/; < .0; 1/; .1 ; !/; .1 ˇ2;˛2/ ˆ Ä ˇ ˆ ˇ ˆ 2;1 ! ˇ .0; 1/; . ; !/; .ˇ1; ˛1/ : x H3;3 x ˇ .! < 0/: .0; 1/; . ; !/;.1 ˇ2;˛2/

(b) If ˛1 <0and ˛2 <0, then for x>0 E ! It ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ 1;2 ˇ.0; 1/; .1 C ; !/; .ˇ1; ˛1/; .ˇ2; ˛2/ ˆx H x! ˇ .! > 0/; <ˆ 4;2 .0; 1/; .1 ; !/ ˆ Ä ˇ ˆ ˇ ˆ 2;1 ˇ .0; 1/; . ; !/; .ˇ1; ˛1/; .ˇ2; ˛2/ :ˆx H x! ˇ .! < 0/: 4;2 .0; 1/; . ; !/

(c) If ˛1 >0and ˛2 >0, then for x>0 E ! It ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ 1;2 ! ˇ.0; 1/; .1 C ; !/ ˆx H2;4 x ˇ .! > 0/; < .0; 1/; .1 ; !/; .1 ˇ1;˛1/; .1 ˇ2;˛2/ ˆ Ä ˇ ˆ ˇ ˆ 2;1 ! ˇ.0; 1/; . ; !/ :x H2;4 x ˇ .! < 0/: .0; 1/; . ; !/;.1 ˇ1;˛1/; .1 ˇ2;˛2/ 144 6 Multi-index Mittag-Lefﬂer Functions

The left-sided Riemann–Liouville fractional derivative of the four-parametric Mittag-Lefﬂer function is given by the following formulas:

(a) If ˛1 <0and ˛2 >0, then for x>0 1E ! D0Ct ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ 1 2;1 ! ˇ .0; 1/; .1 ; !/; .ˇ1; ˛1/ ˆ x H3;3 x ˇ .! > 0/; < .0; 1/; .1 C ; !/; .1 ˇ2;˛2/ ˆ Ä ˇ ˆ ˇ ˆ 1 1;2 ! ˇ .0; 1/; . ; !/;.ˇ1; ˛1/ : x H3;3 x ˇ .! < 0/: .0; 1/; . ; !/; .1 ˇ2;˛2/

(b) If ˛1 <0and ˛2 <0, then for x>0 1E ! D0Ct ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ 1;2 ˇ.0; 1/; .1 ; !/; .ˇ1; ˛1/; .ˇ2; ˛2/ ˆx 1H x! ˇ .! > 0/; <ˆ 4;2 .0; 1/; .1 C ; !/ ˆ Ä ˇ ˆ ˇ ˆ 2;1 ˇ.0; 1/; . ; !/; .ˇ1; ˛1/; .ˇ2; ˛2/ :ˆx 1H x! ˇ .! < 0/: 4;2 .0; 1/; . ; !/

(c) If ˛1 >0and ˛2 >0, then for x>0 Á 1E ! D0Ct ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 " ˇ # ˇ ˆ ˇ .0; 1/; .1 ; !/ ˆ 1 1;2 ! ˇ ˆ x H2;4 x ˇ .! > 0/; <ˆ .0; 1/; .1 C ; !/; .1 ˇ1;˛1/; .1 ˇ2;˛2/ " ˇ # ˆ ˇ ˆ ˇ ˆ 1 2;1 ! ˇ .0; 1/; . ; !/ : x H2;4 x ˇ .! < 0/: .0; 1/; . ; !/; .1 ˇ1;˛1/; .1 ˇ2;˛2/

The right-sided Riemann–Liouville fractional derivative of the four-parametric Mittag-Lefﬂer function is given by the following formulas:

(a) If ˛1 <0and ˛2 >0, then for x>0 E ! Dt ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ 2;1 ! ˇ .0; 1/; .1 ; !/; .ˇ1; ˛1/ ˆ x H3;3 x ˇ .! > 0/; < .0; 1/; .1 ; !/; .1 ˇ2;˛2/ ˆ Ä ˇ ˆ ˇ ˆ 1;2 ! ˇ .0; 1/; . ; !/; .ˇ1; ˛1/ : x H3;3 x ˇ .! < 0/: .0; 1/; . C ; !/; .1 ˇ2;˛2/ 6.2 Mittag-Lefﬂer Functions with 2n Parameters 145

(b) If ˛1 <0and ˛2 <0, then for x>0 E ! Dt ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ 2;1 ˇ.0; 1/; .1 ; !/; .ˇ1; ˛1/; .ˇ2; ˛2/ ˆx H x! ˇ .! > 0/; <ˆ 4;2 .0; 1/; .1 ; !/ ˆ Ä ˇ ˆ ˇ ˆ 1;2 ˇ .0; 1/; . ; !/; .ˇ1; ˛1/; .ˇ2; ˛2/ :ˆx H x! ˇ .! < 0/: 4;2 .0; 1/; . C ; !/

(c) If ˛1 >0and ˛2 >0, then for x>0 E ! Dt ˛1;ˇ1I ˛2;ˇ2 .t / .x/ D 8 Ä ˇ ˆ ˇ ˆ 2;1 ! ˇ.0; 1/; .1 ; !/ ˆx H2;4 x ˇ .! > 0/; < .0; 1/; .1 ; !/; .1 ˇ1;˛1/; .1 ˇ2;˛2/ ˆ Ä ˇ ˆ ˇ ˆ 1;2 ! ˇ.0; 1/; . ; !/ :x H2;4 x ˇ .! < 0/: .0; 1/; . C ; !/;.1 ˇ1;˛1/; .1 ˇ2;˛2/

6.2 Mittag-Lefﬂer Functions with 2n Parameters

6.2.1 Deﬁnition and Basic Properties

R 2 2 C Consider the function deﬁned for ˛i 2 .˛1 CC˛n ¤ 0/ and ˇi 2 .i D 1; ;n2 N/ by

X1 k Q z C E ..˛; ˇ/nI z/ D n .z 2 /: (6.2.1) .˛j k C ˇj / kD0 j D1

When n D 1,(6.2.1) coincides with the Mittag-Lefﬂer function (4.1.1):

X1 zk E..˛;ˇ/ I z/ D E .z/ Á .z 2 C/; (6.2.2) 1 ˛;ˇ .˛kC ˇ/ kD0 and, for n D 2, with the four-parametric function (6.1.1):

X1 zk C E ..˛; ˇ/2I z/ D E˛1;ˇ1I˛2;ˇ2 .z/ Á .z 2 /: .˛1k C ˇ1/ .˛2k C ˇ2/ kD0 (6.2.3) 146 6 Multi-index Mittag-Lefﬂer Functions

First of all we prove that (6.2.1) under the condition ˛1 C ˛2 CC˛n >0is an entire function.

Theorem 6.3. Let n 2 N and ˛i 2 R, ˇi 2 C .i D 1; 2; ;n/be such that

2 2 ˛1 CC˛n ¤ 0; ˛1 C ˛2 CC˛n >0: (6.2.4)

Then E ..˛; ˇ/nI z/ is an entire function of z 2 C of order

1 D (6.2.5) .˛1 C ˛2 CC˛n/ and type

Â Ã ˛ Yn i ˛ CC˛ ˛1CC˛n D 1 n : (6.2.6) j˛ j iD1 i

G Rewrite (6.2.1) as the power series 2 3 1 X1 Yn k 4 5 E ..˛; ˇ/nI z/ D ckz ;ck D .˛j k C ˇj / : (6.2.7) kD0 j D1

According to the asymptotic property (A.1.27)wehave

jc j Yn Yn k ˛j ˛j ˛1C˛2CC˛n j˛j kj D j˛j j k .k !1/: jc C j k 1 j D1 j D1

Then, if ˛1 C ˛2 C ˛n >0, we see that R D1,whereR is the radius of convergence of the power series in (6.2.7). This means that E ..˛; ˇ/nI z/ is an entire function of z. We use (Appendix B,(B.2.3)and(B.2.4)) to evaluate the order and the type of (6.2.1). Applying Stirling’s formula for the Gamma function .z/ at inﬁnity and taking (6.2.7) into account, we have 2 3 Â Ã 1 Yn log D log 4 .˛ k C ˇ /5 c j j k j D1 Â Ã Â Ã Xn 1 Xn n 1 D ˛ k C ˇ log.˛ k/ ˛ k C log.2/CO .k !1/: j j 2 j j 2 k j D1 j D 1 6.2 Mittag-Lefﬂer Functions with 2n Parameters 147

Hence the following asymptotic estimate holds: Â Ã 1 log k log.k/.˛1 C ˛2 CC˛n/.k!1/: (6.2.8) ck

Thus, in accordance with (Appendix B,(B.2.3)), we obtain (6.2.5). Further, according to (6.1.12) we obtain the asymptotic estimate

Yn Yn 1 n=2 ˛j kCˇj ˛j k .˛j k C ˇj / .2/ .˛j k/ 2 e .k !1/: (6.2.9) j D1 j D1

By (6.2.7)and(6.2.9)wehave

Yn 1= 1=k 1= ˛j ˛j lim sup k jckj D lim sup k .j˛j jk/ e k!1 k!1 j D1 Yn Yn ˛1C˛2CC˛n ˛j 1= ˛j D e j˛j j D e j˛j j : j D1 j D1

Substituting this relation into (Appendix B,(B.2.4)) we have 0 1 0 1 1 ˛1CC˛n 1 Yn Yn D @ j˛ j˛j A D .˛ C ˛ CC˛ / @ j˛ j˛j A j 1 2 n j j D1 j D1

Â Ã ˛j Yn ˛ CC˛ ˛1CC˛n D 1 n ; j˛ j j D1 j which proves (6.2.6).F

Remark 6.4. For ˛j >0.j D 1; ;n/ the relations (6.2.5)and(6.2.6)were proved by Kilbas and Koroleva [KilKor05].

Remark 6.5. When n D 1, ˛1 D ˛>0and ˇ1 D ˇ 2 C, relations (6.2.5) and (6.2.6) yield the known order and type of the Mittag-Lefﬂer function E˛;ˇ.z/ in (4.1.1) (Sect. 4.1):

1 D ; D 1: (6.2.10) ˛

R C 2 2 Remark 6.6. When n D 2, ˛j 2 , ˇj 2 .j D 1; 2/ with ˛1 C ˛2 ¤ 0 and ˛1 C ˛2 >0,formulas(6.2.5)and(6.2.6) coincide with (6.1.8)and(6.1.9), respectively. 148 6 Multi-index Mittag-Lefﬂer Functions

6.2.2 Representations in Terms of Hypergeometric Functions

We consider the generalized Mittag-Lefﬂer function E ..˛; ˇ/nI z/ in (6.2.1) under the conditions of Theorem 6.3. First we give a representation of E ..˛; ˇ/nI z/ in terms of the generalized Wright hypergeometric function pq.z/ deﬁned in (Appendix H, (H.12)). By (A.1.17), .1/k D kŠ D .k C 1/ .k 2 N0/ and we can rewrite (6.2.1) in the form

X1 k Q .kC 1/ z C E ..˛; ˇ/nI z/ D n .z 2 /: (6.2.11) .ˇj C ˛j k/ kŠ kD0 j D1

This yields the following representation of E ..˛; ˇ/nI z/ via the generalized Wright hypergeometric function 1n.z/: 2 3 ˇ .1; 1/ ˇ 4 ˇ 5 E ..˛; ˇ/nI z/ D 1n ˇ z .z 2 C/: (6.2.12) .ˇ1;˛1/; ;.ˇn;˛n/

Next we consider the generalized Mittag-Lefﬂer function (6.2.1) with n 2 and ˛j D mj 2 N .j D 1; ;n/:

X1 k Q z E ..m; ˇ/nI z/ D n .mj k C ˇj / kD0 j D1 (6.2.13) X1 .1/ zk Q k C D n .z 2 /: .mj k C ˇj / kŠ kD0 j D1

ˇj According to (A.1.14) with z D k C , m D mj .j D 1; ;n/and (A.1.17)we mi have Ä Â Ã ˇj .mj k C ˇj / D mj k C mj Â Ã mYj 1 m kCˇ 1 ˇj C s D .2/.1mj /=2m j j 2 C k j m sD0 j Â ÃÂ Ã mYj 1 m kCˇ 1 ˇj C s ˇj C s D .2/.1mj /=2m j j 2 j m m sD0 j j k 2 3 Â Ã Â Ã mYj 1 mYj 1 ˇ 1 ˇj C s ˇj C s D mmi k 4.2/.1mj /=2m j 2 5 : j j m m sD0 j sD0 j k 6.2 Mittag-Lefﬂer Functions with 2n Parameters 149

ˇj Then applying (A.1.14) with z D , m D mj ,weget mi Â Ã mYj 1 m k ˇ C s .m k C ˇ / D m j .ˇ / j : j j j j m sD0 j k

Hence 0 1 k B C 1 X1 .1/ B z C 1 E ..m; ˇ/ I z/ D k B C : n Qn m 1 Á Qn Qn Qj @ mj A kŠ kD0 ˇj Cs .ˇj / mj mj j D1 j D1 sD0 k j D1

Therefore, we obtain the following representation of the 2n-parametric Mittag- Lefﬂer function via a generalized hypergeometric function in the case of positive integer ﬁrst parameters ˛j D mj 2 N .j D 1; ;n/

1 E ..m; ˇ/nI z/ D Qn (6.2.14) .ˇj / j D1 0 1 B C B ˇ ˇ C m 1 ˇ ˇ C m 1 z C F B1I 1 ;:::; 1 1 ;:::; n ;:::; n n I C : 1 m1C:::Cmn @ Qn A m1 m1 mn mn mj mj j D1

6.2.3 Integral Representations and Asymptotics

The 2n-parametric Mittag-Lefﬂer function can be introduced either in the form of a series (6.2.1) or in the form of a Mellin–Barnes integral Z 1 .s/.1 s/ E .z/ D .z/s ds.z ¤ 0/: (6.2.15) .˛; ˇ/n 2i Qn L .ˇj ˛j s/ j D1

For Re ˛1C:::C˛n >0one can choose the left loop L1 as a contour of integration in (6.2.15). Calculating this integral by using Residue Theory we immediately obtain the series representation (6.2.1). If ˛j >0I ˇj 2 R .j D 1;:::;n/, then the 2n-parametric Mittag-Leffler C function E..˛; ˇ/nI z/ is an entire function of the complex variable z 2 of finite order 150 6 Multi-index Mittag-Leffler Functions

1 D .˛1 C ˛2 C :::C ˛n/ (6.2.16) and type 2 3 Â Ã ˛j Yn ˛ C ˛ C :::C ˛ ˛1C˛2C:::C˛n D 4 1 2 n 5 : (6.2.17) ˛ j D1 j

This follows directly from the formulas for the order F and the type F of an entire function F.z/ represented as a power series

X1 k F.z/ D ck z ; kD0 namely,

k ln k F D lim sup 1 ; (6.2.18) k!1 ln jck j p 1= F 1= k . F e F / D lim sup.k jckj/; (6.2.19) k!1 and Stirling’s asymptotic formula for the Gamma function [Bat-1, 1.18(1)] Â Ã Â Ã 1 1 1 ln .z/ D z ln z z C ln.2a/ C O .z !1; jargzj 0there exists a positive r" such that ˇ ˇ ˇ ˇ E.˛; ˇ/n .z/ < expf. C "/jzj g; 8z; jzj >r": (6.2.21)

More precisely, the asymptotic behaviour of the function E.˛; ˇ/n .z/ can be described using the representation of the latter in terms of the H-function with special values of parameters (see Sect. 6.2.7 below) and asymptotic results for the H-function (see [KilSai04]).

6.2.4 Extension of the 2n-Parametric Mittag-Lefﬂer Function

An extension of the 2n-parametric Mittag-Lefﬂer function is given by the representation 6.2 Mittag-Lefﬂer Functions with 2n Parameters 151 Z 1 .s/.1 s/ E..˛; ˇ/ I z/ D .z/sds.z ¤ 0/; (6.2.22) n 2i Qn L .ˇj ˛j s/ j D1 where the right loop L D LC1 is chosen as the contour of integration L. By using Stirling’s asymptotic formula for the Gamma function

j.xC iy/j D .2/1=2jxjx1=2exŒ1sign.x/y=2 .x; y 2 RIjxj!1/; (6.2.23) one can show directly that with the above choice of the integration contour the integral (6.2.22) is convergent for all values of parameters ˛1;:::;˛n 2 C, ˇ1;:::;ˇn 2 C such that Re ˛1 C :::C ˛n <0(cf., e.g., [KilKor05]). Under these conditions (the choice of contour and assumption on the parameters) the integral (6.2.22) can be calculated by using Residue Theory. This gives the following Laurent series representation of the extended 2n-parametric Mittag- Lefﬂer function: let ˛j ;ˇj 2 C .j D 1:::n/, z 2 C (z ¤ 0) with Re ˛1 C :::C L L E ˛n <0and D C1, then the function ..˛; ˇ/nI z/ has the Laurent series representation

X1 d Yn 1 E..˛; ˇ/ I z/ D k ;dD : n kC1 k (6.2.24) z .˛j k ˛j C ˇj / kD0 j D1

The series in (6.2.24) is convergent for all z 2 C nf0g. Convergence again follows from the asymptotic properties of the Gamma function which yield the relation

Pn Yn jd j Re .˛j / k Re .˛j / Im .˛j /arg.˛j k/ j D1 j˛j j e k .k !1/: jd C j k 1 j D1

By using the series representation of the extended 2n-parametric Mittag-Lefﬂer function it is not hard to obtain an asymptotic formula for z !1. Namely, if ˛j ;ˇj 2 C .j D 1;:::;n/, z 2 C (z ¤ 0)andRe.˛1 C:::C˛n/<0, with contour of integration in (6.2.22) chosen as L D LC1, then for any N 2 N we have for z !1the asymptotic representation Ä Â Ã XN 1 1 E ..˛; ˇ/nI z/ D Qn 1 C O .z !1/: kC1 z kD0 .˛j k ˛j C ˇj /z j D1

The main term of this asymptotics is equal to 152 6 Multi-index Mittag-Lefﬂer Functions

Ä Â Ã Yn 1 1 E..˛; ˇ/ I z/ D 1 C O .z !1/: n .˛ C ˇ / z j D1 j j

The asymptotics at z ! 0 is more complicated. It can be derived by using the relations of the extended 2n-parametric Mittag-Leffler function with the generalized Wright function and the H-functions (see Sect. 6.2.5 below) and the asymptotics of the latter presented in [KilSai04]. E Another possible way to get the asymptotics of ..˛; ˇ/nI z/ for z ! 0 is to use the following. If ˛j ;ˇj 2 C .j D 1:::n/, z 2 C (z ¤ 0), Re ˛1 C:::C˛n <0, L D LC1, then the extended 2n-parametric Mittag-Leffler function can be presented in terms of the “usual” 2n-parametric Mittag-Leffler function: Â Ã 1 1 E..˛; ˇ/ I z/ D E .˛; ˇ ˛/ I : (6.2.25) n z n z

6.2.5 Relations to the Wright Function and to the H-Function

In this section we present some formulas representing the 2n-parametric Mitttag- E Lefﬂer function E..˛; ˇ/nI z/ and its extension ..˛; ˇ/nI z/ in terms of the generalized Wright function pq and the H-function. E For shortness we use the same notation ..˛; ˇ/nI z/ for the 2n-parametric Mitttag-Lefﬂer function and for its extension. These functions differ in values of the parameters ˛i and in the choice of the contour of integration L in their Mellin– Barnes integral representation. For real values of the parameters ˛j 2 R and complex ˇj 2 C .j D 1;:::;n/ the following representations hold: Pn 1. If ˛j >0;L D L1,then j D1 Ä ˇ ˇ E .1; 1/ ˇ ..˛; ˇ/nI z/ D 1n ˇz I (6.2.26) .ˇ1;˛1/;:::;.ˇn;˛n/

Pn 2. If ˛j <0;L D LC1,then j D1 Ä ˇ ˇ E 1 .1; 1/ ˇ1 ..˛; ˇ/nI z/ D 12 ˇ : (6.2.27) z .ˇ1 ˛1; ˛1/;:::;.ˇn ˛n; ˛n/ z

The above representations can be obtained by comparing the series representation of the corresponding functions. In the case (6.2.27) one can also use the relation (6.2.25). 6.2 Mittag-Lefﬂer Functions with 2n Parameters 153

In the same manner one can obtain the following representations of the 2n- parametric Mittag-Lefﬂer function and its extension in terms of the H-function:

1. If ˛j >0.j D 1;:::;n/,andL D L1,then Ä ˇ ˇ E 1;1 .0; 1/ ˇ ..˛; ˇ/nI z/ D H1;nC1 ˇz I (6.2.28) .0; 1/.1 ˇ1;˛1/;:::;.1 ˇn;˛n/

2. If ˛j >0.j D 1;:::;p;p < n/, ˛j <0.j D p C 1;:::;n/, and either Pn Pn ˛j >0;L D L1,or ˛j <0;L D LC1,then j D1 j D1 Ä ˇ ˇ E 1;1 .0; 1/.ˇpC1; ˛pC1/:::.ˇn; ˛n/ ˇ ..˛; ˇ/nI z/ D HnpC1;pC1 ˇz I .0; 1/.1 ˇ1;˛1/;:::;.1 ˇp;˛p/ (6.2.29)

3. If ˛j <0.j D 1;:::;p;p < n/, ˛j >0.j D p C 1;:::;n/, and either Pn Pn ˛j >0;L D L1,or ˛i <0;L D LC1,then j D1 j D1 Ä ˇ ˇ E 1;1 .0; 1/.ˇ1; ˛1/:::.ˇp; ˛p / ˇ ..˛; ˇ/nI z/ D HpC1;npC1 ˇz I .0; 1/.1 ˇpC1;˛pC1/;:::;.1 ˇn;˛n/ (6.2.30)

4. If ˛j <0, .j D 1;:::;n/and L D LC1,then Ä ˇ ˇ 1;1 .0; 1/.ˇ1; ˛1/;:::;.ˇn; ˛n/ ˇ E..˛; ˇ/ I z/ D H ˇz : (6.2.31) n nC1;1 .0; 1/

6.2.6 Integral Transforms with the Multi-parametric Mittag-Lefﬂer Functions

Here we consider only the case when the parameters ˛i in the definition of the 2n- parametric Mittag-Leffler function and its extension are real numbers. Since the 2n-parametric Mittag-Leffler function is related to the generalized Wright function and to the H-function with special values of parameters (see Sect. 6.2.5), one can use (6.1.34)or(6.1.39) to define the Mellin transform of the function E..˛; ˇ/nI z/ and of its extension E..˛; ˇ/nI z/. Now we present a few results on integral transforms with the 2n-parametric function in the kernel. The transforms are defined by the formula

Z1 E .E.˛; ˇ/nf/.x/D ..˛; ˇ/nIxt/f .t/dt.x>0/; (6.2.32) 0 154 6 Multi-index Mittag-Leffler Functions with the 2n-parametric Mittag-Leffler function in the kernel. These transforms are special cases of more general H-transforms (see Sect. 6.1.7). This can be seen from the definition of the H-transforms (6.1.31) and the following formulas which relate E.˛; ˇ/n-transforms to H-transforms under different assumptions on the parameters.

1. Let ˛j >0.j D 1;:::;n/, L D L1,then

Z1 Ä ˇ ˇ 1;1 ˇ.0; 1/ .E.˛; ˇ/nf /.x/ D H1;nC1 xtˇ f.t/dt: .0; 1/; .1 ˇ1;˛1/;:::;.1 ˇn;˛n/ 0 (6.2.33)

2. Let ˛j >0.j D 1;:::;p;p < n/, ˛j <0.j D p C 1;:::;n/ and either Pn Pn ˛j >0;L D L1 or ˛j <0;L D LC1,then j D1 j D1

.E.˛; ˇ/nf /.x/ Z1 Ä ˇ ˇ 1;1 ˇ .0; 1/; .ˇpC1; ˛pC1/;:::;.ˇn; ˛n/ D HnpC1;pC1 xtˇ f.t/dt: .0; 1/; .1 ˇ1;˛1/;:::;.1 ˇp;˛p / 0 (6.2.34)

3. Let ˛j <0.j D 1;:::;p;p < n/, ˛j >0.j D p C 1;:::;n/ and either Pn Pn ˛j >0;L D L1,or ˛j <0;L D LC1,then j D1 j D1

.E.˛; ˇ/nf/ Z1 Ä ˇ ˇ 1;1 ˇ .0; 1/; .ˇ1; ˛1/;:::;.ˇp; ˛p/ D HpC1;npC1 xtˇ f.t/dt: .0; 1/; .1 ˇpC1;˛pC1/;:::;.1 ˇn;˛n/ 0 (6.2.35)

4. Let ˛j <0,.j D 1;:::;n/and L D LC1,then

Z1 Ä ˇ ˇ 1;1 ˇ .0; 1/; .ˇ1; ˛1/;:::;.ˇn; ˛n/ .E.˛; ˇ/ f /.x/ D H xtˇ f.t/dt: n nC1;1 .0; 1/ 0 (6.2.36) The constant a in the deﬁnition of the H-function takes different values in the above cases: 6.2 Mittag-Lefﬂer Functions with 2n Parameters 155

Pn Pp Pn .1/ a D 2 ˛j I .2/ a D 2 ˛j C ˛j I j D1 j D1 j DpC1 Pp Pn Pn .3/ a D 2 C ˛j ˛j I .4/ a D 2 C ˛j I j D1 j DpC1 j D1 and the constants ; ; ˛; ˇ take the same values in all four cases:

Xn n Xn D ˛ I D ˇ I ˛ D 0I ˇ D 1: j 2 j j D1 j D1

We present results on E.˛; ˇ/n-transforms for two essentially different cases, namely for the case when all ˛j are positive, and for the case when some of them are negative.

A. Let 0<<1, ˛j >0.j D 1;:::;n/, ˇj 2 C .j D 1;:::;n/ be such that Pn Pn Pn n either 0< ˛j <2or ˛j D 2 and 2 C Re ˇj 2 C 2 . j D1 j D1 j D1 L L (a) There exists an injective mapping (transform) E .˛; ˇ/n 2 Œ ;2; 1;2 such that the equality

.s/.1 s/ M M E .˛; ˇ/nf .s/ D Qn . f/.1 s/; .Re s D 1 / .ˇj ˛j s/ j D1 (6.2.37)

holds for any f 2 L;2. Pn Pn n If ˛j D 2, 2 C Re ˇj D 2 C 2 and j D1 j D1

ˇ C k ˇ C l s ¤ 1 ;:::; s ¤ n .k; l D 0; 1; 2; / for Re s D 1 ; ˛1 ˛n (6.2.38)

L L then the mapping E .˛; ˇ/n is bijective from ;2 onto 1;2: (b) For any f; g 2 L;2 the following integration by parts formula holds:

Z1 Z1 f.x/ E .˛; ˇ/ng .x/dx D E .˛; ˇ/nf .x/g.x/dx: (6.2.39) 0 0

L C (c) If f 2 ;2, 2 , h>0, then the value E .˛; ˇ/nf can be represented in the form: 156 6 Multi-index Mittag-Lefﬂer Functions

d E.˛; ˇ/ f .x/ D hx1.C1/=h x.C1/=h n dx Z1 Ä ˇ ˇ 1;2 ˇ .; h/; .0; 1/ H2;nC2 xtˇ f.t/dt; .0; 1/; .1 ˇ1;˛1/;:::;.1 ˇn;˛n/; . 1; h/ 0 (6.2.40)

when Re ./ > .1 /h 1,or

d E.˛; ˇ/ f .x/ Dhx1.C1/=h x.C1/=h n dx 2 3 Z1 ˇ ˇ .0; 1/; .;h/ 2;1 4 ˇ 5 H2;nC2 xtˇ f.t/dt; 0 . 1; h/; .0; 1/; .1 ˇ1;˛1/;:::;.1 ˇn;˛n/ (6.2.41)

when Re ./ < .1 /h 1. (d) The mapping E .˛; ˇ/n does not depend on in the following sense: if two values of the parameter 0<1;2 <1and the corresponding mappings L L E .˛; ˇ/nI1, E .˛; ˇ/nI2 are deﬁned on the spaces T1;2, 2;2, respectively, L L then E .˛; ˇ/nI1f D E .˛; ˇ/nI2f for all f 2 1;2 2;2. Pn Pn Pn (e) If f 2 L;2 and either 0< ˛j <2or ˛j D 2 and 2 C Re ˇj j D1 j D1 j D1 n 2 C 2 , then the mapping (transform) E .˛; ˇ/n coincides with the transform E.˛; ˇ/n given by the formula (6.2.32), i.e. E .˛; ˇ/nf D E.˛; ˇ/nf; 8f 2 L;2.

B. Let 0<<1, ˛j >0.j D 1;:::;jp;j p < n/ and ˛j <0.j D p C Pp Pn 1;j :::;n/, ˇj 2 C .j D 1;:::;n/, be such that either 2 ˛j C ˛j > j D1 j DpC1 Pp Pn Pn Pn n 0 or 2 ˛i C ˛j D 0 and .1 / ˛j C 2 Ä ˇi . j D1 j DpC1 j D1 j D1 L L (a) There exists an injective mapping (transform) E .˛; ˇ/n 2 Œ ;2; 1;2 such that the equality (6.2.37) holds for any f 2 L;2. Pp Pn Pn Pn n If 2 ˛j C ˛j D 0, .1/ ˛j C 2 D ˇj and the parameter j D1 j DpC1 j D1 j D1 s (which determines the line of integration for the inverse Mellin transform L in (6.2.37)) satisfies (6.2.38), then the mapping E .˛; ˇ/n is bijective from ;2 onto L1;2: (b) For any f; g 2 L;2 the integration by parts formula (6.2.39) is satisfied. L C (c) If f 2 ;2, 2 , h>0, then the value E .˛; ˇ/nf can be represented in the form: 6.2 Mittag-Leffler Functions with 2n Parameters 157

d E.˛; ˇ/ f .x/ D hx1.C1/=h x.C1/=h n dx 2 3 Z1 ˇ .; h/; .0; 1/; .ˇ ; ˛ /;:::;.ˇ ; ˛ / 6 ˇ pC1 pC1 n n 7 1;2 4 ˇ 5 HnpC2;pC2 xtˇ f.t/dt; 0 .0; 1/; .1 ˇ1;˛1/;:::;.1 ˇp;˛p/; . 1; h/ (6.2.42)

when Re ./ > .1 /h 1,or

d E.˛; ˇ/ f .x/ Dhx1.C1/=h x.C1/=h n dx Z1 " ˇ # ˇ 2;1 ˇ .0; 1/; .ˇpC1; ˛pC1/;:::;.ˇn; ˛n/; .;h/ HnpC2;pC2 xtˇ f.t/dt; . 1; h/; .0; 1/; .1 ˇ1;˛1/;:::;.1 ˇp;˛p/ 0 (6.2.43)

when Re ./ < .1 /h 1. (d) The mapping E .˛; ˇ/n does not depend on in the following sense: if 0<1;2 <1and the mappings E .˛; ˇ/nI1, E .˛; ˇ/nI2 are deﬁned on L L the spacesT1;2, 2;2, respectively, then E .˛; ˇ/nI1f D E .˛; ˇ/nI2f for all L L f 2 1;2 2;2. Pp Pn Pp Pn (e) If f 2 L;2 and either 2 ˛i C ˛i >0or 2 ˛i C ˛i D 0 and iD1 iDpC1 iD1 iDpC1 Pn Pn n .1 / ˛i C 2 Ä ˇi , then the mapping (transform) E .˛; ˇ/n coincides iD1 iD1 with the transform E.˛; ˇ/n given by the formula (6.2.32), i.e. E .˛; ˇ/nf D L E.˛; ˇ/nf; 8f 2 ;2.

6.2.7 Relations to the Fractional Calculus

In this subsection we present a few formulas relating the 2n-parametric Mittag- Leffler function (with different values of the parameters ˛j ) to the left- and right-sided Riemann–Liouville fractional integral and derivative. For shortness we E use the same notation ..˛; ˇ/nI z/ for the 2n-parametric Mitttag-Leffler function and for its extension. These functions differ in values of the parameters ˛i andinthe choice of the contour of integration L in their Mellin–Barnes integral representation. The results in this subsection are obtained (see [KiKoRo13]) by using known formulas for the fractional integration and differentiation of power-type functions (see [SaKiMa93, (2.44) and formula 1 in Table 9.3]). 158 6 Multi-index Mittag-Leffler Functions

Let ˛j 2 R;˛j ¤ 0.jD 1;:::;n/; ˛1 < 0;:::;˛l <0;˛lC1 > 0;:::;˛n >0 .1Ä l Ä n/ and let the contour L be given by one of the following:

L D L1; if ;˛1 C :::C ˛n >0;or L D LC1; if ;˛1 C :::C ˛n <0:

Let ; ; 2 C be such that Re./ > 0,Re. / > 0 and ! 2 R;.!¤ 0/. Then the following assertions are true. A. Calculation of the left-sided Riemann–Liouville fractional integral. (a) If !>0, then for x>0 1E ! I0Ct ..˛; ˇ/nI t / .x/ Ä ˇ ˇ C1 1;2 ! ˇ .0; 1/; .1 ; !/; .ˇj ; ˛j /1; l D x H2Cl;2Cnl x ˇ : .0; 1/; .1 ; !/; .1 ˇj ;˛j /lC1;n (6.2.44)

(b) If !<0, then for x>0 1E ! I0Ct ..˛; ˇ/nI t / .x/ Ä ˇ ˇ C1 2;1 ! ˇ .0; 1/; . C ; !/;.ˇj ; ˛j /1;l D x H2Cl;2Cnl x ˇ : .0; 1/; . ; !/; .1 ˇj ;˛j /lC1;n (6.2.45)

B. Calculation of the right-sided Liouville fractional integral. (a) If !>0, then for x>0 E ! It ..˛; ˇ/nI t / .x/ Ä ˇ ˇ 1;2 ! ˇ .0; 1/; .1 C ; !/; .ˇj ; ˛j /1;l D x H2Cl;2Cnl x ˇ : .0; 1/; .1 ; !/; .1 ˇj ;˛j /lC1;n (6.2.46)

(b) If !<0, then for x>0 E ! It ..˛; ˇ/nI t / .x/ Ä ˇ ˇ 2;1 ! ˇ .0; 1/; . ; !/; .ˇj ; ˛j /1;l D x H2Cl;2Cnl x ˇ : .0; 1/; . ; !/;.1 ˇj ;˛j /lC1;n (6.2.47) 6.3 Historical and Bibliographical Notes 159

C. Calculation of the left-sided Riemann–Liouville fractional derivative. (a) If !>0, then for x>0 1E ! D0Ct ..˛; ˇ/nI t / .x/ Ä ˇ ˇ 1 1;2 ! ˇ .0; 1/; .1 ; !/; .ˇj ; ˛j /1;l D x H2Cl;2Cnl x ˇ : .0; 1/; .1 C ; !/; .1 ˇj ;˛j /lC1;n (6.2.48)

(b) If !<0, then for x>0 1E ! D0Ct ..˛; ˇ/nI t / .x/ Ä ˇ ˇ 1 2;1 ! ˇ .0; 1/; . ; !/;.ˇj ; ˛j /1;l D x H2Cl;2Cnl x ˇ : .0; 1/; . ; !/; .1 ˇj ;˛j /lC1;n (6.2.49)

D. Calculation of the right-sided Liouville fractional derivative. (a) If !>0, then for x>0 E ! Dt ..˛; ˇ/nIt / .x/ Ä ˇ ˇ 2;1 ! ˇ .0; 1/; .1 ; !/; .ˇj ; ˛j /1;l D x H2Cl;2Cnl x ˇ : .0; 1/; .1 ; !/; .1 ˇj ;˛j /lC1;n (6.2.50)

(b) If !<0then for x>0 E ! Dt ..˛; ˇ/nI t / .x/ Ä ˇ ˇ 1;2 ! ˇ .0; 1/; . ; !/; .ˇj ; ˛j /1;l D x H2Cl;2Cnl x ˇ : .0; 1/; . C ; !/;.1 ˇj ;˛j /lC1;n (6.2.51)

6.3 Historical and Bibliographical Notes

In recent decades, starting from the eighties in the last century, we have observed a rapidly increasing interest in the classical Mittag-Lefﬂer function and its generalizations. This interest mainly stems from their use in the explicit solution of certain classes of fractional differential equations (especially those modelling processes of fractional relaxation, oscillation, diffusion and waves). 160 6 Multi-index Mittag-Lefﬂer Functions

R 2 2 C For ˛1;˛2 2 .˛1 C˛2 ¤ 0/ and ˇ1;ˇ2 2 the four-parametric Mittag-Lefﬂer function is deﬁned by the series

X1 k z C E˛1;ˇ1I˛2;ˇ2 .z/ Á .z 2 /: (6.3.1) .˛1k C ˇ1/ .˛2k C ˇ2/ kD0

For positive ˛1 >0, ˛2 >0and real ˇ1;ˇ2 2 R it was introduced by Djrbashian [Dzh60]. When ˛1 D ˛; ˇ1 D ˇ and ˛2 D 0, ˇ2 D 1, it coincides with the Mittag-Lefﬂer function E˛;ˇ.z/:

X1 zk E .z/ D E .z/ Á .z 2 C/: (6.3.2) ˛;ˇI0;1 ˛;ˇ .˛kC ˇ/ kD0 Generalizing the four-parametric Mittag-Lefﬂer function, Al-Bassam and Luchko [Al-BLuc95] introduced the Mittag-Lefﬂer type function

X1 k z N E..˛; ˇ/nI z/ D Qn .n 2 / (6.3.3) kD0 .˛j k C ˇj / j D1 with 2n real parameters ˛j >0I ˇj 2 R .j D 1;:::;n/ and with complex z 2 C: In [Al-BLuc95] an explicit solution to a Cauchy type problem for a fractional differential equation is given in terms of (6.3.3). The theory of this class of functions was developed in a series of articles by Kiryakova et al. [Kir99,Kir00,Kir08,Kir10a, Kir10b]. Among the results dealing with multi-index Mittag-Leffler functions we point out those which show their relation to a general class of special functions, namely to Fox’s H-function. Representations of the multi-index Mittag-Leffler functions as special cases of the H-function and the generalized Wright function are obtained in [AlKiKa02, Kir10b]. Relations of such multi-index functions to the Erdelyi–Kober (E-K) operators of fractional integration are discussed. The novel Mittag-Leffler functions are also used as generating functions of a class of so-called Gelfond– Leontiev (G-L) operators of generalized differentiation and integration. Laplace- type integral transforms corresponding to these G-L operators are considered too. The multi-index Mittag-Leffler functions (6.3.3) can be regarded as “fractional index” analogues of the hyper-Bessel functions, and the multiple Borel–Dzrbashjan integral transforms (being H-transforms) as “fractional index” analogues of the Obrechkoff transforms (being G-transforms). In a more precise terminology, these are Gelfond–Leontiev (G-L) operators of generalized differentiation and integration with respect to the entire function, a multi-index generalization of the Mittag-Leffler function. Fractional multi-order integral equations

y.z/ Ly.z/ D f.z/; (6.3.4) 6.3 Historical and Bibliographical Notes 161 and initial value problems for the corresponding fractional multi-order differential equations

Dy.z/ y.z/ D f.z/ (6.3.5) are considered. From the known solution of the Volterra-type integral equation with m-fold integration, via a Poisson-type integral transformation P as a transformation (transmutation) operator, the corresponding solution of the integral equation (6.3.4) is found. Then a solution of the fractional multi-order differential equation (6.3.5) comes out, in an explicit form, as a series of integrals involving Fox’s H-functions. For each particularly chosen right-hand side function f.z/, such a solution can be evaluated as an H-function. Special cases of the equations considered here lead to solutions in terms of the Mittag-Lefﬂer, Bessel, Struve, Lommel and hyper-Bessel functions, and some other known generalized hypergeometric functions. In [Kir10b](seealso[Kir10a]) a brief description of recent results by Kiryakova at al. on an important class of “Special Functions of Fractional Calculus” is presented. These functions became important in solutions of fractional order (or multi-order) differential and integral equations, control systems and reﬁned mathematical models of various physical, chemical, economical, management and bioengineering phenomena. The notion “Special Functions of Fractional Calculus” essentially means the Wright generalized hypergeometric function pq , as a special case of the Fox H-function. A generalization of the Prabhakar type function was given by Shukla and Prajapati [ShuPra07]:

X1 ./ zn E; .z/ D E.˛;ˇI ; I z/ D n .n 2 N/: (6.3.6) ˛Iˇ .˛nC ˇ/ kD0 In [SriTom09] the existence of the function (6.3.6) for a wider set of parameters was shown. The definition (6.3.6) was combined with (6.3.3)in[SaxNis10](seealso [Sax-et-al10]). As a result, the following definition of the generalized multi-index Mittag-Leffler function appears:

X1 ./ zn E; .z/ D E ..˛ ;ˇ /m I z/ D n .m 2 N/: .˛j ;ˇj /m ; j j j D1 Qm nD0 .˛j n C ˇj / j D1 (6.3.7) On the basis of the above described results a special H-transform was constructed in [Al-MKiVu02](seealso[KilSai04]). This transform turns out to exhibit many properties similar to the Laplace transform. Moreover, the inverse transform and the operational calculus, which is based on it, are related to the recently introduced multi-index Mittag-Lefﬂer function. Some basic operational properties, complex and real inversion formulas, as well as a convolution theorem, have been derived. 162 6 Multi-index Mittag-Lefﬂer Functions

Further generalizations of the Mittag-Leffler functions have been proposed recently in [Pan-K11, Pan-K12]. The 3m-parametric Mittag-Leffler functions generalizing the Prabhakar three parametric Mittag-Leffler function are introduced by the relation

X1 k . /;m . / :::. / z E j D 1 k m k ; (6.3.8) .˛j /;.ˇj / .˛1k C ˇ1/:::.˛mk C ˇm/ kŠ kD0 where ./k is the Pochhammer symbol, ˛j ;ˇj ;j 2 C;j D 1;:::;m,Re˛j >0. These are entire functions for which the order and the type have been calculated. Representations of the 3m-parametric Mittag-Leffler functions as generalized Wright functions and Fox H-functions have been obtained. Special cases of novel special functions have been discussed. Composition formulas with Riemann– Liouville fractional integrals and derivatives have been given. Analogues of the Cauchy–Hadamard, Abel, Tauber and Hardy–Littlewood theorems for the three multi-index Mittag-Leffler functions have also been presented. The extension of the Mittag-Leffler function to a wider set of parameters by using Mellin–Barnes integrals was realized in a series of papers [KilKor05, KilKor06a, KilKor06b,KilKor06c] (see also the paper [Han-et-al09]). The method of extension of different special functions having a representation via a Mellin–Barnes integral has been developed recently. First of all we have to mention the paper [Han-et-al09]. In this paper the Mittag- Leffler function E˛;ˇ.z/ of two parameters for negative values of the parameter ˛ is introduced. This definition is based on an analytic continuation of the integral representation Z 1 t ˛ˇet E .z/ D dt; z 2 C; (6.3.9) ˛;ˇ 2i t ˛ z Ha where the path of integration Ha is the Hankel path, a loop starting and ending at 1, and encircling the disk jtjÄjzj1=˛ counterclockwise in the positive sense: 0,Eq.(6.3.9) is equivalent to the infinite series representation for the Mittag-Leffler function. This is accomplished by expanding the integrand in Eq. (6.3.9)inpowersofz and integrating term-by-term, making use of Hankel’s contour integral for the reciprocal of the Gamma function (see, e.g., [NIST]). To find a defining equation for E˛;ˇ.z/, the integral representation of the Mittag- Leffler function is rewritten as Z 1 et E .z/ D dt; z 2 C: (6.3.10) ˛;ˇ 2i t ˇ zt ˛Cˇ Ha

By expanding a part of the integrand in Eq. (6.3.10) in partial fractions 6.4 Exercises 163

1 1 1 D ; t ˇ zt ˛Cˇ t ˇ t ˇ z1t ˛Cˇ substituting it into (6.3.10) we get another representation Z Z 1 et 1 et E .z/ D dt dt; z 2 C nf0g: (6.3.11) ˛;ˇ 2i t ˇ 2i t ˇ z1t ˛Cˇ Ha Ha

Thus we arrive at the following deﬁnition of the Mittag-Lefﬂer function E˛;ˇ.z/ for negative values of the parameter ˛: Â Ã 1 1 E .z/ D E : (6.3.12) ˛;ˇ .ˇ/ ˛;ˇ z

General properties of E˛;ˇ.z/ were discussed and many of the common relationships between Mittag-Leffler functions of negative ˛ were compared with their analogous relationships for positive ˛. The special case of E˛.z/ has found application in the analysis of the transient kinetics of a two-state model for anomalous diffusion (see [Shu01]). The Mittag-Leffler functions with negative ˛ and the results of this work are likely to become increasingly important as fractional- order differential equations find more applications. This method of extension was also applied recently in [Kil-et-al12]forthe generalized hypergeometric functions. This paper is devoted to the study of a certain function pFqŒz Á pFq a1; ;ap I b1; ;bqI z (with complex z ¤ 0 and complex parameters aj .j D 1; p/ and bj .j D 1; ;q/), represented by the Mellin–Barnes integral. Such a function is an extension of the classical generalized hypergeometric function pFq Œa1; ;apI b1; ;bqI z defined for all complex z 2 C when pqC1.The series representations and the asymptotic expansions of pFqŒz at infinity and at the origin are established. Special cases have been considered.

6.4 Exercises

6.4.1. Let I0C be the left-sided Riemann–Liouville fractional integral and E ˛1;ˇ1I˛2;ˇ2 .z/ be either the four-parametric Mittag-Lefﬂer function or its extension. In the case ˛1 >0;˛2 <0calculate the following compositions 1E ! .a/ I0Ct ˛1;ˇ1I˛2;ˇ2 .t / .x/ .!; > 0; 0 < x Ä d 0; 0 < x Ä d

6.4.2. Let D0C be the left-sided Riemann–Liouville fractional derivative and E ˛1;ˇ1I˛2;ˇ2 .z/ be either the four-parametric Mittag-Lefﬂer function or its extension. In the case ˛1 >0;˛2 <0calculate the following compositions 1E ! .a/ D0Ct ˛1;ˇ1I˛2;ˇ2 .t / .x/ .!; > 0; 0 < x Ä d 0; 0 < x Ä d

6.4.3. In the case of positive integer ˛1 D m1 and ˛2 D m2 represent the four-

parametric Mittag-Lefﬂer function Em1;ˇ1Im2;ˇ2 .z/ in term of a generalized hypergeometric function pFq with appropriate p; q. 6.4.4. Prove that the Laplace transform of a hyper-Bessel type generalized hypergeometric function 0m is related to the 2n-parametric Mittag-Lefﬂer function as follows [KirLuc10, p. 601] Â Ä Ã Â Ã L 1 1 0m .s/ D E.˛1;ˇ1/;:::;.˛n;ˇn/ : .ˇ1;˛1/;:::;.ˇn;˛n/ s s Ä Qn 6.4.5. Let I .i /;.ıi /f.z/ D I .i /;.ıi / f.z/ be the generalized fractional integral .ˇi /;n .ˇi /;n iD1 of multi-order, where [KirLuc10, pp. 603–604]

Z1 1 I ;ı f.z/ D .1 /ı1 f.z/ 1=ˇ d .ı;ˇ>0;2 R/ ˇ .ı/ 0

is the Erdelyi–Kober fractional integral. Prove the following formulas Á 1 .z/ I .ˇi 1/;.˛i /E .z/ D E .z/ ; .1=˛i /;n .˛i /;.ˇi / .˛i /;.ˇi / Qn .ˇi / iD1 Á 1 D.ˇi 1˛i /;.˛i /E .z/ D .z/E .z/ C : .1=˛i /;n .˛i /;.ˇi / .˛i /;.ˇi / Qn .ˇi ˛i / iD1 Chapter 7 Applications to Fractional Order Equations

In this chapter we consider a number of integral equations and differential equations (mainly of fractional order). In representations of their solution, the Mittag-Lefﬂer function, its generalizations and some closely related functions are used.

7.1 Fractional Order Integral Equations

7.1.1 The Abel Integral Equation

Let us consider the Abel integral equation of the ﬁrst kind in the classical setting (i.e. for 0<˛<1) Z 1 t u./ D f.t/;0<˛<1; 1˛ d (7.1.1) .˛/ 0 .t / where f.t/ is a given function. We easily recognize that this equation can be expressed in terms of a fractional integral, i.e. ˛ I0C u .t/ D f.t/;0<˛ <1: (7.1.2)

Consequently it is solved in terms of a fractional derivative: ˛ u.t/ D D0C f .t/ : (7.1.3)

To this end we need to recall the deﬁnition of a fractional integral and derivative and the property D˛ I ˛ D I. Certainly, the solution (7.1.3) exists if the right-hand side satisﬁes certain conditions (see the discussion below).

© Springer-Verlag Berlin Heidelberg 2014 165 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__7 166 7 Applications to Fractional Order Equations

A formal solution can be obtained for Eq. (7.1.1) (or, what is equivalent, to Eq. (7.1.2)) for any positive value of the parameter ˛. Let us consider Eq. (7.1.2) with an arbitrary positive parameter ˛.Letm D N m˛ Œ˛,i.e.m 1<˛Ä m; m 2 : We apply operator IaC to both sides of Eq. (7.1.2)

m˛ ˛ m˛ I0C I0Cu D I0C f: (7.1.4)

Then using the semigroup property of the fractional integral operators we get m m˛ .I u/.t/ D I0C f .t/; (7.1.5) where I m is the m-times repeated integral. Since m-times differentiation Dm is the left-inverse operator to I m we obtain ﬁnally m m˛ ˛ u.t/ D D I0C f .t/ DW D0C f .t/ ; (7.1.6) with

˛ m m˛ D0C D D I0C :

Thus, the solution to Eq. (7.1.1)hastheform(7.1.6), if it exists. The problem is that ˛ Eqs. (7.1.2)and(7.1.4) are not equivalent since the operator D0C is only left-inverse ˛ to I0C, but not right-inverse. Solvability conditions for the Abel integral equation of the ﬁrst kind are given in [SaKiMa93, pp. 31, 39]: Let ˛ be a positive non-integer (i.e. m 1<˛

Zx 1 f.t/ m fm˛.x/ D dt 2 AC .Œ0; b/ (7.1.7) .m f˛g/ .x t/f˛g a and

.k/ fm˛.0/ D 0; k D 0;1;:::;m 1: (7.1.8)

Condition (7.1.7) means that the function fm˛ is .m 1/-times differentiable and .m1/ the .m 1/-th derivative fm˛ is absolutely continuous on the interval Œ0; b. An alternative approach is to use the Laplace transform for the solution of the Abel integral equation (7.1.1). Note that the operator in the left-hand side of (7.1.2) can be written in the form of the Laplace convolution. Let us for simplicity consider only the case m D 1. 7.1 Fractional Order Integral Equations 167

˛ ˛ 1 1 The relation I u.t/ D ˚˛.t/ u.t/ Qu.s/=s holds with ˚˛.t/ D .˛/ t 1˛ : This gives

uQ.s/ D f.s/Q H)u Q .s/ D s˛ f.s/:Q (7.1.9) s˛ Now one can choose two different ways to get the inverse Laplace transform from (7.1.9), according to the standard rules. (a) Writing (7.1.9)as " # f.s/Q uQ.s/ D s ; (7.1.10) s1˛

we obtain Z 1 d t f./ u.t/ D : ˛ d (7.1.11) .1 ˛/ dt 0 .t /

(b) On the other hand, writing (7.1.9)as

1 f.0C/ uQ.s/ D Œs f.s/Q f.0C/ C ; (7.1.12) s1˛ s1˛ we obtain Z 1 t f 0./ t ˛ u.t/ D C f.0C/ : ˛ d (7.1.13) .1 ˛/ 0 .t / .1 a/

Thus, the solutions (7.1.11)and(7.1.13) are expressed in terms of the Riemann– Liouville and Caputo fractional derivatives D˛ and CD˛ ; respectively, according to properties of fractional derivatives with m D 1: Method (b) requires f.t/ to be differentiable with L-transformable derivative; consequently 0 Äjf.0C/j < 1 : Then it turns out from (7.1.13)thatu.0C/ can be infinite if f.0C/ ¤ 0;being u.t/ D O.t˛/;as t ! 0C : Method (a) requires weaker conditions in that the integral on the right-hand side of (7.1.11)mustvanish as t ! 0C; consequently f.0C/ could be infinite but with f.t/ D O.t/; 0 < <1 ˛ as t ! 0 C : Then it turns out from (7.1.11)thatu.0C/ can be infinite if f.0C/ is infinite, being u.t/ D O.t.˛C//;as t ! 0 C : Finally, let us remark that the case of Eq. (7.1.1) with 0<˛<1replaced by ˛>0can be treated analogously. If m 1<˛Ä m with m 2 N ; then again we have (7.1.2), now with D˛ f.t/given by the formula which can also be obtained by the Laplace transform method. 168 7 Applications to Fractional Order Equations

The Abel Integral Equation of the Second Kind

Let us now consider the Abel equation of the second kind Z t u./ u.t/ C D f.t/; ˛ >0; 2 C : 1˛ d (7.1.14) .˛/ 0 .t /

In terms of the fractional integral operator this equation reads

.1 C I˛/ u.t/ D f.t/; (7.1.15) and consequently it can be formally solved as follows: ! X1 u.t/ D .1 C I ˛/1 f.t/ D 1 C ./n I ˛n f.t/: (7.1.16) nD1

The formula is obtained by using the standard technique of successive approximation. Convergence of the Neumann series simply follows for any function f 2 CŒ0; a (cf., e.g., [GorVes91, p. 130]). Note that

t ˛n1 I ˛n f.t/ D ˚ .t/ f.t/ D C f.t/: ˛n .˛n/

Thus the formal solution reads ! X1 t ˛n1 u.t/ D f.t/C ./n C f.t/: (7.1.17) .˛n/ nD1

Recalling

X1 .t˛/n e .tI / WD E .t˛/ D ; t>0;˛>0;2 C ; (7.1.18) ˛ ˛ .˛nC 1/ nD0

1 X t ˛n1 d ./n C D E .t˛/ D e0 .tI / ; t > 0 : (7.1.19) .˛n/ dt ˛ ˛ nD1

Finally, the solution reads

0 u.t/ D f.t/C e˛.tI / f.t/: (7.1.20)

Of course the above formal proof can be made rigorous. Simply observe that because of the rapid growth of the Gamma function the inﬁnite series in (7.1.17) 7.1 Fractional Order Integral Equations 169 and (7.1.19) are uniformly convergent in every bounded interval of the variable t so that term-wise integrations and differentiations are allowed.1 Alternatively one can use the Laplace transform, which will allow us to obtain the solution in different forms, including the result (7.1.20). Applying the Laplace transform to (7.1.14) we obtain Ä s˛ s˛1 1 C uQ.s/ D f.s/Q H)u Q .s/ D f.s/Q D s f.s/:Q (7.1.21) s˛ s˛ C s˛ C

Now, let us proceed to ﬁnd the inverse Laplace transform of (7.1.21)usingthe Laplace transform pair (see Appendix C)

s˛1 e .tI / WD E .t˛/ : (7.1.22) ˛ ˛ s˛ C

We have

s˛1 e0 .tI / s : (7.1.23) ˛ s˛ C

Therefore the inverse Laplace transform of the right-hand side of (7.1.21)isthe 0 Laplace convolution of e˛.tI / and f , i.e. the solution to the Abel integral equation of the second kind has the form

Zt 0 0 u.t/ D f.t/C e˛.tI / f.t/D f.t/C f.t /e˛.I /d: (7.1.24) 0

Formally, one can apply integration by parts and rewrite (7.1.24):

Zt 0 u.t/ D f.t/C f .t /e˛.I /d C f.0C/e˛.tI / : (7.1.25) 0

Note that this formula is more restrictive with respect to conditions on the given function f . If the function f is continuous on the interval Œ0; a then formula (7.1.20)(or, what is the same, (7.1.24)) gives the unique continuous solution to the Abel integral equation of the second kind (7.1.14) for any real . The existence of the integral in (7.1.20) follows for any absolutely integrable function f .

1In other words, we use that the Mittag-Lefﬂer function is an entire function for all ˛>0. 170 7 Applications to Fractional Order Equations

7.1.2 Other Integral Equations Whose Solutions Are Represented via Generalized Mittag-Lefﬂer Functions

In many physical applications the Abel integral equations of the ﬁrst kind arise in more general forms:

Zx 1 u.t/dt D f.x/; a

Using this notation, Eq. (7.1.26) becomes the Abel integral equation of the ﬁrst kind. Thus by inverse substitutions in (7.1.11)weobtaintheformalsolutionto(7.1.26)in the following form

Zx 1 d h0.t/f .t/ u.x/ D dt; a < x < b: (7.1.28) .1 ˛/ dt .h.x/ h.t//˛ a

Zb 1 d h0.t/f .t/ u.x/ D dt; a < x < b: (7.1.29) .1 ˛/ dt .h.t/ h.x//˛ x

Another method for solving (7.1.26)and(7.1.27)isgivenin[Sri63]. Thus, in the case of Eq. (7.1.26) we multiply both sides of the equation by

1 h0.x/ ;xÄ y Ä b; .1 ˛/ .h.y/ h.x//˛ 7.2 Fractional Ordinary Differential Equations 171 and integrate with respect to x over the interval .a; y/. Then after changing the order of integration we obtain the following relation

Zy Zy Zy 1 h0.x/dx h0.x/f .x/dx u.t/dt D : .˛/.1 ˛/ .h.y/ h.x//˛.h.x/ h.t//1˛ .h.y/ h.x//˛ a t a

We observe that after making a suitable substitution one can show that the inner integral on the left-hand side is equal to .˛/. Hence formula (7.1.28) follows. Solvability conditions of Eqs. (7.1.26)and(7.1.27) in different functional spaces can be derived by using arguments similar to that for the classical Abel integral equation of the ﬁrst kind (see, e.g., [SaKiMa93,p.31]). A number of integral equations which reduce to the Abel integral equation of the ﬁrst or the second kind are presented in the Handbook of Integral Equations [PolMan08]. Among them we single out the following: 10:

Zx p Á 1 C b x t y.t/dt D f.x/; b D const; (7.1.30)

which reduces to the Abel integral equation of the second kind by differentiating with respect to x; 20:

Zx Â Ã 1 b C p y.t/dt D f.x/; b D const; (7.1.31) x t 0

which can be solved as a combination of the Abel integral equations of the ﬁrst and the second kind.

7.2 Fractional Ordinary Differential Equations

7.2.1 Fractional Ordinary Differential Equations with Constant Coefﬁcients

Here we focus on results concerning ordinary fractional differential equations (FDEs) which are solved in an explicit form via the Mittag-Lefﬂer function, its generalizations, and related special functions. We choose the most simple equations in order to reach the main aim of this subsection, namely, to demonstrate the role of the Mittag-Lefﬂer function in the solution of ordinary FDEs. 172 7 Applications to Fractional Order Equations

For a wider exposition presenting a classiﬁcation of equations and initial boundary value problems for FDEs, and different methods of solution, we refer to the monograph [KiSrTr06](seealso[Bal-et-al12, Die10, Pod99] and references therein).

A Cauchy Type Problem for One-Term Equations

Let us start with a simple linear ordinary differential equation with one fractional derivative of Riemann–Liouville type ˛ R DaCy .x/ y.x/ D f.x/.a0I 2 /: (7.2.1)

Standard initial conditions for such an equation are the so-called Cauchy type initial conditions ˛k R DaC y .aC/ D bk .bk 2 ;kD 1;:::;nDŒ˛/; (7.2.2) where Œ indicates the integral part of a number. If we suppose that the right-hand side in (7.2.1) is Hölder-continuous, i.e. f 2 C Œa; b; 0 Ä <1;<˛;then (see [KiSrTr06, p. 172]) the Cauchy type problem (7.2.1)–(7.2.2) is equivalent in the space Cn˛Œa; b to the Volterra integral equation

Zx Zx Xn b .x a/˛j y.t/ 1 f.t/ y.x/ D j C dt C dt: .˛ j C 1/ .˛/ .x t/1˛ .˛/ .x t/1˛ j D1 a a (7.2.3)

Here and in what follows C0Œa; b means simply the set of continuous functions on Œa; b,i.e.CŒa; b. One can solve Eq. (7.2.3) by the method of successive approximation (for the justiﬁcation of this method in the present case, see, e.g. [KiSrTr06, pp. 172, 222]). If we set

Xn b y .x/ D j .x a/˛j ; (7.2.4) 0 .˛ j C 1/ j D1 then we get the recurrent relation

Zx Zx y .t/ 1 f.t/ y .x/ D y .x/ C m1 dt C dt: (7.2.5) m 0 .˛/ .x t/1˛ .˛/ .x t/1˛ a a 7.2 Fractional Ordinary Differential Equations 173

This can be rewritten in terms of the Riemann–Liouville fractional integrals ˛ ˛ ym.x/ D y0.x/ C IaCym1 .x/ C IaCf .x/: (7.2.6)

Performing successive substitution one can obtain from (7.2.6) the following formula for the m-th approximation ym to the solution of (7.2.3)

Zx " # Xn mXC1 k1.x a/˛kj Xm k1 y .x/ D b C .x t/˛k1 f.t/dt: m j .˛k j C 1/ .˛k/ j D1 kD1 a kD1 (7.2.7)

Taking the limit as m !1we get the solution of the integral equation (7.2.3)(and thus of the Cauchy type problem (7.2.1)–(7.2.2))

Zx " # Xn X1 k1.x a/˛kj X1 k1 y.x/ D b C .x t/˛k1 f.t/dt; j .˛k j C 1/ .˛k/ j D1 kD1 a kD1 (7.2.8) or

Zx " # Xn X1 k ˛kC˛j X1 k .x a/ ˛kC˛1 y.x/ D bj C .x t/ f.t/dt: D D .˛kC ˛ j C 1/ D .˛kC ˛/ j 1 k 0 a k 0 (7.2.9)

The latter yields the following representation of the solution to (7.2.1)–(7.2.2)in terms of the Mittag-Lefﬂer function:

Xn Zx ˛j ˛ ˛1 ˛ y.x/ D bj .xa/ E˛;˛j C1 Œ.x a/ C .xt/ E˛;˛ Œ.x a/ f.t/dt: D j 1 a (7.2.10)

The Cauchy Problem for One-Term Equations

Another important problem for linear ordinary differential equations is the Cauchy problem for FDEs with one fractional derivative of Caputo type C ˛ N R DaCy .x/ y.x/ D f.x/.a Ä x Ä bI n 1 Ä ˛

In this case it is possible to pose initial conditions in the same form as that for ordinary differential equations (i.e. by involving usual derivatives) due to the properties of the Caputo derivative (see the corresponding discussion in Appendix E). Under assumption f 2 C Œa; b; 0 Ä <1;<˛;the Cauchy problem (7.2.11)–(7.2.12) is equivalent (see, e.g., [KiSrTr06, pp. 172, 230]) to the Volterra integral equation

Zx Zx Xn1 b y.t/ 1 f.t/ y.x/ D j .x a/j C dt C dt: jŠ .˛/ .x t/1˛ .˛/ .x t/1˛ j D0 a a (7.2.13) By applying the method of successive approximation with initial approximation

Xn1 b y .x/ D j .x a/j (7.2.14) 0 jŠ j D0 we get the solution to the Volterra equation (7.2.13) (and thus to the Cauchy problem (7.2.11)–(7.2.12)) in the form

Zx " # Xn1 X1 k.x a/˛kCj X1 k1 y.x/ D b C .x t/˛k1 f.t/dt: j .˛kC j C 1/ .˛k/ j D0 kD0 a kD1 (7.2.15) The latter yields the following representation of the solution to (7.2.11)–(7.2.12)in terms of the Mittag-Lefﬂer function:

Xn Zx j ˛ ˛1 ˛ y.x/ D bj .x a/ E˛;j C1 Œ.x a/ C .x t/ E˛;˛ Œ.x a/ f.t/dt: j D1 a (7.2.16)

Solution Methods for Multi-term Ordinary FDEs

The Operational Method

The operational calculus for fractional differential equations has been developed in series of articles (see, e.g. [HiLuTo09] and the references therein). The idea of this approach goes back to the work by Mikusinski´ [Mik59] in which the Laplace convolution

Zx .f g/ .x/ D f.x t/g.t/dt (7.2.17)

0 7.2 Fractional Ordinary Differential Equations 175 is interpreted as an algebraic multiplication in a ring of continuous functions on the real half-axis. We brieﬂy describe this approach (for a more detailed exposition we refer to the book [KiSrTr06, Sec. 4.3]). For any 1 the mapping ı,

Zx 1 1 .f ı g/ .x/ D I0C f g .x/ D I0C f.x t/g.t/dt (7.2.18) 0 becomes the convolution (without zero divisors) of the Liouville fractional integral ˛ operator I0C.˛ > 0/ in the space

p C1 WD fy 2 C.0; C1/ W9p>1; y.x/ D x y1.x/; y1.x/ 2 CŒ0; C1/g : (7.2.19) ˛ If ˛>0and 1 Ä <˛C 1, then the fractional integral I0C has the following convolutional representation

x˛ I ˛ f .x/ D .h ı f / .x/; h.x/ D : (7.2.20) 0C .˛ 1/

˛ C By the semigroup property and linearity of the fractional integral I0C, the space 1 with the operations ı and “C” becomes a commutative ring without zero divisors. As in the Mikusinski´ approach, one can see that this ring can be extended to the quotient ﬁeld

P D C1 .C1 nf0g/= (7.2.21) with the standard equivalence relation determined by the convolution. Then the ˛ P algebraic inverse to I0C in the quotient field can be defined as an element S of P which is reciprocal to the element h.x/ in the field P:

I h h S D WD D ; 2 (7.2.22) h .h ı h/ h where I is an identity in P with respect to convolution. For any ˛>0and m 2 N we introduce the space n o m C C ˛ k C ˝˛ . 1/ WD y 2 1 W D0C y 2 1;kD 1;:::;m ; (7.2.23) ˛ k ˛ where D0C is the k-th power of the operator D0C : ˛ k ˛ ˛ D0C ./ D D0C :::D0C ./: 176 7 Applications to Fractional Order Equations

N m C If ˛>0, m 2 and f 2 ˝˛ . 1/, then the following relation

mX1 Á ˛ m m mk ˛ k D0C f D S S F D0C f (7.2.24) kD0

P ˛ ˛ ˛ holds in the ﬁeld .HereF D E I0CD0C is a projector of I0C determined in 1 C ˝˛ . 1/ by the formula Xn D˛k y .0C/ E I ˛ D˛ y .x/ D 0C xk˛;y2 ˝1 .C /; n 1<˛Ä n; 0C 0C .˛ k 1/ ˛ 1 kD1

1 C 1 C 1 C and E W ˝˛ . 1/ ! ˝˛ . 1/ is the identity operator on ˝˛ . 1/. If ˛>0, 1 Ä <˛C 1, m 2 N, ! 2 C, then convolution relations follow from the analytical property in the ﬁeld P (see, e.g., [KiSrTr06, p. 265])

I h D D h E C !h C !2h2 C ::: D x˛E .!x˛/; S ! E !h ˛;˛C1 (7.2.25) I D xm˛Em .!x˛/; (7.2.26) .S !/m ˛;m˛C1

ı where E˛;ˇ, Eˇ; are two- and three-parametric Mittag-Leffler functions, respectively. By using these relations one can represent solutions to the Cauchy type problem for certain multi-term fractional differential equations in terms of the Mittag-Leffler function. Since the application of this method critically depends on the analytical properties of the corresponding differential operator, we have taken the liberty of reproducing an example of such a problem from [KiSrTr06]. For this problem the method gives the solution in a final form. Example ([KiSrTr06, p. 269]). Let ! 2 C, 2 R be arbitrary numbers such that 3 C 1 Ä <3=2. We consider in the space ˝1=2 . 1/ the following Cauchy type problem: Á Á 3=2 0 2 1=2 2 C D0C y .x/ !y .x/ C D0C y .x/ ! y.x/ D f.x/; f 2 1 (7.2.27) Á Á 1=2 1=2 lim D0C y .x/ D 0; y.0/ D 0; lim I0C y .x/ D 0: (7.2.28) x!0C x!0C

Here ˛ D 1=2 and the problem is reduced in the ﬁeld P to the algebraic equation

S 3y !S2y C 2Sy !2y D f: (7.2.29) 7.2 Fractional Ordinary Differential Equations 177

Since the polynomial on the left-hand side possesses a simple decomposition

P.S/ D S 3 !S2 C 2S !2 D .S 2 C 2/.S !/ one can ﬁnd a representation of an algebraic inverse to the differential operator in (7.2.27) as the reciprocal to P.S/ in the ﬁeld P:

I 1 S ! I D C : (7.2.30) P.S/ !2 C 2 S 2 C 2 S 2 C 2 S !

Using the identity (7.2.25) we have the solution of the considered problem in the form Zx y.x/ D K.x t/f.t/dt; (7.2.31)

0 where 1 K.x/ D (7.2.32) !2 C 2 n o 1 2 1 2 1 1 x 2 E1;2. x/ !x E1;2. x/ C x 2 E 1 3 .!x 2 / : 2 ; 2

The Laplace Transform Method

Another useful method which leads to the explicit solution of ordinary fractional differential equations is the Laplace transform method. The direct application of the Laplace transform to the homogeneous multi-term fractional differential equation

Xm C ˛k N Ak D0Cy .x/ C A0y.x/ D 0.x>0;m2 ;0<˛1 <:::<˛m/ kD1 (7.2.33) yields an explicit representation of the fundamental system fyi .x/g;i D 0;:::;l 1; l DŒ˛m; for this equation, i.e. of the solutions to the Cauchy problems

.j / j yi .0/ D ıi ;i;jD 0; 1; : : : ; l 1; (7.2.34) for Eq. (7.2.33). This representation critically depends on the properties of the so-called quasi-polynomial

Xm ˛k P.s/ D A0 C Aks ;s2 C: kD1 178 7 Applications to Fractional Order Equations

Thus, the system of fundamental solutions for the one-term fractional differential equation C ˛ N D0Cy .x/ y.x/ D 0.x>0;l 1<˛Ä l; l 2 / (7.2.35) has the form

i ˛ yi .x/ D x E˛;iC1.x /; i D 0;:::;l 1I (7.2.36) and the system of fundamental solutions for two-term fractional differential equations with derivatives of orders ˛ and ˇ (˛>ˇ>0, l 1<˛Ä l; l 2 N, n 1<ˇÄ n; n 2 N) Á C ˛ C ˇ D0Cy .x/ D0Cy .x/ D 0.x>0/ (7.2.37) consists of two groups of functions (see [KiSrTr06, p. 314])

i ˛ˇ ˛ˇCi ˛ˇ yi .x/ D x E˛ˇ;iC1.x /x E˛ˇ;˛ˇCiC1.x /; i D 0;:::;n1I (7.2.38) i ˛ˇ yi .x/ D x E˛ˇ;iC1.x /; i D n;:::;l 1: (7.2.39)

For multi-term equations with a greater number of derivatives, the system of fundamental solutions has also been found (see [KiSrTr06, pp. 319–321]). The corresponding formulas are rather cumbersome. They have the form of series with coefﬁcients represented in terms of the Wright function 11. The inhomogeneous equation, corresponding to (7.2.33), can also be solved by using the Laplace transform method. The solution is represented in the form

Zx Xl1

y.x/ D G˛1;:::;˛m .x t/f.t/dt C ci yi .x/; (7.2.40) 0 iD0 where fyi .x/g;i D 0;:::;l 1; is the fundamental system of solutions of the homogeneous equations and Â Ä Ã 1 G .x/ D L1 .x/ (7.2.41) ˛1;:::;˛m P.s/ is the function determined by the inverse transform of the reciprocal to the quasi-polynomial (an analogue of Green’s function). In some cases the function

G˛1;:::;˛m .x/ can be represented in an explicit form (see [KiSrTr06]). In a similar way the Cauchy type problem for multi-term equations can be studied using the Laplace transform method. 7.2 Fractional Ordinary Differential Equations 179

7.2.2 Ordinary FDEs with Variable Coefﬁcients

Here we describe certain approaches concerning the solution of ordinary differential equations with variable coefficients. All of them are related to the case when the coefficients in the equations are power-type functions. Let us consider first the following Cauchy type problem for the one-term differential equation with Riemann–Liouville fractional derivative: ˛ ˇ R D0Cy .x/ .x a/ y.x/ D 0.a0;ˇ>f˛gI 2 /; (7.2.42) ˛k R D0C y .aC/ D bk .bk 2 I k D 1;:::;n; nDŒ˛/: (7.2.43)

As in the above considered case of the equations with constant coefﬁcients, one can prove that problem (7.2.42)–(7.2.43) is equivalent in the space Cn˛Œa; b to the Volterra integral equation

Zx Xn b .x a/˛j .x a/ˇy.t/ y.x/ D j C dt: (7.2.44) .˛ j C 1/ .˛/ .x t/1˛ j D1 a

By applying the successive approximation method and using formulas for fractional integrals of power-type functions we can derive the following series representation of the solution to (7.2.42)–(7.2.43): " # Xn ˛j X1 b .x a/ k y.x/ D j 1 C c .x a/˛Cˇ ; (7.2.45) .˛ j C 1/ k;j j D1 kD1 where

Yk .r.˛C ˇ/ j C 1/ c D : k;j .r.˛C ˇ/ C ˛ j C 1/ rD1

Changing in the last product the index r to i D r 1 we arrive at the representation of the solution to (7.2.42)–(7.2.43) in terms of the three-parametric Mittag-Lefﬂer function (Kilbas–Saigo function) E˛;m;l .z/ (see Sect. 5.2)

Xn b .x a/˛j y.x/ D j E .x a/˛Cˇ : (7.2.46) .˛ j C 1/ ˛;1Cˇ=˛;1C.ˇj/=˛ j D1

In this case the Cauchy problem (7.2.47)–(7.2.48) is equivalent in the space Cn1Œa; b to the Volterra integral equation

Zx Xn1 d .x a/j .x a/ˇy.t/ y.x/ D j C dt: (7.2.49) jŠ .˛/ .x t/1˛ j D0 a

Here, the successive approximation method gives the following representation (see [KiSrTr06, p. 233]) of the solution to (7.2.46)–(7.2.47) in terms of the three- parametric Mittag-Lefﬂer function (Kilbas–Saigo function):

Xn1 d y.x/ D j .x a/j E .x a/˛Cˇ : (7.2.50) jŠ ˛;1Cˇ=˛;.ˇCj/=˛ j D0

The Cauchy type problem for the inhomogeneous equation corresponding to Eq. (7.2.42) is treated via the differentiation formulas (of integer and fractional order) for the three-parametric Mittag-Lefﬂer function derived in [KilSai95b, GoKiRo98]. Let

˛>0;n 1<˛f˛gI fr ;r 2 R;r > 1; r D 1;:::;k:

Let us consider the Cauchy type problem for the inhomogeneous fractional differential equation with quasi-polynomial free term

Xk ˛ ˇ r R D0Cy .x/ D .x a/ y.x/ C fr .x a/ .a

This problem has a unique solution in the space Iloc.a; b/ of functions locally integrable on .a; b/ (see [KiSrTr06, pp. 251–252])

Xn b y.x/ D y C j .x a/˛j E .x a/˛Cˇ ; 0 .˛ j C 1/ ˛;1Cˇ=˛;.ˇj/=˛ j D1 (7.2.53) 7.2 Fractional Ordinary Differential Equations 181 where

Xk fr .r C 1/ y D .x a/˛Cr E .x a/˛Cˇ : 0 . C ˛ C 1/ ˛;1Cˇ=˛;.ˇCr /=˛ rD1 r (7.2.54) The Mellin transform method is applied to solve the following fractional differential equations with power-type coefﬁcients (which are sometimes called Euler type fractional equations, see, e.g. [Zhu12])

Xm Á ˛Ck ˛Ck Akx D0C y .x/ D f.x/.x>0;˛>0/: (7.2.55) kD0

In this case one can use the following property of the Mellin transform ÁÁ .1 s/ Mx˛Ck D˛Cky .s/ D .My/ .s/: 0C .1 s ˛ k/

Hence by applying the Mellin transform to (7.2.55), one obtains the solution in the form of a Mellin convolution (see, e.g. [KiSrTr06, p. 330])

y.x/ D G˛.t/f .xt/dt; (7.2.56) 0 where the Mellin fractional analogue of Green’s function G˛.x/ isgivenbythe formula Â Ä Ã Xm 1 1 .s/ G˛.x/ D M ;P˛.s/ D Ak : (7.2.57) P˛.1 s/ .s ˛ k/ kD0

In some cases it is possible to determine an explicit representation of the analogue of Green’s function (and thus an explicit solution to Eq. (7.2.55)) (see, e.g. [KilZhu08, KilZhu09a, KilZhu09b]). For instance, in the case m D 1, the corresponding analogue of Green’s function G˛;.x/ for the equation ˛C1 ˛C1 ˛ ˛ R x D0C y .x/ C x D0Cy .x/ D f.x/ .˛>0;2 / (7.2.58) has the form Ä ˇ ˇ ˛ .1 / 1 .1 ;1/ ˇ G˛.x/ D G˛;.x/ D x x 12 ˇ x : .˛C 1 / .˛; 1/; .2 ;1/ (7.2.59) 182 7 Applications to Fractional Order Equations

7.2.3 Other Types of Ordinary Fractional Differential Equations

In [KiSrTr06] fractional ordinary differential equations with another type of fractional derivative have been considered. Here we focus on only a few of them which are solved in an explicit form. Let us consider the following Cauchy type problem: D˛ R aCy .x/ y.x/ D f.x/.a 0; 2 /; (7.2.60) D˛k R aC y .aC/ D bk .bk 2 ;kD 1;:::;n; nDŒ˛/ (7.2.61)

C R C D˛ where f 2 ;log Œa; b WD fg W .a; b ! W Œlog .x=a/g.x/ 2 Œa; b and aC is the Hadamard fractional derivative Â Ã Zx Á d n 1 x n˛1 y.t/dt D˛ y .x/ D x log : (7.2.62) aC dx .n ˛/ t t a

Employing the same technique as for one-term ordinary fractional differential equations with Riemann–Liouville fractional derivative, one can show that the Cauchy type problem (7.2.60)–(7.2.61) is equivalent in the space Cn˛;logŒa; b to the Volterra integral equation

n Á X b x ˛k y.x/ D k log .˛ k C 1/ a kD1 Zx Á Zx Á x ˛1 dt 1 x ˛1 dt C log y.t/ C log f.t/ : (7.2.63) .˛/ t t .˛/ t t a a

By applying the method of successive approximation we obtain (see [KiSrTr06, p. 234]) the unique solution to the Volterra integral equation (and thus to the Cauchy type problem (7.2.60)–(7.2.61)) in terms of the two-parametric Mittag- Lefﬂer function

n Á h Á i X x ˛k x ˛ y.x/ D b log E log k a ˛;˛kC1 a kD1 Zx Á h Á i x ˛1 x ˛ C log E log f.t/dt: (7.2.64) t ˛;˛kC1 t a 7.3 Differential Equations with Fractional Partial Derivatives 183

The solution to the fractional ordinary differential equation

Xm ˛k Ak .D y/.x/CA0y.x/ D f.x/.0<˛1 <:::<˛m;Ak 2 R;k D 1;:::;m/ kD1 (7.2.65) with the Riesz fractional derivative

ZC1 1 l y .x/ .D˛y/.x/ D t t.l>˛/ 1C˛ d (7.2.66) d1.l; ˛/ jtj 1 is given (see, e.g., [KiSrTr06, pp. 344–345]) in term of the Fourier convolution

ZC1 y.x/ GF .x t/f.t/ t: D ˛1;:::;˛m d (7.2.67) 1

GF Here ˛1;:::;˛m is a fractional analogue of Green’s function, which has the following form in the case of an equation with constant coefﬁcients:

ZC1 1 1 GF .x/ D Ä cos xd: (7.2.68) ˛1;:::;˛m Pm ˛ 0 Akjj k C A0 kD1 Finally, we single out an important class of so-called sequential fractional differential equations

Xn1 n˛ k˛ DaCy .x/ C ak .x/ DaCy .x/ D f.x/ (7.2.69) kD0 treated, e.g., in [KiSrTr06, Ch. 7] by different methods. In the case of constant coefﬁcients the solution of the corresponding Cauchy type problem is given in terms of the ˛-exponential function

z ˛1 ˛ e˛ D z E˛;˛.z /:

7.3 Differential Equations with Fractional Partial Derivatives

Partial fractional differential equations are of great theoretical and practical importance (see, e.g., [KiSrTr06, Ch. 6], [KilTru02] and the references therein). Recently a number of books presenting results in this area have been published (e.g., 184 7 Applications to Fractional Order Equations

[Die10,Mai10,Psk06,Tar10,Uch13a,Uch13b]). One can ﬁnd there different aspects of the theory and applications of partial fractional differential equation. Indeed, we have to note that this branch of analysis is far from complete. Furthermore, many partial differential equations serve to describe some models in mechanics, physics, chemistry, biology etc. Some of these equations will be discussed in the following two chapters. In this section we present a few results concerning the simplest fractional partial differential equations. Our main focus will be on the one-dimensional diffusion- wave equation (with Riemann–Liouville fractional derivative, see Sect. 7.3.1,and with Caputo fractional derivative, see Sect. 7.3.2).Themainideaistopresentto the reader elements of the techniques developed for fractional partial differential equations. Some more speciﬁc equations dealing with certain models are presented in the next two chapters.

7.3.1 Cauchy-Type Problems for Differential Equations with Fractional Partial Derivatives

The simplest partial differential equation with Riemann–Liouville fractional derivative is the so-called fractional diffusion equation

@2u D˛ u .x; t/ D 2 .x 2 RI t>0I >0I ˛>0/: (7.3.1) 0C;t @x2

˛ Here D0C;t is the Riemann–Liouville fractional derivative of order ˛ with respect to time t. Partial differential equations where the Riemann–Liouville fractional derivative are in time are supplied by initial conditions known as Cauchy-type conditions. Let us consider Eq. (7.3.1)for0<˛<2. The Cauchy-type initial conditions then have the form ˛k R D0C;t u .x; 0C/ D fk .x/; x 2 ; (7.3.2) where k D 1 if 0<˛<1, and two conditions with k D 1; 2 if 1<˛<2.2 The problem (7.3.1)–(7.3.2) is usually solved by the method of integral transforms. Let us apply to Eq. (7.3.1) successively the Laplace transform with respect to the time variable t

2Note that for ˛ D 1 Eq. (7.3.1) becomes the standard diffusion equation and initial condition (7.3.2) becomes the standard Cauchy condition. 7.3 Differential Equations with Fractional Partial Derivatives 185

Z1 st .Lt u/.x;s/D u.x; t/e dt.x2 RI s>0/; 0 and the Fourier transform with respect to the spatial variable x

ZC1 ix .Fxu/. ;t/ D u.x; t/e dx. 2 RI t>0/: 1

The Laplace transform of the Riemann–Liouville fractional derivative satisﬁes the relation

Xl Á L ˛k ˛ L j 1 ˛j R t D0C;t u .x; s/ D s . t u/.x;s/ s D0C;t u .x; 0C/; .x 2 /; j D1 (7.3.3) where l 1<˛Ä l, l 2 N. In the considered case (0<˛<2)wetakeinto account the initial conditions (7.3.2) we get from (7.3.1) Â Ã Xk @2 s˛ .L u/.x;s/ D sj 1f .x/ C 2 L u : (7.3.4) t j @x2 t j D1

Here either k D 1 or k D 2. For the Fourier transform the following relation is known Â Ä Ã @2u F . ; t/ Dj j2 .F u/. ;t/: (7.3.5) x @x2 x

Thus applying the Fourier transform to (7.3.4)wegetfork D 1 or k D 2

Xk sj 1 .F L u/. ;s/ D . 2 RI s>0/: (7.3.6) x t s˛ C 2j j2 j D1

In order to obtain an explicit solution to problem (7.3.1)–(7.3.2)weusethe inverse Fourier and the inverse Laplace transform and corresponding tables of these transforms. The ﬁnal result reads (see, e.g., [KiSrTr06, Thm. 6.1]): Let 0<˛<2and >0. Then the formal solution of the Cauchy-type problem (7.3.1)–(7.3.2) is represented in the form

Xk ZC1 ˛ u.x; t/ D Gj .x ;t/fj ./d; (7.3.7) j D11 186 7 Applications to Fractional Order Equations where k D 1 if 0<˛<1, and k D 2 if 1<˛<2, Â Ã 1 ˛ ˛ jxj G˛.x; t/ D t ˛=2j ' ; j C 1I t ˛=2 ;.j D 1; 2/ (7.3.8) j 2 2 2 where '.a;bI z/ is the classical Wright function (see deﬁnition (F. 2 . 1 )). The formal solution (7.3.7) becomes the real one if the integrals on the right-hand side of (7.3.7) converge.

7.3.2 The Cauchy Problem for Differential Equations with Fractional Partial Derivatives

As an example we consider here the Cauchy problem for the partial fractional differential equation with Caputo derivative

@2u CD˛ u .x; t/ D 2 .x 2 RI t>0I >0I 0<˛<2/: (7.3.9) 0C;t @x2 This equation is a particular case of the so-called fractional diffusion-wave equation C ˛ 2 Rn D0C;t u .x;t/D xu.x;t/ .x 2 I t>0I >0I 0<˛<2/: (7.3.10)

Equation (7.3.9) is supplied by the Cauchy condition(s)

@ku .x; 0/ D f .x/ .x 2 R/; (7.3.11) @xk k where k D 0,if0<˛<1, and two conditions with k D 0; 1,if1<˛<2;the 0-th order derivative means the value of the solution u at the points .x; 0/.3 To solve the Cauchy problem (7.3.11) for the fractional differential equation (7.3.9) we use the same method as in the previous subsection. We ﬁrst apply the Laplace integral transform with respect to the time variable t using the relation

Xk1 @j u L CD˛ u .x; s/ D s˛ .L u/.x;s/ s˛j 1 .x; 0/; (7.3.12) t 0C;t t @tj j D0 and then the Fourier transform with respect to the spatial variable x using the relation (7.3.5). Then in view of the Cauchy initial condition(s) we obtain from Eq. (7.3.9)

3Note once again that for ˛ D 1 Eq. (7.3.9) becomes the standard diffusion equation and initial condition (7.3.11) becomes the standard Cauchy condition. 7.4 Historical and Bibliographical Notes 187

Xk1 s˛j 1 .F L u/. ;s/ D .F f /. /: (7.3.13) x t s˛ C 2j j2 x k j D0

By using the inverse Fourier and the inverse Laplace transform and corresponding tables of these transforms we get the ﬁnal result in the form (see, e.g., [KiSrTr06, Thm. 6.3]): Let 0<˛<2and >0. Then the formal solution of the Cauchy problem (7.3.9), (7.3.11) is represented in the form

Xk1 ZC1 ˛ u.x; t/ D Gj .x ;t/fj ./d; (7.3.14) j D01 where k D 1 if 0<˛<1, and k D 2 if 1<˛<2, Â Ã 1 ˛ ˛ jxj G˛.x; t/ D t j ˛=2' ;j C 1 I t ˛=2 ;.j D 0; 1/ (7.3.15) j 2 2 2 where '.a;bI z/ is the classical Wright function (see deﬁnition (F. 2 . 1 )). The formal solution (7.3.14) becomes the real one if the integrals on the right-hand side of (7.3.14) converge.

7.4 Historical and Bibliographical Notes

The most simple integral equations of fractional order, namely the Abel integral equations of the first kind, were investigated by Abel himself [Abe26]. Abel integral equations of the second kind were studied by Hille and Tamarkin [HilTam30]. In this paper for the first time the solution was represented via the Mittag-Leffler function. The interested reader is referred to [SaKiMa93, CraBro86, GorVes91], and [Gor96, Gor98] for historical notes and detailed analyses with applications. It is well known that Niels Henrik Abel was led to his famous equation by the mechanical problem of the tautochrone, that is by the problem of determining the shape of a curve in the vertical plane such that the time required for a particle to slide down the curve to its lowest point is equal to a given function of its initial height (which is considered as a variable in an interval Œ0; H ). After appropriate changes of variables he obtained his famous integral equation of the first kind with ˛ D 1=2. He did, however, solve the general case 0<˛<1. As a special case Abel discussed the problem of the isochrone, in which it is required that the time taken for the particle to slide down is independent of the initial height. Already in his earlier publication [Abe23] he had recognized the solution as a derivative of non-integer order. We point out that integral equations of Abel type, including the simplest (7.1.1)and(7.1.14), have found so many applications in diverse fields that it is almost impossible to provide an exhaustive list of them. 188 7 Applications to Fractional Order Equations

Abel integral equations occur in many situations where physical measurements are to be evaluated. In many of these the independent variable is the radius of a circle or a sphere and only after a change of variables does the integral operator take the form J ˛ ; usually with ˛ D 1=2 ; and the equation is of the first kind. Applications are, e.g. in evaluation of spectroscopic measurements of cylindrical gas discharges, the study of the solar or a planetary atmosphere, the investigation of star densities in a globular cluster, the inversion of travel times of seismic waves for the determination of terrestrial sub-surface structure, spherical stereology. Descriptions and analyses of several problems of this kind can be found in the books by Gorenflo and Vessella [GorVes91] and by Craig and Brown [CraBro86], see also [Gor96]. Equations of the first and of the second kind, depending on the arrangement of the measurements, arise in spherical stereology. See [Gor98] where an analysis of the basic problems and many references to the previous literature are given. Another field in which Abel integral equations or integral equations with more general weakly singular kernels are important is that of inverse boundary value problems in partial differential equations, in particular parabolic ones in which the independent variable naturally has the meaning of time. A number of integral equations similar to the Abel integral equation of the second kind are discussed in the book by Davis [Dav36, Ch. 6]. In this part of the historical overview of solutions of linear and non-linear fractional differential equations we partly follow the survey papers [KilTru01, KilTru02], and the books [KiSrTr06]and[Die10]. The paper of O’Shaughnessay [O’Sha18] was probably the first where the methods for solving the differential equation of half-order y .D1=2y/.x/ D (7.4.1) x were considered. Two solutions of such an equation

y.x/ D x1=2e1=x (7.4.2) and a series solution Z p x y.x/ D 1 i x1=2e1=x C x1=2e1=x t 3=2e1=t dt 1 (7.4.3) p p D 1 i x1=2 2x1 C i x3=2 C were suggested by O’Shaughnessay and discussed later by Post [Pos19].Their arguments were formal and based on an analogy with the Leibnitz rule, which for the Riemann–Liouville fractional derivative has the form

X1 .˛C 1/ D˛ .fg/ .x/ D .D˛k f /.x/g.k/.x/: (7.4.4) 0C .˛ k C 1/kŠ 0C kD0 7.4 Historical and Bibliographical Notes 189

As was proved later (see, e.g. [MilRos93, pp. 195–199]), (7.4.2) is really a solution of Eq. (7.4.1) y .D1=2y/.x/ D (7.4.5) 0C x

1=2 with the Riemann–Liouville fractional derivative D0C y. ˛ ˛ As for (7.4.3), it is not a solution of Eq. (7.4.1) with D D DaC, a being any real constant, because O’Shaughnessay and Post made a mistake while using the relation for the composition of I 1=2D1=2y. Such a relation for the Riemann– ˛ Liouville derivative DaCy has the form

Xn .x a/˛k .I ˛ D˛ y/.x/ D y.x/ B ; (7.4.6) aC aC k .˛ k C 1/ kD1 where

.nk/ n˛ Bk D yn˛ .a/; yn˛.x/ D .IaC y/.x/; .˛ 2 C;nD ŒRe.˛/ C 1/; (7.4.7) in particular,

.x a/˛1 .I ˛ D˛ y/.x/ D y.x/ B ;BD y .a/; (7.4.8) aC aC .˛/ 1˛ for 0

˛ with the Riemann–Liouville fractional derivative DaCy.x/. He had assumed that the corresponding variations are equal to zero and obtained the differential equation with Cauchy conditions

˛ .k/ dF Á FŒDaCy.x/I x D 0; y .a/ D bk .k D 1; 2; ;n/: (7.4.9)

Fujiwara [Fuj33] considered the differential equation of fractional order Á ˛ ˛ .D˛ y/.x/ D y.x/ (7.4.10) C x 190 7 Applications to Fractional Order Equations with the Hadamard fractional derivative of order ˛>0deﬁned as followed Â Ã Z d n 1 x y.t/dt .D˛ y/.x/ D .n D Œ˛ C 1/; C ˛nC1 (7.4.11) dx .n ˛/ 0 t.log.x=t// with n 2 N Df1;2;:::g and ˛ … N. He obtained a formal solution of (7.4.10)in the form of the Mellin–Barnes integral Z 1 Ci1 y.x/ D Œ .s/xs ˛ds.>0/; (7.4.12) 2i i1 and proved that y.x/ has the following asymptotics at zero

y.x/ Axe=x; (7.4.13) with 1 ˛ 1 A D p .2/.˛1/=2;D ;D ˛: (7.4.14) ˛ 2

Pitcher and Sewell [PitSew38] ﬁrst considered the non-linear FDE

˛ R .DaCy/.x/ D f.x;y.x// .0 < ˛ < 1; a 2 / (7.4.15)

˛ with the Riemann–Liouville fractional derivative DaCy provided that f.x;y/ is bounded in the special region G lying in R R .R D .1; 1// and satisﬁes the Lipschitz condition with respect to y:

jf.x;y1/ f.x;y2/jÄAjy1 y2j; (7.4.16) where the constant A>0does not depend on x. They tried to prove the uniqueness of a continuous solution y.x/ of such an equation on the basis of the corresponding result for the non-linear integral equation Z 1 x fŒt;y.t/dt y.x/ D 0.x>aI 0<˛<1/: 1˛ (7.4.17) .˛/ a .x t/

But the result of Pitcher and Sewell given in [PitSew38, Theorem 4.2] is not correct since they made the same mistake as O’Shaughnessay [O’Sha18]andPost[Pos19] ˛ ˛ by using the relation IaCDaCy D y instead of (7.4.8). However, the paper of Pitcher and Sewell [PitSew38] contained the idea of the reduction of the fractional differential equation (7.4.15) to the Volterra integral equation (7.4.17). 7.4 Historical and Bibliographical Notes 191

Al-Bassam [Al-B65] ﬁrst considered the following Cauchy-type problem

˛ .DaCy/.x/ D f.x;y.x// .0 < ˛ Ä 1/; (7.4.18) ˛1 1˛ R .DaC y/.x/jxDa Á .IaC y/.x/jxDa D b1;b1 2 ; (7.4.19) in the space of continuous functions CŒa;b provided that f.x;y/ is a real- valued, continuous and Lipschitzian function in a domain G R R such that ˛ sup.x;y/2G jf.x;y/jDb0 < 1: Applying the operator IaC to both sides of (7.4.18), using the relation (7.4.8) and the initial conditions (7.4.19), he reduced (7.4.18)– (7.4.19) to the Volterra non-linear integral equation Z b .x a/˛1 1 x fŒt;y.t/dt y.x/ D 1 C .x > aI 0<˛Ä 1/: 1˛ (7.4.20) .˛/ .˛/ a .x t/

Using the method of successive approximations he established the existence of the continuous solution y.x/ of Eq. (7.4.20). Furthermore, he was probably the ﬁrst to indicate that the method of contracting mapping can be applied to prove the uniqueness of this solution y.x/ of (7.4.20), and gave such a formal proof. Al-Bassam also indicated – but did not prove – the equivalence of the Cauchy- type problem (7.4.18)–(7.4.19) and the integral equation (7.4.20), and therefore his results on the existence and uniqueness of the continuous solution y.x/, formulated in [Al-B65, Theorem 1], could be true only for the integral equation (7.4.20). We also note that the conditions suggested by Al-Bassam are not suitable to solve the Cauchy-type problem (7.4.18)–(7.4.19) in the simplest linear case when f Œx;y.x/ D y.x/. The same remarks apply to the existence and uniqueness results formulated without proof in [Al-B65, Theorems 2, 4, 5, 6] for a more general Cauchy-type problem of the form (7.4.18)–(7.4.19) with real ˛>0:

˛ .DaCy/.x/ D f.x;y.x// .n 1<˛Ä n; n DŒ˛/; (7.4.21) ˛k R .DaC y/.x/jxDa D bk;bk 2 .k D 1; 2; ;n/; (7.4.22) where the corresponding Volterra equation has the form (7.4.20): Z Xn b .x a/˛k 1 x fŒt;y.t/dt y.x/ D k C .x > aI n 1<˛Ä n/ .˛ k C 1/ .˛/ .x t/1˛ kD1 a (7.4.23) for the system of the equations (7.4.18) and for non-linear fractional equations more general than (4.4.18) Á n˛ ˛ 2˛ .n1/˛ .DaCy/.x/ D f x;y.x/;.DaCy/.x/; .DaCy/.x/; ;.DaC y/.x/ .0 < ˛ Ä 1/ (7.4.24) 192 7 Applications to Fractional Order Equations and linear fractional equations

Xn .nk/˛ ck.x/.DaC y/.x/ D f.x/; .0<˛ Ä 1/ (7.4.25) kD0 for continuous f.x;x1;x2; ;xm/ and f.x/, pk.x/ .0 Ä k Ä m/ and under the initial conditions

k˛1 .DaC y/.x/jxDa D bk .k D 1; 2; ;n/: (7.4.26)

The above and some other results were presented in Al-Bassam [Al-B82,Al-B86, Al-B87]. Cauchy-type problems for non-linear ordinary differential equations of fractional order have been studied by many authors (in particular, developing Al-Bassam’s method). An extended bibliography on subject is presented in the survey paper [KilTru01], and in the book [KiSrTr06]. Cauchy-type problems for linear ordinary differential equations of fractional order were investigated mainly by using the method of reduction to Volterra integral equations. First, we have to mention the paper by Barrett [Barr54], which ﬁrst considered the Cauchy-type problem for the linear differential equation with the Riemann–Liouville fractional derivative on a ﬁnite interval .a; b/ of the real axis

˛ C .DaCy/.x/ y.x/ D f.x/.n 1 Ä Re.˛/ < nI 2 /; (7.4.27)

˛k C .DaC y/.x/jxDaC D bk 2 .k D 1; 2; ;n/ (7.4.28) n D ŒReT.˛/ C 1; ˛ ¤ n 1. He proved that if f.x/ belongs to L.a; b/ or L.a; b/ C.a;b, then the problem has a unique solution y.x/ in some subspaces of L.a; b/ and this solution is given by

Xn ˛k ˛ y.x/ D bk.x a/ E˛;˛kC1 ..x a/ / kD1 Z x ˛1 ˛ C .x t/ E˛;˛ ..x t/ /f.t/dt (7.4.29) a where E˛;ˇ.z/ is the two-parametric Mittag-Lefﬂer function. Barrett’s argument ˛ ˛ was based on the formula (7.4.6) for the product IaCDaCf .From(7.4.29) Barrett obtained the unique solution

Xn ˛k ˛ y.x/ D bk.x a/ E˛;˛kC1 ..x a/ / (7.4.30) kD1 of the Cauchy-type problem for the homogeneous equation .f .x/ D 0/ correspond- ingto(7.4.27): 7.4 Historical and Bibliographical Notes 193

˛ .DaCy/.x/ y.x/ D 0.n 1 Ä Re.˛/ < n/; (7.4.31)

˛k .DaC y/.x/jxDaC D bk 2 C .k D 1; 2; ;n/: (7.4.32)

He also proved the uniqueness of the solution y.x/ of the simplest such Cauchy-type problem (7.4.31)–(7.4.32) with 0<˛<1and D1. Barrett implicitly used the method of reduction of the Cauchy-type problem (7.4.27)–(7.4.28) to the Volterra integral equation of the second kind and the method of successive approximations. Dzhrbashyan and Nersesyan [DzhNer68] studied the linear differential equation of fractional order

Xn2 n nk1 .D y/.x/ Á .D y/.x/ C ak .x/.D y/.x/ C an.x/y.x/ D f.x/ kD0 (7.4.33) with sequential fractional derivatives .D y/.x/ and .D nk1 y/.x/ .k D 0; 1; ;n 1/ deﬁned in terms of the Riemann–Liouville fractional derivatives. Here, the term “sequential” means that the orders of the derivatives are related in the following manner:

Xk k D ˛j 1.kD 0; 1; ;n/I 0<˛j Ä 1.j D 0; 1; ;n/; j D0 (7.4.34)

.˛k D k k1 .k D 1; 2; ;n/; ˛0 D 0 C 1/ :

They proved that for ˛0 >1 ˛n the Cauchy-type problem

k .D y/.x/ D f.x/; .D y/.x/jxD0 D bk .k D 0; 1; ;n 1/ (7.4.35) has a unique continuous solution y.x/ on an interval Œ0; d provided that the functions pk.x/ .0 Ä k Ä n 1/ and f.x/ satisfy some additional conditions. In particular, when pk.x/ D 0.kD 0; 1; ;n/, they obtained the explicit solution Z Xn1 x bkx k 1 y.x/ D C .x t/ n1f.t/dt (7.4.36) .1C k/ . n/ kD0 a of the Cauchy-type problem

n k .D y/.x/ D f.x/; .D y/.x/jxD0 D bk .k D 0; 1; ;n 1/: (7.4.37)

Laplace transform methods for ordinary fractional differential equations have successfully been used by many authors. Maravall [Mara71] was probably the ﬁrst who suggested a formal approach based on the Laplace transform to obtain the explicit solution of a particular case of the equation 194 7 Applications to Fractional Order Equations

Xm ˛k ck .DaCy/.x/ C c0y.x/ D f.x/.0

˛k where m 1, DaCy.k D 1; 2; ;m/, are the Riemann–Liouville fractional derivatives, and ck ¤ 0, .k D 0; 2; ;m/, are real or complex constants. However, since this paper was published in Spanish, it was practically unknown. Later the method of Laplace transforms was used in a different form, based on the main properties of the Laplace transform (see, e.g., [Doe74]and[DitPru65]) and the analytic properties of so-called characteristic quasi-polynomials. Special attention was paid to this approach in recent FDA-Congresses and FDTAs symposiums (see [Adv-07, NewTr-10] and references therein, as well as [KilTru01], and the books [KiSrTr06, Cap-et-al10]). The operational calculus method for ordinary differential equations of fractional order is based on the interpretation of the Laplace convolution Z x .f g/.x/ D f.x t/g.t/dt (7.4.39) 0 as a multiplication of elements f and g in the ring of continuous functions on the half-axis RC (see, e.g., [Mik59]). In [LucSri95], the operational calculus for the ˛ Riemann–Liouville fractional derivative D0Cy was constructed. This method was generalized and developed by several authors (see [KilTru01]). This calculus was applied to the solution of Cauchy-type problems for fractional differential equations of a special kind. The idea of the composition method for ordinary differential equations of fractional order is based on the known formula for the Riemann–Liouville fractional derivative

.ˇ/ .D˛ .t a/ˇ1/.x/ D .x a/ˇ˛1 .Re.ˇ/ > Re.˛/ > 0/ (7.4.40) aC .ˇ ˛/

(see [SaKiMa93, (2.26) and (2.35)]). These arguments lead us to the conjecture that compositions of fractional derivatives and integrals with elementary functions can give exact solutions of differential and integral equations of fractional order. Moreover, from here we deduce another possibility concerning such results for compositions of fractional calculus operators with special functions. It allows us to find the exact solutions of new classes of differential and integral equations of fractional order. This method was developed by A. Kilbas with co-authors and uses composition of Riemann–Liouville fractional operators with different type of special functions (in particular, the three parametric Mittag-Leffler function, or Kilbas–Saigo function, see, e.g. [KilSai95a]and[KiSrTr06] and references therein). The above investigations were devoted to the solution of the fractional differ- ˛ ential equations with the Riemann–Liouville fractional derivative DaCy on a finite interval Œa; b of the real axis R. Such equations with the Caputo fractional derivative 7.4 Historical and Bibliographical Notes 195

C ˛ DaCy have not been studied extensively. Gorenﬂo and Mainardi [GorMai96] applied the Laplace transform to solve the fractional differential equation C ˛ D0Cy .x/ y.x/ D f.x/ .x>0I a>0I >0/ with the Caputo fractional derivative of order ˛>0and with the initial conditions

.k/ y .0/ D bk .k D 0;1;:::;n 1I n 1<˛Ä nI n 2 N/:

They discussed the key role of the Mittag-Leffler function for the cases 1<˛<2 and 2<˛<3. In relation to this, see also the papers by Gorenflo and Mainardi [GorMai97], Gorenflo and Rutman [GorRut94], and Gorenflo et al. [GoMaSr98]. Luchko and Gorenflo [LucGor99] used the operational method to prove that the above Cauchy problem has a unique solution in terms of the Mittag-Leffler functions in a special space of functions on the half-axis RC. They also obtained the explicit solution to the Cauchy problem for the more general fractional differential equation

Xm C ˛ C ˛k D0Cy .x/ ck D0Cy .x/ D f.x/.˛>˛1 >::: >˛m 0/ kD1 via certain multivariate Mittag-Leffler functions. It was probably Dzhrbashyan [Dzh70] who first considered the Dirichlet-type problems for the integro-differential equations of fractional order. The problem is to find the solution y.x/ (in L.0; T / or in L2.0; T /) on a finite interval .0; T / of the following equation

.D y/.x/ Œ C q.x/y.x/ D 0.0

1˛0 1˛1 a.I0C y/.x/jxD0 C b.I0C y/.x/jxD0 D 0; (7.4.42) and

1˛0 1˛1 c.I0C y/.x/jxDT C d.I0C y/.x/jxDT D 0; (7.4.43) with Lipschitzian q.x/ and real a; b; c and d such that a2Cb2 D 1 and c2Cd 2 D 1. When ˛0 D ˛1 D ˛2 D 1 the problem (7.4.41), (7.4.42)–(7.4.43) is reduced to the Sturm–Liouville problem for the ordinary differential equation of second order: y00.x/ŒCq.x/y.x/ D 0.0

7.5 Exercises

7.5.1. Show that the integral equation ([Dav36, p. 280]) Z x1˛ x u.t/dt u.x/ D ˛ (7.5.1) .2 ˛/ .1 ˛/ 0 .x t/

has a solution u.x/ D E .xˇ/; ˇ D 1 ˛: (4.5.1a) ˇ 7.5.2. Show that the equation ([Dav36, p. 282]) Z x tu.t/dt u.x/ C D f.x/ 1=2 (7.5.2) 0 .x t/

is equivalent to Z x u.x/ t.x C t/u.t/dt D F.x/; (7.5.2a) 2 0

where p 1p F.x/ D f.x/ xD1=2f.x/C D3=2f.x/: (7.5.2b) 0C 2 0C 7.5.3. Show that the equation ([Dav36, p. 282]) Z x t 2u.t/dt u.x/ C D f.x/ 2=3 (7.5.3) 0 .x t/

is equivalent to Z 3.1=3/ x u.x/ C t 2.44x4 C 40x3t C 75x2t 240xt3 C 44t 4/u.t/dt D F.x/; 243 0 (7.5.3a) where h i 2 1=3 2 4=3 4 7=3 F.x/ D f.x/C .1=3/ x D0C f.x/ 3 xD0C f.x/C 9 xD0C f.x/ h 2 4 2=3 3 5=3 34 2 8=3 C .2=3/ x D0C f.x/ 2x D0C f.x/C 9 x D0C f.x/ i 16 11=3 352 14=3 3 xD0C f.x/C 81 D0C f.x/ : (7.5.3b) 198 7 Applications to Fractional Order Equations

7.5.4. Solve the following integral equation ([PolMan08, Eq. 3.1–6.44])

Z1 y.xt/ p dt D f.x/; 0

Hint. Reduce to the Abel integral equation of the ﬁrst kind. 7.5.5. Solve the following integral equation ([PolMan08, Eq. 3.1–6.46])

Z1 t y.xt/ dt D f.x/; 0

Hint. Reduce to the Abel integral equation of the ﬁrst kind. 7.5.6. Find an explicit solution to the so-called test Abel integral equation of the second kind ([GorVes91, (7.2.8)])

Zx u.t/dt u.x/ D 1 C dt D f.x/; 0

Zt 2c p c './ '.t/ D p t p p d; t 0: (7.5.7a) t 0

7.5.8. Solve the following fractional differential equation using the Laplace transform ([KiSrTr06, p. 281]) Á y D1=2y .x/ D .x > 0/: (7.5.8a) 0C x 7.5.9. Solve the Cauchy problem for the fractional differential equation with Caputo derivative ([KiSrTr06, p. 282, (5.1.20)]) C ˛ N D0Cy .x/ D f.x/; .x>0; m 1<˛Ä mI m 2 /; (7.5.9a)

y.0/ D y0.0/ D :::D y.m1/.0/ D 0: (7.5.9b)

Hint. Use the Laplace transform. 7.5.10. Solve the Cauchy problem for the fractional differential equation with Caputo derivative ([KiSrTr06, p. 282, (5.1.21)]) 7.5 Exercises 199 0 C ˛ R y .x/ C a D0Cy .x/ D f.x/; y.0/D c0 2 .x>0;0<˛<1/: (7.5.10a) Hint. Use the Laplace transform. 7.5.11. Solve the Cauchy problem for the fractional differential equation with Caputo derivative ([KiSrTr06, p. 282, (5.1.22)]) 00 C ˛ y .x/ C a D0Cy .x/ D f .x/; .x > 0; 0 < ˛ < 1/ (7.5.11a) 0 y.0/ D c0;y .0/ D c1I c0;c1 2 R: (7.5.12a)

Hint. Use the Laplace transform. 7.5.12. Show that the family of functions ([KiSrTr06, p. 284, (5.2.5–6)])

˛j ˛ yj .x/ D x E˛;˛C1j .x /.jD 1;:::;l/ (7.5.12a)

forms a fundamental system of solutions of the fractional differential equation ˛ N R D0Cy .x/ y.x/ D 0.x>0;l 1<˛Ä lI l 2 I 2 /: (7.5.12b) Hint. Use the Laplace transform. 7.5.13. Prove that if ˛ 1 C j ˇ then the functions ([KiSrTr06, p. 286, (5.2.30– 31)])

˛j ˛ˇ yj .x/ D x E˛ˇ;˛C1j .x / (7.5.13a)

form a system of linear independent solutions of the fractional differential equations Á ˛ ˇ D0Cy .x/ D0Cy .x/ D 0 .x > 0; l 1<˛Ä lI l 2 NI 2 R;˛>ˇ>0/: (7.5.13b)

Hint. Use the Laplace transform. 7.5.14. Show that the general solution to the equation ([KiSrTr06, p. 295, (5.2.83– 84)]) ˛ R D0Cy .x/ y.x/ D f.x/ .x>0;˛>0I 2 / (7.5.14a)

can be represented in the form of the Laplace convolution:

Zx ˛1 ˛ y.x/ D .x t/ E˛;˛ Œ.x t/ f.t/dt: (7.5.14b) 0 200 7 Applications to Fractional Order Equations

7.5.15. Solve the Cauchy type problem ([KiSrTr06, p. 295, 310, (5.2.83), (5.2.172)]) ˛k R D0C y .0C/ D bk .bk 2 I k D 1;:::;lI l 1<˛Ä l/ (7.5.15a)

for the fractional differential equation ˛ R D0Cy .x/ y.x/ D f.x/ .x>0;˛>0I 2 /: (7.5.15b)

Answer.

Zx Xl ˛1 ˛ ˛j ˛ y.x/ D .x t/ E˛;˛ Œ.x t/ f.t/dt C bj x E˛;˛C1j Œx : 0 j D1

7.5.16. Show that the particular solution to the inhomogeneous fractional differential equation ([KiSrTr06, p. 331, (5.4.14–5.4.16–17)]) ˛C1 ˛C1 ˛ ˛ R x D0C y .x/ C x D0Cy .x/ D f.x/ .˛>0;2 I x>0/ (7.5.16a) can be represented in the form

Z1 ˛; y.x/ D G1 .t/f .xt/dt; (7.5.16b) 0

where Ä ˇ .1 / .1 ;1/ ˇ G˛;.x/ D x˛ x1 ˇx : 1 .˛C 1 / 1 2 .˛; 1/; .2 ;1/ (7.5.16c)

Hint. Use the Mellin transform. Chapter 8 Applications to Deterministic Models

Here we present material illuminating the role of the Mittag-Leffler function and its generalizations in the study of deterministic models. It has already been mentioned that the Mittag-Leffler function is closely related to the Fractional Calculus (being called ‘The Queen Function of the Fractional Calculus’). This is why we focus our attention here to fractional (deterministic) models. We start with a technical Sect. 8.1 in which the fractional differential equations, related to the fractional relaxation and oscillation phenomena, are discussed in full detail. In the second part of the chapter (Sect. 8.2) some examples of physical and mechanical models involving fractional derivatives are briefly outlined. The main focus is on the problems of fractional visco-elasticity. For other deterministic fractional models we derive only the corresponding fractional differential equation. This section is intended to show how fractional models can appear and which features of fractional objects are useful for such modelling.

8.1 Fractional Relaxation and Oscillations

We now analyze the most simple differential equations of fractional order which have appeared in applications. For this purpose, we choose some examples which, by means of fractional derivatives, generalize the well-known ordinary differential equations related to relaxation and oscillation phenomena. In the ﬁrst subsection we treat the simplest types, which we refer to as the simple fractional relaxation and oscillation equations. Then, in the next subsection we consider the types, somewhat more cumbersome, which we refer to as the composite fractional relaxation and oscillation equations.

© Springer-Verlag Berlin Heidelberg 2014 201 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__8 202 8 Applications to Deterministic Models

8.1.1 Simple Fractional Relaxation and Oscillation

The classical phenomena of relaxation and oscillation in their simplest form are known to be governed by linear ordinary differential equations, of order 1 and 2 respectively, that hereafter we recall with the corresponding solutions. Let us denote by u D u.t/ the ﬁeld variable and by q.t/ a given continuous function, with t 0: The relaxation differential equation reads as

u0.t/ Du.t/ C q.t/; (8.1.1)

C whose solution, under the initial condition u.0 / D c0; is Z t t u.t/ D c0 e C q.t /e d: (8.1.2) 0

The oscillation differential equation reads as

u00.t/ Du.t/ C q.t/; (8.1.3)

C 0 C whose solution, under the initial conditions u.0 / D c0 and u .0 / D c1; is Z t u.t/ D c0 cos t C c1 sin t C q.t / sin d: (8.1.4) 0 generalization of Eqs. (8.1.1)and(8.1.3) is obtained by replacing the ordinary derivative with a fractional one of order ˛: In order to preserve the type of initial conditions required in the classical phenomena, we agree to replace the ﬁrst and second derivative in (8.1.1)and(8.1.3) with a Caputo fractional derivative of order ˛ with 0<˛<1and 1<˛<2;respectively. We agree to refer to the corresponding equations as the simple fractional relaxation equation and the simple fractional oscillation equation. Generally speaking, we consider the following differential equation of fractional order ˛>0; ! mX1 t k D˛ u.t/ D D˛ u.t/ u.k/.0C/ Du.t/ C q.t/; t > 0: (8.1.5) kŠ kD0

Here m is a positive integer uniquely deﬁned by m 1<˛Ä m; which provides .k/ C the number of the prescribed initial values u .0 / D ck ;kD 0;1;2;:::;m 1: Implicit in the form of (8.1.5) is our desire to obtain solutions u.t/ for which the u.k/.t/ are continuous for t 0; k D 0;1;:::;m 1: In particular, the cases of fractional relaxation and fractional oscillation are obtained for m D 1 and m D 2; respectively. We note that when ˛ D m is an integer, then equation (8.1.5) reduces to an ordinary differential equation whose solution can be expressed in terms of m 8.1 Fractional Relaxation and Oscillations 203 linearly independent solutions of the homogeneous equation and of one particular solution of the inhomogeneous equation. We summarize this well-known result as follows Z mX1 t u.t/ D ckuk.t/ C q.t /uı./ d: (8.1.6) kD0 0

k .h/ C uk.t/ D J u0.t/ ; uk .0 / D ıkh;h;kD 0; 1; : : : ; m 1; (8.1.7)

0 uı.t/ Du0.t/ ; (8.1.8)

Rt where J k is a k-times repeated integral, J 1u.t/ D J u.t/ D u./d. Thus, the m 0 functions uk.t/ represent the fundamental solutions of the differential equation of order m; namely those linearly independent solutions of the homogeneous equation which satisfy the initial conditions in (8.1.7). The function uı.t/; with which the free term q.t/ appears convoluted, represents the so-called impulse-response solution, namely the particular solution of the inhomogeneous equation with all ck Á 0; k D 0;1;:::;m 1; and with q.t/ D ı.t/: In the cases of ordinary t relaxation and oscillation we recognize that u0.t/ D e D uı.t/ and u0.t/ D cos t; u1.t/ D J u0.t/ D sin t D cos .t =2/ D uı.t/; respectively. Remark 8.1. The more general equation ! mX1 t k D˛ u.t/ u.k/.0C/ D ˛ u.t/ C q.t/; > 0; t > 0; (8.1.9) kŠ kD0 can be reduced to (8.1.5) by a change of scale t ! t= . We prefer, for ease of notation, to discuss the “dimensionless” form (8.1.5). Let us now solve (8.1.5) by the method of Laplace transforms. For this purpose we can use the Caputo formula directly or, alternatively, reduce (8.1.5) with the prescribed initial conditions to an equivalent (fractional) integral equation and then treat the integral equation by the Laplace transform method. Here we prefer to follow the second approach. Then, applying the operator of fractional integration I ˛ to both sides of (8.1.5) we obtain

mX1 t k u.t/ D c I ˛ u.t/ C I ˛ q.t/: (8.1.10) k kŠ kD0

The application of the Laplace transform yields

mX1 c 1 1 uQ.s/ D k uQ.s/ C q.s/;Q skC1 s˛ s˛ kD0 204 8 Applications to Deterministic Models hence

mX1 s˛k1 1 uQ.s/ D c C q.s/:Q (8.1.11) k s˛ C 1 s˛ C 1 kD0 Introducing the Mittag-Lefﬂer type functions

s˛1 e .t/ Á e .tI 1/ WD E .t ˛ / ; (8.1.12) ˛ ˛ ˛ s˛ C 1

s˛k1 u .t/ WD J ke .t/ ;kD 0; 1; : : : ; m 1; (8.1.13) k ˛ s˛ C 1 we ﬁnd, from inversion of the Laplace transforms in (3.10), Z mX1 t 0 u.t/ D ck uk.t/ q.t /u0./ d: (8.1.14) kD0 0

To ﬁnd the last term in the right-hand side of (8.1.14), we have to use the well-known C C rule for the Laplace transform of the derivative, noting that u0.0 / D e˛.0 / D 1; and Â Ã 1 s˛1 D s 1 u0 .t/ De0 .t/ : (8.1.15) s˛ C 1 s˛ C 1 0 ˛

The formula (8.1.14) encompasses the solutions (8.1.2)and(8.1.4) found for ˛ D 1; 2;respectively. When ˛ is not an integer, namely for m1<˛

.h/ C uk .0 / D ıkh;h;kD 0;1;:::;m 1; (8.1.16) and therefore they represent the fundamental solutions of the fractional equation 0 (8.1.5), in analogy with the case ˛ D m. Furthermore, the function uı.t/ De˛.t/ represents the impulse-response solution. Hereafter, we are going to compute and exhibit the fundamental solutions and the impulse-response solution for the cases (a) 0<˛<1and (b) 1<˛<2, pointing out the comparison with the corresponding solutions obtained when ˛ D 1 and ˛ D 2. We now infer the relevant properties of the basic functions e˛.t/ directly from their representation as a Laplace inverse integral Z 1 s˛1 e .t/ D st ds ; ˛ e ˛ (8.1.17) 2i Br s C 1 8.1 Fractional Relaxation and Oscillations 205 in detail for 0<˛Ä 2; without having to make a detour into the general theory of Mittag-Lefﬂer functions in the complex plane. In (8.1.17) Br denotes the Bromwich path, i.e. a line Re fsgD with a value 1,andImfsg running from 1 to C1. For reasons of transparency, we separately discuss the cases

.a/0<˛<1 and .b/1<˛<2; recalling that in the limiting cases ˛ D 1; 2, we know e˛.t/ as an elementary t function, namely e1.t/ D e and e2.t/ D cos t: For ˛ not an integer the power function s˛ is uniquely deﬁned as s˛ Djsj˛ ei arg s , with

e˛.t/ D f˛.t/ C g˛ .t/ ; t 0; (8.1.18) with Z 1 s˛1 f .t/ WD e st ds; (8.1.19) ˛ 2i s˛ C 1 Ha. / where now the Hankel path Ha. / denotes a loop comprising a small circle jsjD with ! 0 and two sides of the cut negative real semi-axis, and Ä X ˛1 X s0 t s 1 s0 t g˛.t/ WD e h Res D e h : (8.1.20) s˛ C 1 0 ˛ h sh h

0 ˛1 ˛ Here sh are the relevant poles of s =.s C 1/. In fact the poles turn out to be sh D exp Œi.2h C 1/=˛ with unit modulus; they are all simple but the only relevant 0 ones are those situated in the main Riemann sheet, i.e. the poles sh with argument 0 such that : As a consequence,

g˛.t/ Á 0; hence e˛.t/ D f˛.t/ ; if 0<˛<1: (8.1.21) 206 8 Applications to Deterministic Models

0 If 1<˛<2;then there exist precisely two relevant poles, namely s0 D exp.i=˛/ 0 0 and s1 D exp.i=˛/ D s0 ; which are located in the left half-plane. Then one obtains h Ái 2 g .t/ D et cos .=˛/ cos t sin ; if 1<˛<2: (8.1.22) ˛ ˛ ˛

We note that this function exhibits oscillations with circular frequency !.˛/ D sin .=˛/ and with an exponentially decaying amplitude with rate .˛/ D j cos .=˛/j: Remark 8.2. One easily recognizes that (8.1.22) is also valid for 2 Ä ˛<3:In the classical case ˛ D 2 the two poles are purely imaginary (coinciding with ˙i)sothat we recover the sinusoidal behaviour with unitary frequency. In the case 2<˛<3; however, the two poles are located in the right half-plane, so providing ampliﬁed oscillations. This instability, which is common to the case ˛ D 3; is the reason why we limit ourselves to consider ˛ in the range 0<˛Ä 2:

In addition to the basic fundamental solutions u0.t/ D e˛.t/ we need to compute 1 the impulse-response solutions uı.t/ DD e˛.t/ for cases (a) and (b) and, only 1 in case (b), the second fundamental solution u1.t/ D J e˛.t/: For this purpose we note that in general it turns out that Z 1 k rt J f˛.t/ D e K˛;k.r/ dr; (8.1.23) 0 with

.1/k r˛1k sin .˛/ K .r/ WD .1/k rk K .r/ D ; (8.1.24) ˛;k ˛ r2˛ C 2r˛ cos .˛/ C 1 where K˛.r/ D K˛;0.r/; and h Á i 2 J kg .t/ D et cos .=˛/ cos t sin k : (8.1.25) ˛ ˛ ˛ ˛

This can be done in direct analogy to the computation of the functions e˛.t/,the k Laplace transform of J e˛.t/ being given by (8.1.13). For the impulse-response solution we note that the effect of the differential operator D1 is the same as that of the virtual operator J 1. In conclusion we can resume the solutions for the fractional relaxation and oscillation equations as follows: (a) 0<˛<1; Z t u.t/ D c0 u0.t/ C q.t /uı./ d; (8.1.26) 0 8.1 Fractional Relaxation and Oscillations 207

where 8 Z ˆ 1 ˆ rt <ˆ u0.t/ D e K˛;0.r/ dr; 0 Z (8.1.27) ˆ 1 ˆ rt : uı.t/ D e K˛;1.r/ dr; 0

C C with u0.0 / D 1; uı.0 / D1I (b) 1<˛<2; Z t u.t/ D c0 u0.t/ C c1 u1.t/ C q.t /uı./ d; (8.1.28) 0

where 8 Z h Ái ˆ 1 2 ˆ rt t cos .=˛/ ˆ u0.t/ D e K˛;0.r/ dr C e cos t sin ; ˆ 0 ˛ ˛ ˆ <ˆ Z h Á i 1 2 u .t/ D e rt K .r/ dr C et cos .=˛/ cos t sin ; ˆ 1 ˛;1 ˛ ˛ ˛ ˆ 0 ˆ Z ˆ 1 h Á i ˆ rt 2 t cos .=˛/ : uı.t/ D e K˛;1.r/ dr e cos t sin C ; 0 ˛ ˛ ˛ (8.1.29)

C 0 C C 0 C C with u0.0 / D 1; u0.0 / D 0; u1.0 / D 0; u1.0 / D 1; uı.0 / D 0; 0 C uı.0 / DC1:

In Fig. 8.1 we show the plots of the spectral distributions K˛.r/ for ˛ D 0:25 ; 0:50 ; 0:75 ; 0:9; and K˛.r/ for ˛ D 1:25 ; 1:50 ; 1:75 ; 1:9: ab

Kα(r) −Kα(r) 1 0.5 α=0.90 α=1.25

0.5 α =0.75 α=1.50

α=0.50 α=1.75 α=0.25 α r =1.90 r 0 0.5 1 1.5 2 0 0.5 1 1.5 2

Fig. 8.1 K˛.r/ for 0<˛<1(left)and1<˛<2(right) 208 8 Applications to Deterministic Models ab 1 1 α α eα(t)=Eα(−t ) eα(t)=Eα(−t ) 0.8

0.6 0.4

0.2 t 0.5 0 α=1.25 5 10 15 α=0.25 −0.2 α=1.5 −0.4 α=0.50 α=1.75 −0.6 α=0.75 −0.8 α=2 α=1 t −1 0 5 10 15

Fig. 8.2 The solution u0.t/ D e˛ .t/ for 0<˛Ä 1 (left)and1<˛Ä 2 (right)

In Fig. 8.2 we show the plots of the basic fundamental solution for the following cases: (left) ˛ D 0:25 ; 0:50 ; 0:75 ; 1; and (right) ˛ D 1:25 ; 1:50 ; 1:75 ; 2; obtained from the first formula in (8.1.27)and(8.1.29), respectively. We have verified that our present results confirm those obtained by Blank [Bla97] by a numerical treatment and those obtained by Mainardi [Mai96a]byananalytical treatment, valid when ˛ is a rational number. Of particular interest is the case ˛ D 1=2 where we recover a well-known formula from the theory of the Laplace transform, p p 1 e .t/ WD E . t/ D e t erfc. t/ ; (8.1.30) 1=2 1=2 s1=2 .s1=2 C 1/ where erfc denotes the complementary error function. We now point out that in both cases (a) and (b) (in which ˛ is just not integer) i.e. for fractional relaxation and fractional oscillation, all the fundamental and impulse- response solutions exhibit an algebraic decay as t !1; as discussed below. Let us start with the asymptotic behaviour of u0.t/: To this end we first derive an asymptotic series for the function f˛.t/, valid for t !1: Using the identity

1 sN˛ D 1 s˛ C s2˛ s3˛ CC.1/N 1 s.N 1/˛ C .1/N ; s˛ C 1 s˛ C 1 in formula (8.1.19) and the Hankel representation of the reciprocal Gamma function, we (formally) obtain the asymptotic expansion (for non-integer ˛)

XN t n˛ f .t/ D .1/n1 C O t .N C1/˛ ; as t !1: (8.1.31) ˛ .1 n˛/ nD1

The validity of this asymptotic expansion can be established rigorously using the (generalized) Watson lemma, see [BleHan86]. We can also start from the spectral 8.1 Fractional Relaxation and Oscillations 209

ab x 10−3 0.2 1 α=1.25 α=1.75

gα(t) gα(t) 0.1 0.5

t t 0 0 0 5 10 30 40 50 60

eα(t) −0.1 −0.5 fα(t) eα(t) fα(t)

−0.2 −1

Fig. 8.3 Decay of the solution u0.t/ D e˛ .t/ for ˛ D 1:25; 1:75 representation and expand the spectral function for small r: Then the (ordinary) Watson lemma yields (8.1.31). We note that this asymptotic expansion coincides with that for u0.t/ D e˛.t/, having assumed 0<˛<2(˛ ¤ 1). In fact the contribution of g˛.t/ is identically zero if 0<˛<1and exponentially small as t !1if 1<˛<2:The asymptotic expansions of the solutions u1.t/ and uı.t/ are obtained from (8.1.31) by integrating or differentiating term-by-term with respect to t: In particular, taking the leading term in (8.1.31), we obtain the asymptotic representations

t ˛ t 1˛ t ˛1 u .t/ ; u .t/ ; u .t/ ; as t !1: 0 .1 ˛/ 1 .2 ˛/ ı .˛/ (8.1.32)

They yield the algebraic decay of the fundamental and impulse-response solutions. In Fig. 8.3 we show some plots of the basic fundamental solution u0.t/ D e˛.t/ for ˛ D 1:25 ; 1:75: Here the algebraic decay of the fractional oscillation can be recognized and compared with the two contributions provided by f˛ (monotonic behaviour) and g˛.t/ (exponentially damped oscillation).

The Zeros of the Solutions of the Fractional Oscillation Equation

Now we carry out some investigations concerning the zeros of the basic fundamental solution u0.t/ D e˛.t/ in the case (b) of fractional oscillations. For the second fundamental solution and the impulse-response solution the analysis of the zeros can be easily carried out analogously. Recalling the ﬁrst equation in (8.1.29), the required zeros of e˛.t/ are the solutions of the equation h Ái 2 e .t/ D f .t/ C e t cos .=˛/ cos t sin D 0: (8.1.33) ˛ ˛ ˛ ˛ 210 8 Applications to Deterministic Models

We first note that the function e˛.t/ exhibits an odd number of zeros, in that e˛.0/ D 1; and, for sufficiently large t, e˛.t/ turns out to be permanently negative, as shown in (8.1.32) by the sign of .1 ˛/: The smallest zero lies in the first positivity interval of cos Œt sin .=˛/; hence in the interval 0

2 e t cos .=˛/ jf .t/j : (8.1.34) ˛ ˛ When t is sufﬁciently large, the zeros are expected to be found approximately from the equation

2 t ˛ e t cos .=˛/ ; (8.1.35) ˛ j.1 ˛/j obtained from (8.1.33) by ignoring the oscillation factor of g˛.t/ (see (8.1.22)) and taking the first term in the asymptotic expansion of f˛.t/ (see (8.1.31)and(8.1.32)). This approximation turns out to be useful when ˛ ! 1C and ˛ ! 2.For˛ ! 1C; only one zero is present, which is expected to be very far from the origin in view of the large period of the function cos Œt sin .=˛/: In fact, since there is no zero for ˛ D 1, and by increasing ˛ more and more zeros arise, we are sure that only one zero exists for ˛ sufficiently close to 1. Putting ˛ D 1 C , the asymptotic C position T of this zero can be found from the relation (8.1.35) in the limit ! 0 : Assuming in this limit a first-order approximation, we get Â Ã 2 T log ; (8.1.36) which shows that T tends to infinity slower than 1= ; as ! 0: For ˛ ! 2 , there is an increasing number of zeros up to infinity since e2.t/ D cos t has infinitely many zeros (tn D .n C 1=2/ ; n D 0; 1; : : : ). Putting now ˛ D 2 ı the asymptotic position T for the largest zero can be found again from (8.1.35) in the limit ı ! 0C: Assuming in this limit a first-order approximation, we get Â Ã 12 1 T log : (8.1.37) ı ı

C Now, for ı ! 0 the length of the positivity intervals of g˛.t/ tends to and, as long as t Ä T; there are two zeros in each positivity interval. Hence, in the limit ı ! 0C; there is on average one zero per interval of length ; so we expect that N T=: For the above considerations on the zeros of the oscillating Mittag-Lefﬂer function we were inspired by the paper of Wiman [Wim05b] who at the beginning 8.1 Fractional Relaxation and Oscillations 211

Table 8.1 The intervals of N ˛ T amplitude ˛ D 0:01 where the transitions of number of 1 3 1:40 1:41 1:730 5:726 zeros occur, and the location 3 5 1:56 1:57 8:366 13:48 T of the largest zeros. 5 7 1:64 1:65 14:61 20:00 7 9 1:69 1:70 20:80 26:33 9 11 1:72 1:73 27:03 32:83 11 13 1:75 1:76 33:11 38:81 13 15 1:78 1:79 39:49 45:51 15 17 1:79 1:80 45:51 51:46

N = number of zeros, ˛ = fractional order, T location of the largest zero of the 20th century, after having treated the Mittag-Lefﬂer function in the complex plane, considered the position of the zeros of the function on the negative real semi- axis (without providing any details). Our expressions for T differ from those of Wiman in numerical factors; however, the results of our numerical studies conﬁrm and illustrate the validity of our analysis. Here, we analyse the phenomenon of the transition of the (odd) number of zeros as 1:4 Ä ˛ Ä 1:8: For this purpose, in Table 8.1 we report the intervals of amplitude ˛ D 0:01 where these transitions occur, and the location T of the largest zeros (evaluated within a relative error of 0:1 %) found at the two extreme values of the above intervals. We recognize that the transition from 1 to 3 zeros occurs as 1:40 Ä ˛ Ä 1:41, that the transition from 3 to 5 zeros occurs as 1:56 Ä ˛ Ä 1:57,andso on. The last transition in the considered range of ˛ is from 15 to 17 zeros, and it occurs as 1:79 Ä ˛ Ä 1:80:

8.1.2 The Composite Fractional Relaxation and Oscillations

In this subsection we consider the following fractional differential equations for t 0; equipped with suitable initial conditions,

du d˛u C a C u.t/ D q.t/; u.0C/ D c ;0<˛<1; (8.1.38) dt dt ˛ 0

d2v d˛v C a C v.t/ D q.t/; v.0C/ D c ; v0.0C/ D c ;0<˛<2; (8.1.39) dt 2 dt ˛ 0 1 where a is a positive constant. The unknown functions u.t/ and v.t/ (the field variables) are required to be sufficiently well behaved to be treated with their derivatives u0.t/ and v0.t/ ; v00.t/ by the technique of the Laplace transform. The given function q.t/ is assumed to be continuous. In the above equations the ˛ fractional derivative of order ˛ is assumed to be provided by the operator D, 212 8 Applications to Deterministic Models the Caputo derivative, in agreement with our choice in the previous subsection. Note that in (8.1.39) we distinguish the cases .a/0<˛<1;.b/ 1<˛<2 and ˛ D 1: Equations (8.1.38)and(8.1.39) will be referred to as the composite fractional relaxation equation and the composite fractional oscillation equation, respectively, to be distinguished from the corresponding simple fractional equations. Here we also apply the method of the Laplace transform to solve the fractional differential equations and get some insight into their fundamental and impulse- response solutions. However, in contrast with the previous subsection, we now find it more convenient to apply the corresponding formula for the Laplace transform of fractional and integer derivatives directly, instead of reducing the equations with the prescribed initial conditions as equivalent (fractional) integral equations to be treated by the Laplace transform. Let us apply the Laplace transform to the composite fractional relaxation equation (8.1.38). This leads us to the transformed algebraic equation

1 C as˛1 q.s/Q uQ.s/ D c0 C ; 0<˛<1; (8.1.40) w1.s/ w1.s/ where

˛ w1.s/ WD s C as C 1; (8.1.41) and a>0:Putting

1 C as˛1 1 u0.t/ Qu0.s/ WD ; uı.t/ Quı.s/ WD ; (8.1.42) w1.s/ w1.s/ and recognizing that

C u0.0 / D lim s uQ0.s/ D 1; uQı.s/ DŒs uQ0.s/ 1 ; (8.1.43) s!1 we conclude that Z t 0 u.t/ D c0 u0.t/ C q.t /uı./ d; uı.t/ Du0.t/ : (8.1.44) 0

Thus u0.t/ and uı.t/ are respectively the fundamental solution and impulse-response solution to Eq. (8.1.38). Let us ﬁrst consider the problem of ﬁnding u0.t/ as the inverse Laplace transform of uQ0.s/: We easily see that the function w1.s/ has no zero in the main sheet of the Riemann surface including the sides of the cut (simply show that Im fw1.s/g does not vanish if s is not a real positive number), so that the inversion of the Laplace transform uQ0.s/ can be carried out by deforming the original Bromwich path into the Hankel path Ha. / introduced in the previous subsection, i.e. into the loop constituted by a small circle jsjD with ! 0 and by the two borders of the cut negative real axis. As a consequence we write 8.1 Fractional Relaxation and Oscillations 213 Z 1 1 C as˛1 u .t/ D st 0 e ˛ (8.1.45) 2i Ha. / s C as C 1

It is now an exercise in complex analysis to show that the contribution from the Hankel path Ha. / as ! 0 is provided by Z 1 rt .1/ u0.t/ D e H˛;0 .rI a/ dr; (8.1.46) 0 with ˇ ˛1 ˇ .1/ 1 1 C as ˇ H˛;0 .rI a/ D Im ˇ w1.s/ sDr ei (8.1.47) 1 ar˛1 sin .˛/ D : .1 r/2 C a2 r2˛ C 2.1 r/ar˛ cos .˛/

.1/ For a>0and 0<˛<1the function H˛;0.rI a/ is positive for all r>0since it has the sign of the numerator; in fact in (8.1.47) the denominator is strictly positive, 2 ˙i being equal to jw1.s/j as s D r e : Hence, the fundamental solution u0.t/ has .1/ the peculiar property of being completely monotone,andH˛;0 .rI a/ is its spectral 0 function. Now the determination of uı.t/ Du0.t/ is straightforward. We see that the impulse-response solution uı.t/ is also completely monotone since it can be represented by Z 1 rt .1/ uı.t/ D e H˛;1.rI a/ dr; (8.1.48) 0 with spectral function

1 ar˛1 sin .˛/ H .1/ .rI a/ D rH.1/.rI a/ D : ˛;1 ˛;0 .1 r/2 C a2 r2˛ C 2.1 r/ar˛ cos .˛/ (8.1.49)

Both solutions u0.t/ and uı.t/ turn out to be strictly decreasing from 1 towards 0 as t runs from 0 to 1: Their behaviour as t ! 0C and t !1can be found by means of a proper asymptotic analysis. The behaviour of the solutions as t ! 0C can be determined from the behaviour of their Laplace transforms as Re fsg!C1as is well known from the theory of the Laplace transform, see, e.g. [Doe74]. We obtain as Re fsg!C1; 1 2 3C˛ 1 .2˛/ 2 uQ0.s/ D s s CO s ; uQı.s/ D s as CO s ; (8.1.50) so that 214 8 Applications to Deterministic Models

t 1˛ u .t/ D 1 t C O t 2˛ ; u .t/ D 1 a C O.t/; as t ! 0C : 0 ı .2 ˛/ (8.1.51) The spectral representations (8.1.46)and(8.1.48) are suitable to obtain the asymptotic behaviour of u0.t/ and uı.t/ as t !C1; by using the Watson lemma. In fact, expanding the spectral functions for small r and taking the dominant term in the corresponding asymptotic series, we obtain

t ˛ t ˛1 u .t/ a ; u .t/ a ; as t !1: (8.1.52) 0 .1 ˛/ ı .˛/

We note that the limiting case ˛ D 1 can easily be treated by extending the validity of (8.1.40)–(8.1.44)to˛ D 1; as is legitimate. In this case we obtain

1 u .t/ D e t=.1Ca/ ; u .t/ D e t=.1Ca/ ;˛D 1: (8.1.53) 0 ı 1 C a

t In the case a Á 0 we recover the standard solutions u0.t/ D uı.t/ D e : We conclude with some considerations on the solutions when the order ˛ is a rational number. If we take ˛ D p=q; where p; q 2 N are assumed (for convenience) to be relatively prime, a factorization in (8.1.41) is possible by using the pro- cedure indicated by Miller and Ross [MilRos93]. In these cases the solutions can be expressed in terms of a linear combination of q Mittag-Lefﬂer functions of fractional order 1=q, which, in turn, can be expressed in terms of incomplete gamma functions. Here we illustrate the factorization in the simplest case ˛ D 1=2 and provide the solutions u0.t/ and uı.t/ in terms of the functions e˛.tI / (with ˛ D 1=2), introduced in the previous subsection. In this case, in view of the application to the Basset problem equation (8.1.38) deserves particular attention.1 For ˛ D 1=2 we can write

1=2 1=2 1=2 2 1=2 w1.s/ D s C as C 1 D .s C/.s /; ˙ Da=2 ˙ .a =4 1/ : (8.1.54)

Here ˙ denote the two roots (real or conjugate complex) of the second degree polynomial with positive coefﬁcients z2 C az C 1; which, in particular, satisfy the following binary relations

2 1=2 2 1=2 C D 1; C C Da; C D 2.a =4 1/ D .a 4/ : (8.1.55) We recognize that we must treat separately the following two cases

1Basset considered in [Bas88] a model of a quiescent ﬂuid which leads in modern language to Eq. (8.1.38) with ˛ D 1=2. For arbitrary 0<˛<1the (generalized) Basset problem is discussed in [Mai97]. 8.1 Fractional Relaxation and Oscillations 215

.i/ 0 < a < 2 ; or a>2; and .ii/aD 2; which correspond to two distinct roots (C ¤ ), or two coincident roots (C Á D1), respectively. For this purpose, we write 8 A A <ˆ .i/ C C ; 1 C as1=2 1=2 1=2 1=2 1=2 M.s/Q WD D s .s C/ s .s / 1=2 ˆ 2 s C as C 1 : .ii/1 C ; .s1=2 C 1/2 s1=2.s1=2 C 1/2 (8.1.56) and 8 A A <ˆ .i/ C C ; 1 1=2 1=2 1=2 1=2 N.s/Q WD D s .s C/ s .s / 1=2 ˆ 1 s C as C 1 : .ii/ ; .s1=2 C 1/2 (8.1.57) where

˙ A˙ D˙ : (8.1.58) C

Using (8.1.55) we note that

AC C A D 1; AC C A C D 0; AC C C A Da: (8.1.59)

Recalling the Laplace transform pairs we obtain p p .i/ A E1=2 .C t/ C AC Ep1=2 . t/; u0.t/ D M.t/ WD p (8.1.60) .ii/.1 2t/E1=2 . t/C 2 t=; and p p .i/ AC E1=2.C t/C A Ep1=2. t/; uı.t/ D N.t/ WD p (8.1.61) .ii/.1C 2t/E1=2. t/ 2 t=: p In (8.1.60)and(8.1.61p) the functions e1=2.tI˙/ D E1=2.˙ t/ and e1=2.t/ D e1=2.tI 1/ D E1=2. t/ are presented. In particular, the solution of the Basset problem can be easily obtainedR from (8.1.44) with q.t/ D q0 by using (8.1.60) t and (8.1.61) and noting that 0 N./d D 1 M.t/: Denoting this solution by uB .t/ we get

uB .t/ D q0 .q0 c0/M.t/: (8.1.62) 216 8 Applications to Deterministic Models

When a Á 0, i.e. in the absence of the term containing the fractional derivative (due to the Basset force), we recover the classical Stokes solution, which we denote by uS .t/;

t uS .t/ D q0 .q0 c0/ e :

In the particular case q0 D c0; we get the steady-state solution uB .t/ D uS .t/ Á q0: For vanishing initial condition c0 D 0; we have the creep-like solutions

t uB .t/ D q0 Œ1 M.t/ ; uS .t/ D q0 1 e :

In this case it is instructive to compare the behaviour of the two solutions as t ! 0C and t !1: Recalling the general asymptotic expressions of u0.t/ D M.t/ in (8.1.51)and(8.1.52) with ˛ D 1=2; we recognize that 3=2 2 C uB .t/ D q0 t C O t ; uS .t/ D q0 t C O t ; as t ! 0 ; and h p i uB .t/ q0 1 a= t ; uS .t/ q0 Œ1 EST ; as t !1; where EST denotes exponentially small terms. In particular we note that the normalized plot of uB.t/=q0 remains under that of uS .t/=q0 as t runs from 0 to 1: The reader is invited to convince himself of the following fact. In the general case 0<˛<1 the solution u.t/ has the particular property of being equal to 1 for all t 0 if q.t/ has this property and u.0C/ D 1; whereas q.t/ D 1 for all t 0 and u.0C/ D 0 implies that u.t/ is a creep function tending to 1 as t !1: Let us now apply the Laplace transform to the fractional oscillation equation (8.1.39). This leads us to the transformed algebraic equations

s C as˛1 1 q.s/Q .a/ vQ.s/ D c0 C c1 C ; 0<˛<1; (8.1.63) w2.s/ w2.s/ w2.s/ or

s C as˛1 1 C as˛2 q.s/Q .b/ vQ.s/ D c0 C c1 C ; 1<˛<2; (8.1.64) w2.s/ w2.s/ w2.s/ where

2 ˛ w2.s/ WD s C as C 1; (8.1.65) and a>0:Putting

s C as˛1 vQ0.s/ WD ;0<˛<2; (8.1.66) w2.s/ 8.1 Fractional Relaxation and Oscillations 217 we have

C 1 0 v0.0 / D lim s vQ0.s/ D 1; DŒs vQ0.s/ 1 v0.t/ ; (8.1.67) s!1 w2.s/ and Z ˛2 t 1 C as vQ0.s/ D v0./ d: (8.1.68) w2.s/ s 0

Thus we can conclude that Z t 0 0 .a/ v.t/ D c0 v0.t/ c1 v0.t/ q.t /v0./ d; 0<˛<1; (8.1.69) 0 or Z Z t t 0 .b/ v.t/ D c0 v0.t/Cc1 v0./ d q.t/v0./ d; 1<˛<2: (8.1.70) 0 0

0 In both equations the term v0.t/ represents the impulse-response solution vı.t/ for the composite fractional oscillation equation (8.1.39), namely the particular solution of the inhomogeneous equation with c0 D c1 D 0 and with q.t/ D ı.t/: For the fundamental solutions of (8.1.39) we have two distinct pairs of solutions according to the case (a) and (b) which read Z t 0 .a/ fv0.t/ ; v1a.t/ Dv0.t/g ;.b/ fv0.t/ ; v1b.t/ D v0./ dg : (8.1.71) 0

We ﬁrst consider the particular case ˛ D 1 for which the fundamental and impulse response solutions are known in terms of elementary functions. This limiting case can also be treated by extending the validity of (8.1.63)and(8.1.69)to˛ D 1; as is legitimate. From

s C a s C a=2 a=2 vQ .s/ D D ; 0 s2 C asC 1 .s C a=2/2 C .1 a2=4/ .s C a=2/2 C .1 a2=4/ (8.1.72) we obtain the basic fundamental solution 8 h i a ˆ eat=2 cos.!t/ C sin.!t/ if 02; 2 218 8 Applications to Deterministic Models where p p ! D 1 a2=4 ; D a2=4 1: (8.1.74)

By a differentiation of (8.1.73) we easily obtain the second fundamental solution 0 v1a.t/ and the impulse-response solution vı.t/ since v1a.t/ D vı.t/ Dv0.t/: We point out that all the solutions exhibit an exponential decay as t !1: Let us now consider the problem of ﬁnding v0.t/ as the inverse Laplace transform of vQ0.s/; Z ˛1 1 st s C as v0.t/ D e ds; (8.1.75) 2i Br w2.s/ where Br denotes the usual Bromwich path. Using a result of Beyer and Kempﬂe [BeyKem95] we know that the function w2.s/ (for a>0and 0<˛<2;˛¤ 1) has exactly two simple, conjugate complex zeros on the principal branch in the Ci open left half-plane, cut along the negative real axis, say sC D e and s D e i with >0and =2 < < : This enables us to repeat the considerations carried out for the simple fractional oscillation equation to decompose the basic fundamental solution v0.t/ into two parts according to v0.t/ D f˛.tI a/ C g˛.tI a/: In fact, the evaluation of the Bromwich integral (8.1.75) can be achieved by adding the contribution f˛.tI a/ from the Hankel path Ha. /,as ! 0; to the residual contribution g˛.tI a/ from the two poles s˙: As an exercise in complex analysis we obtain Z 1 rt .2/ f˛.tI a/ D e H˛;0 .rI a/ dr; (8.1.76) 0 with spectral function ˇ ˛1 ˇ .2/ 1 s C as ˇ H˛;0 .rI a/ D Im ˇ w2.s/ sDr ei (8.1.77) 1 ar˛1 sin .˛/ D : .r2 C 1/2 C a2 r2˛ C 2.r2 C 1/ a r˛ cos .˛/

2 Since in (8.1.77) the denominator is strictly positive, being equal to jw2.s/j as ˙i .2/ s D r e ; the spectral function H˛;0 .rI a/ turns out to be positive for all r>0 for 0<˛<1and negative for all r>0for 1<˛<2:Hence, in case (a) the function f˛.t/ and in case (b) the function f˛.t/ is completely monotone; in both cases f˛.t/ tends to zero as t !1; from above in case (a) and from below in case (b), according to the asymptotic behaviour

t ˛ f .tI a/ a ; as t !1; 0<˛<1;1<˛<2; (8.1.78) ˛ .1 ˛/ as derived by applying the Watson lemma in (8.1.76) and considering (8.1.77). The other part, g˛.tI a/; is obtained as 8.2 Examples of Applications of the Fractional Calculus in Physical Models 219 Ä s C as˛1 g˛ .tI a/ D e sC t Res C conjugate complex w2.s/ sC ( ) (8.1.79) s C as˛1 C C sC t D 2 Re ˛1 e : 2sC C a˛sC

Thus this term exhibits an oscillatory character with exponentially decreasing amplitude like exp . tj cos j/: Then we recognize that the basic fundamental solution v0.t/ exhibits a ﬁnite number of zeros and that, for sufﬁciently large t; it turns out to be permanently positive if 0<˛<1and permanently negative if 1<˛<2with an algebraic decay provided by (8.1.78). For the second fundamental solutions v1 a.t/ ; v1 b.t/ and for the impulse-response solution vı.t/; the corresponding analysis is straightforward in view of their connection with v0.t/, pointed out in (8.1.70)and(8.1.71). The algebraic decay of all the solutions as t !1; for 0<˛<1 and 1<˛<2;is henceforth resumed in the relations

t ˛ t ˛1 t 1˛ v .t/ a ; v .t/ D v .t/ a ; v .t/ a : 0 .1 ˛/ 1 a ı .˛/ 1 b .2 ˛/ (8.1.80)

In conclusion, except in the particular case ˛ D 1; all the present solutions of the composite fractional oscillation equation exhibit similar characteristics as the corresponding solutions of the simple fractional oscillation equation, namely a ﬁnite number of damped oscillations followed by a monotonic algebraic decay as t !1:

8.2 Examples of Applications of the Fractional Calculus in Physical Models

Here we present a few physical models involving the fractional calculus. An interest in such models is growing rapidly nowadays and several books on the subject have recently appeared. It would be impossible to discuss general fractional models in detail here. Instead we select some models which are related to the above discussed equations (or their simple generalizations) and which demonstrate the essential role of the Mittag-Lefﬂer function in fractional modelling.

8.2.1 Linear Visco-Elasticity

Visco-Elastic Models

Let us ﬁrst introduce some notation. We denote the stress by D .x;t/ and the strain by D .x; t/ where x and t are the space and time variables, respectively. 220 8 Applications to Deterministic Models

For the sake of convenience, both stress and strain are intended to be normalized, i.e. scaled with respect to a suitable reference state f ; g: According to the linear theory of viscoelasticity, assuming the presence of sufﬁciently small strains, the body may be considered as a linear system with the stress (or strain) as the excitation function (input) and the strain (or stress) as the response function (output). To formulate general stress-strain relations (or constitutive equations), two fundamental hypotheses are required: (i) invariance under time translation and (ii) causality; the former means that a time shift in the input results in an equal shift in the output, the latter that the output for any instant t1 depends on the values of the input only for t Ä t1. A fundamental role is played by the step response, i.e. the response function, expressed by the Heaviside function .t/: ( 0 if t<0; .t/ D 1 if t>0:

Two magnitudes are deﬁned in this way:

.t/ D .t/ H) .t/ D J.t/; (8.2.1)

.t/ D .t/ H) .t/ D G.t/ : (8.2.2)

The functions J.t/ and G.t/ are referred to as the creep compliance and relaxation modulus respectively, or, simply, the material functions of the viscoelastic body. The general stress-strain relation is expressed through a linear hereditary integral of Stieltjes type, namely Z t .t/ D J.t /d ./: (8.2.3) 1 Z t .t/ D G.t /d ./ : (8.2.4) 1

In the classical Hook model for an elastic body we have

.t/ D m .t/; (8.2.5) and thus J.t/ D 1=m; G.t/ D m. In the classical Newton model for an ideal ﬂuid we have d .t/ D b ; (8.2.6) 1 dt and thus J.t/ D t=b1;G.t/D b1ı.t/. 8.2 Examples of Applications of the Fractional Calculus in Physical Models 221

Power-Law Creep and the Scott Blair Model

Based on certain rheological experiments Scott Blair [ScoB-Cop42a] argued that the material properties are determined by various states between an elastic solid and a viscous ﬂuid, rather than a combination of an elastic and a viscous element as proposed by Maxwell. The conclusion was that these materials satisfy a law intermediate between Hook’s law and Newton’s law: d .t/ D b ;0<<1: (8.2.7) 1 dt This yields the power-type behaviour of the creep function

t b J.t/ D ) G.t/ D 1 t : (8.2.8) b1.1C / .1 /

We point out that (8.2.7) is the differential form of the Nutting equation [Nut21] (for more details see [RogMai14]). By using a power-type law for the creep functions one can rewrite the constitutive relation (8.2.7) in a form involving either a fractional integral (see also the contribution by Rabotnov [Rab80], who presents the constitutive relation in the form of a more cumbersome integral equation)

Zt 1 ./d 1 .t/ D D I .t/; 1 1 t (8.2.9) b1.1C / .t / b1 1 or a fractional derivative (see also the pioneering contribution by Gerasimov [Ger48])

Zt b ./P d .t/ D 1 D b D .t/: (8.2.10) .1 / .t / 1 1 t 1

Here the fractional integrals and derivatives have Liouville (or Liouville–Weyl) form, i.e. with integration from 1. We note that the fractional derivative in (8.2.10) is similar to the Caputo derivative. Moreover, if we consider causal histories (i.e. starting from t D 0), then in (8.2.10) the Liouville fractional derivative will be replaced by the Caputo fractional derivative. The use of fractional calculus in linear viscoelasticity leads us to generalize the classical mechanical models, in that the basic Newton element (dashpot) is substituted by the more general Scott Blair element (of order ), sometimes referred to as pot. In fact, we can construct the class of these generalized models from Hooke and Scott Blair elements, disposed singly and in branches of two (in series or in parallel). The material functions are obtained using the combination rule; their 222 8 Applications to Deterministic Models determination is made easy if we take into account the following correspondence principle between the classical and fractional mechanical models, as introduced in [CapMai71a], which is empirically justiﬁed. Let us also present some constitutive equations and material functions which correspond to the most popular fractional models in visco-elasticity.

d fractional Newton .ScottBlair/ model W .t/ D b ; 1 dt 8 t <ˆJ.t/ D ; b1 .1C / ˆ t (8.2.11) :G.t/ D b I 1 .1 /

d fractional Voigt model W .t/ D m .t/C b ; 1 dt 8 ˆ 1

where . / D b1=m;

d d fractional Maxwell model W .t/ C a D b ; 1 dt 1 dt 8 ˆ a 1 t

where . / D a1;

fractional Zener model W Ä Ä d d 1 C a .t/ D m C b .t/ ; 1 dt 1 dt ( J.t/ D J C J Œ1 E Œ.t= / ; g 1 (8.2.14) G.t/ D Ge C G1 E Œ.t= / ; 8.2 Examples of Applications of the Fractional Calculus in Physical Models 223 where 8 ˆ a1 1 a1 b1

fractional antiZener model W Ä Ä d d d.C1/ 1 C a .t/ D b C b .t/ ; 1 dt 1 dt 2 dt .C1/ 8 t <ˆJ.t/ D J C J Œ1 E Œ.t= / ; C .1C / 1 ˆ t (8.2.15) :G.t/ D G C G E Œ.t= /; .1 / 1 where 8 ˆ 1 a b b <ˆ 1 2 2 JC D ;J1 D 2 ; D ; b1 b1 b1 b1 ˆ b b b :ˆ 2 1 2 G D ;G1 D 2 ; D a1 : a1 a1 a1

These equations can be treated in the same way as described in Sect. 8.1.2 for composite type fractional equations, i.e. by the Laplace transform method (for details see [Mai10]). The most general form is the so-called fractional operator equation " # " # Xp Xq d k d k 1 C ak .t/ D m C bk .t/ ; (8.2.16) dt k dt k kD1 kD1 with k D k C 1; and

8 X ˆ t <ˆJ.t/ D Jg C Jn f1 E Œ.t= ;n/ g C JC ; .1C / n ˆ X t :ˆG.t/ D G C G E Œ.t= / C G ; e n ;n .1 / n where all the coefﬁcients are non-negative. 224 8 Applications to Deterministic Models

8.2.2 Other Deterministic Fractional Models

Here we present a few other formulations of fractional deterministic models. Our choice is determined by two considerations. We try to avoid heavy machinery here in order to follow the main idea of this chapter, to give readers an impression of how the fractional models appear and what the advantages are of using the fractional approach. Second, since the subject we are touching is very large and growing rapidly, we understand that any survey text on the applications of the fractional calculus will soon become incomplete after publication. We note also that the material presented here is known and widely spread in the literature.

The Fractional Newton Equation

Let us consider motion of a body in an incompressible viscous Newtonian fluid. In the absence of external forces, the unsteady flow of such a fluid is governed by the Navier–Stokes system of equations. Following [Uch13b, Sec. 7.3] we write down this (simplified) system in the case of the problem of entrainment of the fluid by a large sized plate moving in the xOz-plane along the x-axis with the given velocity V.t/.

@v @2v D ;0

d2vO .i!/ vO.z;!/D .z;!/; (8.2.18) dz2 supplied by the boundary conditions

vO.z;!/D V.!/;O vO.1;!/D 0: (8.2.19)

Its solution has the form ( r ) i! vO.z;!/D V.!/O exp z : (8.2.20)

By introducing the shear stress (the viscous source per unit surface) .z;t/ D @v @z .z;t/,weget 8.2 Examples of Applications of the Fractional Calculus in Physical Models 225

p .O z;!/D .i!/1=2 vO.z;!/; (8.2.21) and thus, by applying the inverse Fourier transform, we obtain the following fractional differential equation Á p 1=2 .z;t/ D Dt v .z;t/; (8.2.22) where the fractional derivative can be understood either in the Riemann–Liouville or Caputo sense. The physical interpretation of this result is that the shear stress observed at the moment t at the point .x; z/ is a result of the contribution of the ﬂuid particles at the points .x0; z/, where they were located at t 0

The Fractional Ohm Law

Here we also follow the presentation given in [Uch13b, Sec. 11.4]. The polarization P.t/ of a dielectric, placed in an electric ﬁeld, consists of two parts, P1.t/ which is proportional to the intensity of the ﬁeld at time t,

P1.t/ D 1E.t/; and the retarded part

Zt 0 0 0 P2.t/ D K.t t /E.t /dt : 1

Let us denote by 2E the extreme value of E2 with constant E and t !1.In the classical relaxation theory it is assumed that the rate of the change of P2 is proportional to the difference between its extreme and current values:

dP 1 2 D . E P /: dt 2 2 The simplest macroscopic way to implement a fractional derivative into the relaxation problem is the fractional generalization of the classical law relating the current i.t/ and voltage u.t/ ˛ i.t/ D K˛ aDt u .t/: (8.2.23)

This relation describes an element of a chain which, in a sense, possesses an intermediate property between those for an ideal dielectric with capacity C D K1 and a simple conductor with resistance R D 1=K0. 226 8 Applications to Deterministic Models

If u.t 0/ varies continuously and rather rapidly approaches 0 as t 0 !1,then ˛ we take a D1in (8.2.23), understanding by 1Dt u the Liouville form of either the Riemann–Liouville or Caputo fractional derivative. Taking into account the presence of the active resistance in the chain, we write the Kirchhoff law

i.t/R C u.t/ D E.t/; where E.t/ is the electromotive force (EMF). Inserting relation (8.2.23) we obtain ˛ E K˛R1Dt C 1 u .t/ D .t/: (8.2.24)

If we assume E.t/ D ei!t then the solution to (8.2.24) coincides with the Cole–Cole response function fO˛ (see [ColCol41, ColCol42]):

˛ Œ.i!/ C 1 fO˛.i!/ D 1; 0 < ˛ < 1: (8.2.25)

Fractional Equations for Heat Transfer

See [Uch13b, Sec. 10.2]. The classical theory of heat conduction is based on Fourier’s law, relating the heat ﬂux q and the gradient of temperature T

q DrT (8.2.26) with conductivity coefﬁcient . There exist two fractional generalizations of Fourier’s law (see, e.g., [Uch13b, p. 164]) ˛ 1 C 0 0Dt q DrT; 0 < ˛ < 1; (8.2.27) Â Ã @ 1 C q D D1˛rT; 1 < ˛ Ä 2: (8.2.28) @t 0 t

The standard Fourier law leads to the following heat transfer equation (e.g. in the one-dimensional case, when we consider the heat transfer in a long rod)

@T @2T D : (8.2.29) @t @x2 This relation can be generalized, taking into account the dependence of the heat ﬂux on the history of the gradient of the temperature (some authors call this generalization the Maxwell–Cattaneo–Lykov equation). The one-dimensional equation of this type has the form:

@T @2T @2T C D : (8.2.30) @t @t2 @x2 8.3 Historical and Bibliographical Notes 227

In the three-dimensional case we have @T @2T C D 4T: (8.2.31) @t @t2 Considering two extremal possibilities for the relaxation time we have 8 <ˆ @T D 4T; ! 0.diffusion equation/I @T @2T @t C D 4T ) 2 ˆ @t @t : @2T @t2 D 4T; !C1.wave equation/; which is commonly rewritten as

@T 1 D 4T; D 1; 2: @t

Assuming again the intermediate situation .0; 2, we obtain the time-fractional generalization of the heat transfer equation

1 0Dt T D 4T; .0; 2: (8.2.32)

The fractional Fourier law allows us to formulate a fractional generalization of the Maxwell–Cattaneo–Lykov equation. Taking the gradient of both sides of Eqs. (8.2.27)and(8.2.28) and applying the heat balance equation

@T rq D ; @t we get, respectively,

@T C D1C˛T D 4T; 0 < ˛ < 1; (8.2.33) @t 0 0 t

@T @2T C D D1˛4T; 1 < ˛ < 2: (8.2.34) @t @t2 0 t

8.3 Historical and Bibliographical Notes

8.3.1 General Notes

One of the first investigations of differential equations of fractional order was made by Barrett [Barr54]. He considered differential equations with a fractional derivative of Riemann–Liouville type of arbitrary order ˛; Re ˛>0. n boundary conditions (n D Re ˛ C 1) in the form of the values at the initial points of the fractional 228 8 Applications to Deterministic Models derivatives of order ˛k; k D 1;2;:::;n;are posed. It was shown that in a suitable class of functions the solution is unique and is represented via the Mittag-Leffler function. As former applications in physics we would like to highlight the contributions by K.S. Cole [Col33], quoted by H.T. Davis [Dav36, p. 287] in connection with nerve conduction, and by F.M. de Oliveira Castro [Oli39], K.S. Cole and R.H. Cole (1941–1942) [ColCol41, ColCol42], and B. Gross (1947) [Gro47] in connection with dielectric and mechanical relaxation, respectively. Subsequently, in 1971, Caputo and Mainardi [CapMai71a] proved that the Mittag-Leffler function is present whenever derivatives of fractional order are introduced in the constitutive equations of a linear viscoelastic body. Since then, several other authors have pointed out the relevance of the Mittag-Leffler function for fractional viscoelastic models, see Mainardi [Mai97, Mai10].

8.3.2 Notes on Fractional Differential Equations

The fractional differential equation in (8.1.38) with ˛ D 1=2 corresponds to the Basset problem, a classical problem in fluid dynamics concerning the unsteady motion of a particle accelerating in a viscous fluid under the action of gravity. The fractional differential equation in (8.1.39) with 0<˛<2models an oscillation process with fractional damping term. It was formerly treated by Caputo, who provided a preliminary analysis using the Laplace transform. The special cases ˛ D 1=2 and ˛ D 3=2; but with the standard definition D˛ for the fractional derivative, were discussed by Bagley [Bag90]. Beyer and Kempfle [BeyKem95] considered (8.1.39)for1 0:

8.3.3 Notes on the Fractional Calculus in Linear Viscoelasticity

The First Generation of Pioneers of Fractional Calculus in Viscoelasticity

Linear viscoelasticity is certainly the ﬁeld in which the most extensive applications of fractional calculus have appeared, in view of its ability to model hereditary phenomena with long memory. During the twentieth century a number of authors have (implicitly or explicitly) used the fractional calculus as an empirical method of describing the properties of viscoelastic materials. 8.3 Historical and Bibliographical Notes 229

In the first half of that century the early contributors were: Gemant in the UK and USA, see [Gem36, Gem38], Scott Blair in the UK, see [ScoB44, ScoB47, ScoB49], and Gerasimov and Rabotnov in the Soviet Union, see [Ger48, Rab48a]. In 1950 Gemant published a series of 16 articles entitled Frictional Phenomena in the Journal of Applied Physics between 1941 and 1943, which were collected in a book of the same title [Gem50]. In his eighth chapter-paper [Gem42,p. 220], he referred to his previous articles [Gem36, Gem38] justifying the necessity of fractional differential operators to compute the shape of relaxation curves for some elasto-viscous fluids. Thus, the words fractional and frictional were coupled, presumably for the first time, by Gemant. Scott Blair used the fractional calculus approach to model the observations made in [Nut21, Nut43, Nut46] that the stress relaxation phenomenon could be described by fractional powers of time. He noted that time derivatives of fractional order would simultaneously model the observations of Nutting on stress relaxation and those of Gemant on frequency dependence. It is quite instructive to quote Scott Blair from [Sti79]: I was working on the assessing of firmness of various materials (e.g. cheese and clay by experts handling them) these systems are of course both elastic and viscous but I felt sure that judgments were made not on an addition of elastic and viscous parts but on something in between the two so I introduced fractional differentials of strain with respect to time. Later, in the same letter Scott Blair added: I gave up the work eventually, mainly because I could not find a definition of a fractional differential that would satisfy the mathematicians. The 1948 papers by Gerasimov and Rabotnov were published in Russian, so their contents remained unknown to the majority of western scientists until the translation into English of the treatises by Rabotnov, see [Rab69, Rab80]. Whereas Gerasimov explicitly used a fractional derivative to define his model of viscoelasticity (akin to the Scott Blair model), Rabotnov preferred to use the Volterra integral operators with weakly singular kernels that could be interpreted in terms of fractional integrals and derivatives. Following the appearance of the books by Rabotnov it has become common to speak about Rabotnov’s theory of hereditary solid mechanics. The relation between Rabotnov’s theory and the models of fractional viscoelasticity is briefly described in [RosShi07]. According to these Russian authors, Rabotnov could express his models in terms of the operators of the fractional calculus, but he considered these operators only as a mathematical abstraction.

The Second Generation of Pioneers of Fractional Calculus in Viscoelasticity

In the late sixties, formerly Caputo, see [Cap66, Cap67, Cap69] then Caputo and Mainardi, see [CapMai71a, CapMai71b], explicitly suggested that derivatives of fractional order (of Caputo type) could be successfully used to model dissipation in seismology and in metallurgy. In relation to this, one of the authors (Mainardi) recalls a correspondence carried out between himself (as a young post-doc student) and the Russian Academician Rabotnov, concerning two courses on Rheology held at CISM (International Centre 230 8 Applications to Deterministic Models for Mechanical Sciences, Udine, Italy) in 1973 and 1974, where Rabotnov was an invited speaker but did not participate, see [Rab73,Rab74]. Rabotnov recognized the relevance of the review paper [CapMai71b], writing in his unpublished 1974 CISM Lecture Notes: That’s why it was of great interest for me to know the paper of Caputo and Mainardi from the University of Bologna published in 1971. These authors have obtained similar results independently without knowing the corresponding Russian publications. . . Then he added: The paper of Caputo and Mainardi contains a lot of experimental data of different authors in support of their theory. On the other hand a great number of experimental curves obtained by Postnikov and his coworkers and also by foreign authors can be found in numerous papers of Shermergor and Meshkov. Unfortunately, the eminent Russian scientist did not cite the 1971 paper by Caputo and Mainardi (presumably for reasons independent from his wishes) in the Russian and English editions of his later book [Rab80]. Nowadays, several articles (originally in Russian) by Shermergor, Meshkov and their associated researchers have been re-printed in English in the Journal of Applied Mechanics and Technical Physics (the English translation of Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki), see e.g. [She66, MePaSh66, Mes67, MesRos68, Mes70,Zel-et-al70,GonRos73] available at the URL: http://www.springerlink.com/. In relation to this we cite the recent review papers [Ros10, RosShi10], where the works of the Russian scientists on fractional viscoelasticity are examined. The beginning of the modern applications of fractional calculus in linear viscoelasticity is generally attributed to the 1979 PhD thesis by Bagley (under the supervision of Prof. Torvik), see [Bag79], followed by a number of relevant papers, e.g., [BagTor79, BagTor83a, BagTor83b]and[TorBag84]. However, for the sake of completeness, we should also mention the 1970 PhD thesis of Rossikhin under the supervision of Prof. Meshkov, see [Ros70], and the 1971 PhD thesis of Mainardi under the supervision of Prof. Caputo, summarized in [CapMai71b]. To date, applications of the fractional calculus in linear and non-linear viscoelasticity have been considered by a great and increasing number of authors to whom we have tried to refer in the huge (but not exhaustive) bibliography of the book [Mai10].

8.4 Exercises

8.4.1. Solve the initial-boundary value problem ([DebBha07, p. 313])

u.x; 0/ D f.x/; x 2 R; (8.4.1a) u.x; t/ ! 0; as jxj!1;t>0; (8.4.1b)

for the linear inhomogeneous fractional Burgers equation

@˛u @u @2u C c D q.x;t/; x 2 R; t>0 .0<˛Ä 1/: (8.4.1c) @t˛ @x @x2 8.4 Exercises 231

Answer.

Z1 1 2 ˛ ikx u.x; t/ D p f.k/EO ˛;1.a t /e dk 2 1

Zt Z1 1 ˛1 2 ˛ ikx Cp d q.k;tO /E˛;˛.a t /e dk; 2 0 1

where a2 D .ick C k2/. 8.4.2. Solve the initial-boundary value problem ([DebBha07, p. 314])

@u u.x; 0/ D f.x/; x 2 R; .x; 0/ D g.x/; x 2 R; (8.4.2a) @t u.x; t/ ! 0; as jxj!1;t>0; (8.4.2b)

for the linear inhomogeneous fractional Klein–Gordon equation

@˛u @2u c2 C d 2u D q.x;t/; x 2 R;t>0 @t˛ @x2 .1 < ˛ Ä 2; c D const;dD const/: (8.4.2c)

Answer.

Z1 1 2 ˛ ikx u.x; t/ D p f.k/EO ˛;1.a t /e dk 2 1 Z1 1 2 ˛ ikx Cp tg.k/EO ˛;2.a t /e dk 2 1 Zt Z1 1 ˛1 2 ˛ ikx Cp d q.k;tO /E˛;˛.a t /e dk; 2 0 1

where a2 D .c2k2 C d 2/. 8.4.3. Solve the initial-boundary value problem ([DebBha07, p. 314])

u.x; 0/ D f.x/; x 2 R; (8.4.3a) u.x; t/ ! 0 as jxj!1;t >0; (8.4.3b) 232 8 Applications to Deterministic Models

for the linear inhomogeneous fractional KdV equation

@˛u @u @3u C c C b D q.x;t/; x 2 R;t>0; (8.4.3c) @t˛ @x @x3

with constant b; c and 0<˛Ä 1. Answer.

ZC1 1 2 ˛ ikt u.x; t/ D p f.k/EO ˛;1.a t /e dk 2 1 Zt ZC1 1 ˛ 2 ˛ ikt Cp d q.k;tO /E˛;˛.a t /e dk; (8.4.3d) 2 0 1 where a2 D ick ik3b . 8.4.4. Solve in terms of an H-function the initial-boundary value problem ([Uch13b, pp. 96–97])

u.y; 0/ D 0; y > 0; (8.4.4a) u.0; t/ D 1t>0; (8.4.4b) u ! 0; as y !1; (8.4.4c)

for the Maxwell type fractional equation Â Ã @u @2u C ˛ D˛C1u D ˇ1 Dˇ1 ;y2 R ;t>0: (8.4.4d) @t 0 t 0 t @y 2 C

Answer.

X1 .y/n u.y; t/ D 1 C n.1C˛ˇ/=2t n.ˇ˛/=2n nŠ nD1 Ä ˇ ˛ ˇ 1;1 t ˇ .n=2; 0/ H ˇ : (8.4.4e) 1;3 ˛ .0; 1/ .n=2; 1/ .n n.ˇ ˛/=2; ˛/

8.4.5. Consider the fractional differential model describing the motion of a visco-elastic media between two parallel plates (the lower x D 0 is immovable, and the upper x D a is moving in the Oy-direction according to the law '.t/, '.0/ D 0, '.0/P D 0). Solve the initial-boundary value problem ([Uch08, p. 303], [Ger48]) 8.4 Exercises 233

@y.x; t / y.x;0/ D 0; j D 0; (8.4.5a) @t tD0 y.0;t/ D 0; y.a; t/ D '.t/; (8.4.5b)

for such a motion described by the equation

@2y @ D (8.4.5c) @t2 @x assuming the visco-elastic media satisﬁes the fractional constitutive equation (with ˛ D 1=2)

˛ .t/ D ˛ 0Dt .t/: (8.4.5d)

Find the stress .x;t/ on the upper plane x D a. Answer.

Zt Â Ã 2 X1 .1/n nx X1 n 2k y.x;t/ D sin n n c1=2a nD1 0 kD1 .1/k.t /3k=21 './d; .3k=2/ (8.4.5e)

t ˛=2 .a;t/ D c ˛ ˛ .1 ˛=2/ 9 = X1 X1 Œ.2k C 2/c aj C2 ˛ t j.1˛=2/˛=2 ; j Š ..1 j /.1 ˛=2// ; kD0 j D0 (8.4.5f)

2 2 where c˛ D =˛. Chapter 9 Applications to Stochastic Models

This chapter is devoted to the application of the Mittag-Lefﬂer function and related special functions in the study of certain stochastic processes. As this topic is so wide, we restrict our attention to some basic ideas. For more complete presentations of the discussed phenomena we refer to some recent books and original papers which are mentioned in Sect. 9.6.

9.1 Introduction

The structure of the chapter and the notions and phenomena discussed in each part of it are presented in Sect. 9.1. We start in Sect. 9.2 with a description of an approach to generalizing the Poisson probability distribution due to Pillai [Pil90]. Taking into account the complete monotonicity of the Mittag-Leffler function, Pillai introduced in [Pil90] a probability distribution which he called the Mittag-Leffler distribution. In Sect. 9.3 we present a short introduction to renewal theory and continuous time random walk (CTRW) since these notions (renewal processes and CTRW) are very important for understanding the ideas behind the fractional generalization of stochastic processes. The concept of a renewal process has been developed as a stochastic model for describing the class of counting processes for which the times between successive events are independent identically distributed (i.i.d.) non- negative random variables, obeying a given probability law. Section 9.4 is devoted to a generalization of the standard Poisson process by replacing the exponential function (as waiting time density) by a function of Mittag-Leffler type. Thus the corresponding renewal process can be called the fractional Poisson process or the Mittag-Leffler waiting time process. In this way we discuss how the standard Poisson process is generalized to a fractional process

© Springer-Verlag Berlin Heidelberg 2014 235 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2__9 236 9 Applications to Stochastic Models and describe the differences between them. We also discuss here the concept of thinning of a renewal process. Section 9.5 presents an introduction to the theory of fractional diffusion processes. As a bridge between the simple renewal process and space-time diffusion we consider first the notion of a renewal process with reward. This leads us to the formulation of a fractional master equation which is then reduced to a space-time fraction diffusion equation. We discuss a few properties of the latter, starting with a presentation of the fundamental solution to the space-time fractional diffusion equation. Taking the diffusion limit of the Mittag-Leffler renewal process we derive the space-time fractional diffusion equation. In this connection the rescaling concept is introduced. Another property mentioned here is the possibility of interpreting the space-time fractional diffusion process as a subordination process. As a by-product of the rescaling-respeeding concept we also obtain the asymptotic universality of the Mittag-Leffler waiting time law. We conclude with Sect. 9.6 which presents some historical and bibliographical notes focussing, as in the main text, on those works which concern applications of the Mittag-Leffler function and related special functions. We also point out several notions which have been given different names in recently published papers and books. These have arisen since the discussed theory is not yet complete and has attracted great interest and rapid development because of its applications.

9.2 The Mittag-Lefﬂer Process According to Pillai

We sketch here the theory of a process that has been devised by Pillai [Pil90]asan increasing Lévy process on the spatial half-line x 0 happening in natural time t 0. Switching notation from t to x, from ˚ to F , from to f , and from ˇ to ˛, we consider the probability distribution function

˛ F˛.x/ D 1 E˛.x /; x 0; 0<˛ Ä 1 (9.2.1) and its density

d f .x/ D E .x˛/; x 0; 0<˛ Ä 1: (9.2.2) ˛ dx ˛

Their Laplace transforms (denoting by the Laplace parameter corresponding to x) are

1 ˛1 1 1 FQ ./ D D ; fQ ./ D : (9.2.3) ˛ 1 C ˛ .1 C ˛ / ˛ 1 C ˛ 9.2 The Mittag-Lefﬂer Process According to Pillai 237

According to Feller [Fel71], the distribution F˛.x/ is inﬁnite divisible if its density can be written as

fQ˛./ D exp .g˛.// ; 0 where g˛./ is (in a more modern terminology) a Bernstein function, meaning that g˛./ is non-negative and has a completely monotone derivative. Here we have ˛ g˛./ D log.1 C / 0 so

˛˛1 g0 ./ D : ˛ 1 C ˛

0 As a consequence the derivative g˛./ is a completely monotone function, being a product of two completely monotone functions, and thus it follows that g˛./ is a Bernstein function. Now, following Pillai, we can deﬁne a stochastic process x D x.t/ on the half-line x 0 happening in time t 0 by its density f˛.x; t/ (density in x evolving in t)taking

1 fQ .; t/ D D .1 C ˛/t D .fQ .//t : (9.2.4) ˛ .1 C ˛/t ˛

For the Laplace inversion of fQ˛.; t/ we write for >1 ! X1 t fQ .; t/ D ˛t .1 C ˛ /t D ˛.tCk/ : (9.2.5) ˛ k kD0

Then, using the correspondence

x˛.tCk/1 ˛.tCk/ ; (9.2.6) .˛.t C k// we get ! X1 t x˛.tCk/1 f .x; t/ D : (9.2.7) ˛ k .˛.t C k// kD0

Hence, by integration ! X1 t x˛.tCk/ F .x; t/ D : (9.2.8) ˛ k .˛.t C k/ C 1/ kD0

Manipulation of binomial coefﬁcients yields 238 9 Applications to Stochastic Models ! t t.t C 1/:::.t C k 1/ .t C k/ D .1/k D .1/k ; (9.2.9) k kŠ kŠ .t/ so that ﬁnally we obtain

X1 .t C k/ f .x; t/ D .1/k x˛.tCk/1 ; (9.2.10) ˛ kŠ.t/.˛.t C k// kD0 and

X1 .t C k/ F .x; t/ D .1/k x˛.tCk/ : (9.2.11) ˛ kŠ.t/.˛.t C k/ C 1/ kD0

9.3 Elements of Renewal Theory and Continuous Time Random Walk (CTRW)

9.3.1 Renewal Processes

The General Renewal Process

For the reader’s convenience, we present a brief introduction to renewal theory. For more details see, e.g., the classical treatises by Cox [Cox67], Feller [Fel71], and the more recent book by Ross [Ros97]. By a renewal process we mean an inﬁnite sequence 0 D t0

The Counting Number Process and Its Inverse

We are interested in the counting number process x D N D N.t/ 9.3 Elements of Renewal Theory and Continuous Time Random Walk (CTRW) 239

N.t/ WD max fnjtn Ä tg D n for tn Ä t

pn.t/ WD PŒN.t/ D n ; n D 0; 1; 2; : (9.3.2)

We denote by p.x;t/ the sojourn density for the counting number having the value x. For this process the expectation is Z X1 1 m.t/ WD hN.t/i D npn.t/ D xp.x;t/dx; (9.3.3) nD0 0

X1 since naturally p.x;t/ D pn.t/ ı.x n/. This provides the mean number of nD0 events in the half-open interval .0; t, and is called the renewal function, see e.g. [Ros97]. We will also look at the process t D t.N/, the inverse of the process N D N.t/, that we call the Erlang process. This gives the time t D tN of the N -th renewal. We are now looking for the Erlang probability densities

qn.t/ D q.t;n/; n D 0; 1; 2; : : : (9.3.4)

For every n the function qn.t/ D q.t;n/ is a density in the time variable having value t in the instant of the n-th event. Clearly, this event occurs after n (original) waiting times have passed, so that

n n qn.t/ D .t/ with Laplace transform qQn.s/ D ..s/Q /; (9.3.5) where n.t/ D Œ.t/ ::: Œ.t/ is the multiple Laplace convolution in R with n identical terms. In other words, the function qn.t/ D q.t;n/ is a probability density in the variable t 0 evolving in the discrete variable x D n D 0; 1; 2; : : :.

9.3.2 Continuous Time Random Walk (CTRW)

A continuous time random walk (CTRW) is given by an inﬁnite sequence of spatial positions 0 D x0;x1;x2; , separated by (i.i.d.) random jumps Xj D xj xj 1, whose probability density function w.x/ is given as a non-negative function or generalized function (interpretable asZ a measure) with support on the real axis 1 1

For deﬁniteness, we take ˚.t/ to be right-continuous and .t/ left-continuous. When the non-negative random variable represents the lifetime of a technical system, it is common to call ˚.t/ WD P .T Ä t/ the failure probability and .t/ WD P .T > t/ the survival probability, because ˚.t/ and .t/ are the respective probabilities that the system does or does not fail in .0; t. These terms, however, are commonly adopted for any renewal process. In the Fourier–Laplace domain we have

1 .s/Q .s/Q D ; (9.3.7) s and the famous Montroll–Weiss solution formula for a CTRW, see [MonWei65, Wei94].

X1 1 .s/Q n 1 .s/Q 1 uQO.; s/ D .s/Q wO ./ D : (9.3.8) s s Q nD0 1 .s/wO ./

In our special situation the jump density has support only on the positive semi-axis x 0 and thus, by replacing the Fourier transform by the Laplace transform we obtain the Laplace–Laplace solution

X1 1 .s/Q n 1 .s/Q 1 uQ.; s/ D .s/Q wQ ./ D : (9.3.9) s s Q nD0 1 .s/wQ ./

Recalling the deﬁnition of convolutions, in the physical domain we have for the solution u.x; t/ the Cox–Weiss series,see[Cox67, Wei94], ! X1 u.x; t/ D n wn .x; t/ : (9.3.10) nD0

This formula has an intuitive meaning: Up to and including instant t, there have occurred 0 jumps, or 1 jump, or 2 jumps, or :::, and if the last jump has occurred at instant t 0

The Integral Equation of CTRW

By natural probabilistic arguments we arrive at the integral equation for the probability density p.x;t/ (a density with respect to the variable x) of the particle being at point x at instant t; Z ÄZ t C1 p.x;t/ D ı.x/ .t/ C .t t 0/ w.x x0/p.x0;t0/ dx0 dt 0 : 0 1 (9.3.11)

Here Z 1 .t/ D .t0/ dt 0 (9.3.12) tC is the survival function (or survival probability). It denotes the probability that at instant t the particle is still sitting in its starting position x D 0: Clearly, (9.3.11) satisﬁes the initial condition p.x;0C/ D ı.x/. Note that the special choice

w.x/ D ı.x 1/ (9.3.13) gives the pure renewal process, with position x.t/ D N.t/, denoting the counting function, and with jumps all of length 1 in the positive direction happening at the renewal instants. For many purposes the integral equation (9.3.11) of CTRW can easily be treated by using the Laplace and Fourier transforms. Writing these as Z 1 L ff.t/I sg D f.s/Q WD e st f.t/dt; 0 Z C1 F fg.x/I g DOg./ WD eCix g.x/ dx; 1 in the Laplace–Fourier domain Eq. (9.3.11) reads as

1 .s/Q p.;s/QO D C .s/Q wO ./ p.;s/QO : (9.3.14) s Formally introducing in the Laplace domain the auxiliary function

1 .s/Q .s/Q 1 H.s/Q D D ; hence .s/Q D ; (9.3.15) s .s/Q .s/Q 1 C sH.s/Q and assuming that its Laplace inverse H.t/ exists, we get, following [Mai-et-al00], in the Laplace–Fourier domain the equation 242 9 Applications to Stochastic Models h i H.s/Q sp.;s/QO 1 D ŒwO ./ 1 p.;s/QO ; (9.3.16) and in the space-time domain the generalized Kolmogorov–Feller equation Z Z t C1 0 @ 0 0 0 0 0 H.t t / 0 p.x;t / dt Dp.x;t/ C w.x x /p.x ;t/dx ; 0 @t 1 (9.3.17) with p.x;0/ D ı.x/. If the Laplace inverse H.t/ of the formally introduced function H.s/Q does not exist, we can formally set K.s/Q D 1=H.s/Q and multiply (9.3.16)byK.s/Q . Then, if K.t/ exists, we get in place of (9.3.17) the alternative form of the generalized Kolmogorov–Feller equation Z Ä Z @ t C1 p.x;t/ D K.t t 0/ p.x;t0/ C w.x x0/p.x0;t0/ dx0 dt 0 ; @t 0 1 (9.3.18) with p.x;0/ D ı.x/. There are some interesting special choices of the memory function H.t/.We start the discussion with the following.

.i/H.t/D ı.t/ corresponding to H.s/Q D 1; (9.3.19) giving the exponential waiting time with

1 d .s/Q D ;.t/D e t D e t ;.t/D e t : (9.3.20) 1 C s dt

In this case we obtain in the Fourier–Laplace domain

sp.;s/QO 1 D ŒwO ./ 1 p.;s/QO ; (9.3.21) and in the space-time domain the classical Kolmogorov–Feller equation Z @ C1 p.x;t/ Dp.x;t/ C w.x x0/p.x0;t/dx0 ;p.x;0/D ı.x/ : @t 1 (9.3.22)

The other highly relevant choice is

t ˇ .ii/H.t/D ;0<ˇ<1corresponding to H.s/Q D sˇ1; (9.3.23) .1 ˇ/ that we will discuss in Sect. 9.4. 9.3 Elements of Renewal Theory and Continuous Time Random Walk (CTRW) 243

9.3.3 The Renewal Process as a Special CTRW

An essential trick in what follows is that we treat renewal processes as continuous time random walks with waiting time density .t/ and special jump density w.x/ D ı.x 1/ corresponding to the fact that the counting number N.t/ increases by 1 at each positive event instant tn.WethenhavewQ ./ D exp./ and get for the counting number process N.t/ the sojourn density in the transform domain (s 0, 0),

X1 1 .s/Q n 1 .s/Q 1 p.;s/Q D .s/Q en D : (9.3.24) s s Q nD0 1 .s/e

From this formula we can ﬁnd formulas for the renewal function m.t/ and the probabilities Pn.t/ D PfN.t/ D ng. Because N.t/ assumes as values only the non-negative integers, the sojourn density p.x;t/ vanishes if x is not equal to one of these, but has a delta peak of height Pn.t/ for x D n (n D 0; 1; 2; 3; ). Hence

X1 p.x;t/ D Pn.t/ ı.x n/ : (9.3.25) nD0 Rewriting Eq. (9.3.24), by inverting with respect to ,as

X1 n .t/ ı.x n/ ; (9.3.26) nD0 we identify n Pn.t/ D .t/ : (9.3.27)

According to the theory of the Laplace transform we conclude from Eqs. (9.3.2)and (9.3.25) !ˇ ˇ @ X1 ˇ X1 m.t/ D p.;t/Q j D nP .t/ en ˇ D nP .t/ ; (9.3.28) @ D0 n ˇ n nD0 D0 nD0 a result naturally expected, and

X1 X1 n .s/Q m.s/Q D n PQ .s/ D .s/Q n .s/Q D ; (9.3.29) n Q nD0 nD0 s 1 .s/ thereby using the identity

X1 z nzn D ; jzj <1: .1 z/2 nD0 244 9 Applications to Stochastic Models

Thus we have found in the Laplace domain the reciprocal pair of relationships

.s/Q s m.s/Q m.s/Q D ; .s/Q D ; (9.3.30) s.1 .s//Q 1 C s m.s//Q telling us that the waiting time density and the renewal function mutually determine each other uniquely. The ﬁrst formula of Eq. (9.3.30) can also be obtained as the value at D 0 of the negative derivative for D 0 of the last expression in Eq. (9.3.24). Equation (9.3.30) implies the reciprocal pair of relationships in the physical domain

R1 m.t/ D Œ1 C m.t t 0/ .t 0/ dt 0 ; 0 (9.3.31) R1 m0.t/ D Œ1 C m0.t t 0/ .t 0/ dt 0 : 0

The ﬁrst of these equations is usually called the renewal equation. Considering, formally, the counting number process N D N.t/ as a CTRW (with jumps ﬁxed to unit jumps 1), N running increasingly through the non-negative integers x D 0; 1; 2; : : :, happening in natural time t 2 Œ0; 1/, we note that in the Erlang process t D t.N/,therolesofN and t are interchanged. The new “waiting time density” is now w.x/ D ı.x 1/, the new “jump density” is .t/. It is illuminating to look at the relationships for t 0, n D 0; 1; 2; : : :, between the counting number probabilities Pn.t/ and the Erlang densities qn.t/. n For Eq. (9.3.5)wehaveqn.t/ D .t/, and then by (9.3.27) Z t 0 0 Pn.t/ D . qn/.t/ D qn.t / qnC1.t/ dt : (9.3.32) 0

We can also express the qn in anotherZ way in terms of the Pn. Introducing the t 0 0 cumulative probabilities Qn.t/ D qn.t / dt ,wehave 0 ! Xn X1 Qn.t/ D P Tk Ä t D P .N.t/ n/ D Pk.t/ ; (9.3.33) kD1 kDn and ﬁnally

d d X1 q .t/ D Q.t/ D P .t/ : (9.3.34) n dt dt k kDn

Xn All this is true for n D 0 as well, by the empty sum convention Tk D 0 for kD1 n D 0. 9.4 The Poisson Process and Its Fractional Generalization 245

9.4 The Poisson Process and Its Fractional Generalization (the Renewal Process of Mittag-Lefﬂer Type)

9.4.1 The Mittag-Lefﬂer Waiting Time Density

Returning to the integral equation for the probability density of CTRW (9.3.11)(see Sect. 9.3.2) we note that besides the classical special case (9.3.19) there exist another one.

t ˇ .ii/H.t/D ; 0<ˇ<1; corresponding to H.s/Q D sˇ1 ; .1 ˇ/ (9.4.1) giving the Mittag-Lefﬂer waiting time density with

1 d .s/Q D ;.t/D E .t ˇ / D ML.t/; .t/ D E .t ˇ/: 1 C sˇ dt ˇ ˇ (9.4.2)

In this case we obtain in the Fourier–Laplace domain h i sˇ1 sp.;s/QO 1 D ŒwO ./ 1 p.;s/QO ; (9.4.3) and in the space-time domain the time fractional Kolmogorov–Feller equation Z C1 ˇ 0 0 0 C t D p.x;t/ Dp.x;t/ C w.x x /p.x ;t/dx ;p.x;0/ D ı.x/ ; 1 (9.4.4)

ˇ where t D denotes the fractional derivative of order ˇ in the Caputo sense, see Appendix A. The time fractional Kolmogorov–Feller equation can also be expressed via the 1ˇ Riemann–Liouville fractional derivative t D ,thatis Ä Z C1 @ 1ˇ 0 0 0 p.x;t/ D t D p.x;t/ C w.x x /p.x ;t/dx ; (9.4.5) @t 1 with p.x;0C/ D ı.x/. The equivalence of the two forms (9.4.4)and(9.4.5) is easily proved in the Fourier–Laplace domain by multiplying both sides of Eq. (9.4.3)by the factor s1ˇ. We note that the choice .i/ may be considered as a limit of the choice .ii/ as ˇ D 1. In fact, in this limit we find H.s/Q Á 1 so H.t/ D t 1= .0/ Á ı.t/ (according to a formal representation of the Dirac generalized function [GelShi64]), so that Eqs. (9.3.16)and(9.3.17) reduce to (9.3.21)and(9.3.22), respectively. In this 246 9 Applications to Stochastic Models case the order of the Caputo derivative reduces to 1 and that of the R-L derivative to 0, whereas the Mittag-Leffler waiting time law reduces to the exponential. In the sequel we will formally unite the choices (i)and(ii) by defining what we call the Mittag-Leffler memory function 8 < t ˇ ML ; if 0<ˇ<1; H .t/ D : .1 ˇ/ (9.4.6) ı.t/ ; if ˇ D 1; whose Laplace transform is

HQ ML.s/ D sˇ1 ;0<ˇÄ 1: (9.4.7)

Thus we will consider the whole range 0<ˇÄ 1 by extending the Mittag-Lefﬂer waiting time law in (9.4.2) to include the exponential law (9.3.20).

9.4.2 The Poisson Process

The most celebrated renewal process is the Poisson process characterized by a waiting time pdf of exponential type,

.t/ D et ;>0;t 0: (9.4.8)

The process has no memory. Its moments turn out to be

1 1 1 hT iD ; hT 2iD ; ::: ; hT niD ; ::: ; (9.4.9) 2 n and the survival probability is

.t/ WD P .T > t/ D et ;t 0: (9.4.10)

We know that the probability that k events occur in the interval of length t is

.t/k P .N.t/ D k/ D et ;t 0; k D 0; 1; 2; : : : : (9.4.11) kŠ

The probability distribution related to the sum of k i.i.d. exponential random variables is known to be the so-called Erlang distribution (of order k). The corresponding density (the Erlang pdf ) is thus

.t/k1 f .t/ D et ;t 0; k D 1;2;::: ; (9.4.12) k .k 1/Š 9.4 The Poisson Process and Its Fractional Generalization 247 so that the Erlang distribution function of order k turns out to be Z t Xk1 .t/n X1 .t/n F .t/ D f .t 0/ dt 0 D 1 et D et ;t 0: k k nŠ nŠ 0 nD0 nDk (9.4.13)

In the limiting case k D 0 we recover f0.t/ D ı.t/; F0.t/ Á 1; t 0. The results (9.4.11)–(9.4.13) can easily be obtained by using the technique of the Laplace transform sketched in the previous section, noting that for the Poisson process we have:

1 .s/Q D ; .s/Q D ; (9.4.14) C s C s and for the Erlang distribution:

k Œ.s/Q k k fQ .s/ D Œ.s/Q k D ; FQ .s/ D D : (9.4.15) k . C s/k k s s. C s/k

We also recall that the survival probability for the Poisson renewal process obeys the ordinary differential equation (of relaxation type)

d .t/ D.t/ ; t 0 I .0C/ D 1: (9.4.16) dt

9.4.3 The Renewal Process of Mittag-Lefﬂer Type

A “fractional” generalization of the Poisson renewal process is simply obtained by generalizing the differential equation (9.4.16), replacing there the ﬁrst derivative ˇ with the integro-differential operator t D that is interpreted as the fractional derivative of order ˇ in the Caputo sense. We write, taking for simplicity D 1,

ˇ C t D .t/ D.t/; t > 0; 0 < ˇ Ä 1 I .0 / D 1: (9.4.17)

We also allow the limiting case ˇ D 1 where all the results of the previous section (with D 1) are expected to be recovered. In fact, taking D 1 is simply a normalized way of scaling the variable t. We call this renewal process of Mittag-Lefﬂer type the fractional Poisson process. To analyze this we work in the Laplace domain where we have

sˇ1 1 .s/Q D ; .s/Q D : (9.4.18) 1 C sˇ 1 C sˇ 248 9 Applications to Stochastic Models

If there is no danger of misunderstanding we will not decorate and with the index ˇ. The special choice ˇ D 1 gives us the standard Poisson process with 1.t/ D 1.t/ D exp.t/. Whereas the Poisson process has ﬁnite mean waiting time (that of its standard version is equal to 1), the fractional Poisson process (0<ˇ<1) does not have this property. In fact, Z ˇ 1 sˇ1 ˇ 1; ˇ D 1; hT iD t.t/ t D ˇ ˇ D d ˇ 2 ˇ (9.4.19) 0 .1 C s / sD0 1 ;0<ˇ<1:

Let us calculate the renewal function m.t/. Inserting .s/Q D 1=.1 C sˇ/ into Eq. (9.3.24) and taking w.x/ D ı.x 1/ as in Sect. 9.3.3, we ﬁnd for the sojourn density of the counting function N.t/ the expressions

sˇ1 sˇ1 X1 en p.;s/Q D D ; (9.4.20) 1 C sˇ e 1 C sˇ .1 C sˇ/n nD0 and ˇ p.;t/Q D Eˇ .1 e /t ; (9.4.21) and then ˇ @ ˇ ˇ 0 ˇ ˇ m.t/ D p.;t/Q jD0 D e t Eˇ .1 e /t : (9.4.22) @ D0

0 Using Eˇ.0/ D 1= .1 C ˇ/ now yields 8 < t; ˇD 1; m.t/ D t ˇ (9.4.23) : ;0<ˇ<1: .1C ˇ/

This result can also be obtained by plugging .s/Q D 1=.1 C sˇ/ into the ﬁrst equation in (9.3.30) which yields m.s/Q D 1=sˇC1 and then by Laplace inversion equation (9.4.23). Using the general Taylor expansion

X1 E.n/ E .z/ D ˇ .z b/n ; (9.4.24) ˇ nŠ nD0 in Eq. (9.4.21) with b Dt ˇ we get 9.4 The Poisson Process and Its Fractional Generalization 249

X1 t nˇ p.;t/Q D E.n/.t ˇ / en ; nŠ ˇ nD0 (9.4.25) X1 t nˇ p.x;t/ D E.n/.t ˇ /ı.x n/ ; nŠ ˇ nD0 and, by comparison with Eq. (9.3.25), the counting number probabilities

t nˇ P .t/ D PfN.t/ D ngD E.n/.t ˇ /: (9.4.26) n nŠ ˇ Observing from Eq. (9.4.20)that

1 sˇ1 sˇ1 X en p.;s/Q D D ; (9.4.27) 1 C sˇ e 1 C sˇ .1 C sˇ/n nD0 and inverting with respect to ,

sˇ1 X1 ı.x n/ p.x;s/Q D ; (9.4.28) 1 C sˇ .1 C sˇ/n nD0 we ﬁnally identify

sˇ1 t nˇ PQ .s/ D E.n/.t ˇ/ D P .t/ : (9.4.29) n .1 C sˇ/nC1 nŠ ˇ n

En passant we have proved an often cited special case of an inversion formula due to Podlubny [Pod99, Eq. (1.80)]. For the Poisson process with intensity >0we have a well-known inﬁnite system of ordinary differential equations (for t 0), see e.g. Khintchine [Khi60],

d P .t/ D et ; P .t/ D .P .t/ P .t// ; n 1; (9.4.30) 0 dt n n1 n with initial conditions Pn.0/ D 0, n D 1;2;:::, which is sometimes even used to deﬁne the Poisson process. We have an analogous system of fractional differential equations for the fractional Poisson process. In fact, from Eq. (9.4.30)wehave

sˇ1 .1 C sˇ/ PQ .s/ D D PQ .s/ : (9.4.31) n .1 C sˇ/n n1

Hence

ˇ s PQn.s/ D PQn1.s/ PQn.s/ ; (9.4.32) 250 9 Applications to Stochastic Models so in the time domain

ˇ ˇ P0.t/ D Eˇ.t /; Dt Pn.t/ D Pn1.t/ Pn.t/ ; n 1; (9.4.33)

ˇ with initial conditions Pn.0/ D 0, n D 1;2;:::,where Dt denotes the time-fractional derivative of Caputo type of order ˇ. It is also possible to introduce and define the fractional Poisson process by this difference-differential system. Let us note that by solving the system (9.4.33), Beghin and Orsingher in [BegOrs09] introduce what they call the “first form of the fractional Poisson process”, and in [Mee-et-al02] Meerschaert et al. show that this process is a renewal process with Mittag-Leffler waiting time density as in (9.4.17), hence is identical with the fractional Poisson process. Up to now we have investigated the fractional Poisson counting process N D N.t/ and found its probabilities Pn.t/ in Eq. (9.4.26). To get the corresponding Erlang probability densities qn.t/ D q.t;n/, densities in t, evolving in n D 0; 1; 2 : : :, we find by Eq. (9.3.34) via telescope summation

t nˇ1 q .t/ D ˇ E.n/ t ˇ ;0<ˇÄ 1: (9.4.34) n .n 1/Š ˇ

We leave it as an exercise to the reader to show that in Eq. (9.4.25) interchange of differentiation and summation is allowed. Remark. With ˇ D 1 we get the corresponding well-known results for the standard Poisson process. The counting number probabilities are

t n P .t/ D et ;nD 0; 1; 2; : : : t 0; (9.4.35) n nŠ and the Erlang densities

t n1 q .t/ D et ;nD 1;2;3;:::; t 0: (9.4.36) n .n 1/Š

By rescaling of time we obtain

.t/n P .t/ D et ;nD 0;1;2;:::; t 0; (9.4.37) n nŠ for the classical Poisson process with intensity and

.t/n1 q .t/ D et ;nD 1;2;3;:::; t 0 (9.4.38) n .n 1/Š for the corresponding Erlang process. 9.4 The Poisson Process and Its Fractional Generalization 251

9.4.4 Thinning of a Renewal Process

We are now going to give an account of the essentials of the thinning of a renewal process with power law waiting times, thereby leaning on the presentation of Gnedenko and Kovalenko [GneKov68] but for reasons of transparency not deco- rating the power functions by slowly varying functions. Compare also Mainardi, Gorenﬂo and Scalas [MaGoSc04a] and Gorenﬂo and Mainardi [GorMai08]. Again with the tn in strictly increasing order, the time instants of a renewal process, 0 D t0

With the modiﬁed waiting time T we have P.T Ä t/ D P.T Ä t=/ D F.t=/, hence the density f.t=/=, and analogously for the density of the sum of n waiting times fn.t=/=. The density of the waiting time of the sum of n waiting times of the rescaled (and thinned) process now turns out as 252 9 Applications to Stochastic Models

X1 n1 gq; .t/ D qp fn.t=/= : (9.4.41) nD1

n In the Laplace domain we have fQn.s/ D .f.s//Q , hence (using p D 1 q)

X1 q f.s/Q gQ .s/ D qpn1 .f.s//Q n D : (9.4.42) q Q nD1 1 .1 q/f.s/

By rescaling we get

X1 q f.s/Q gQ .s/ D qpn1 .f.s//Q n D : (9.4.43) q; Q nD1 1 .1 q/f.s/

Being interested in stronger and stronger thinning (infinite thinning) let us consider a scale of processes with the parameters q of thinning and of rescaling tending to zero under a scaling relation q D q./ yet to be specified. Let us consider two cases for the (original) waiting time distribution, namely, case (A) of a finite mean waiting time and case (B) of a power law waiting time. We assume in case (A) Z 1 WD t 0 f.t0/ dt 0 < 1 setting ˇ D 1; (9.4.44) 0 or Z 1 c .t/ D f.t0/ dt 0 t ˇ for t !1 with 0<ˇ<1: (9.4.45) t ˇ

In case (B) we set c D : .ˇC 1/ sin.ˇ/

From Lemma 9.2 in the next Sect. 9.5 we know that f.s/Q D 1 sˇ C o.sˇ/ for 0

q D ˇ (9.4.46) for ﬁxed s the Laplace transform (9.4.43) of the rescaled density gq; .t/ of the thinned process tends to g.s/Q D 1=.1 C sˇ/ corresponding to the Mittag-Lefﬂer density

d g.t/ D E .t ˇ/ D ML.t/ : (9.4.47) dt ˇ ˇ 9.5 The Fractional Diffusion Process 253

Thus, the thinned process converges weakly to the Mittag-Lefﬂer renewal process described in Mainardi, Gorenﬂo and Scalas [MaGoSc04a] (called the fractional Poisson process in Laskin [Lai93]) which in the special case ˇ D 1 reduces to the Poisson process.

9.5 The Fractional Diffusion Process

9.5.1 Renewal Process with Reward

The renewal process can be accompanied by a reward which means that at every renewal instant a space-like variable makes a random jump from its previous position to a new point in “space”. Here “space” is used in a very general sense. In the insurance business, for example, the renewal points are instants where the company receives a payment or must give away money to some claim of a customer, so space is money. In such a process occurring in time and in space, also referred to as a compound renewal process, the probability distribution of jump widths is as relevant as that of the waiting times. Let us denote by Xn the jumps occurring at instants tn ;nD 1; 2; 3; : : : . Let us assume that Xn are i.i.d. (real, not necessarily positive) random variables with probability density w.x/, independent of the waiting time density .t/.Ina physical context the Xns represent the jumps of a diffusing particle (the walker), and the resulting random walk model is known as a continuous time random walk (abbreviated CTRW) in that the waiting time is assumed to be a continuous random variable.

9.5.2 Limit of the Mittag-Lefﬂer Renewal Process

In a CTRW we can, with positive scaling factors h and , replace the jumps X by jumps Xh D hX and the waiting times T by waiting times T D T. This leads to the rescaled jump density wh.x/ D w.x=h/=h and the rescaled waiting time density .t/ D .t=/= and correspondingly to the transforms wO h./ DOw.h/, Q .s/ D .s/Q . For the sojourn density uh; .x; t/, the density in x evolving in t, we obtain from Eq. (9.3.8) in the transform domain (the Montroll–Weiss formula)

1 .s/Q 1 uQOh; .; s/ D ; (9.5.1) s 1 .s/Q wO .h/ where, if w.x/ has support on x 0, we can work with the Laplace transform instead of the Fourier transform (replace the O by Q ). If there exists between h and 254 9 Applications to Stochastic Models

a scaling relation R (to be introduced later) under which u.x; t/ tends as h ! 0, ! 0 to a meaningful limit v.x; t/ D u0;0.x; t/, then we call the process x D x.t/ with this sojourn density a diffusion limit. We ﬁnd it via

vQO.; s/ D lim uQOh; .; s/ ; (9.5.2) h;!0.R/ and Fourier–Laplace (or Laplace–Laplace) inversion. In recent decades power laws in physical (and also economical and other) processes and situations have become increasingly popular for modelling slow (in contrast to fast, mostly exponential) decay at infinity. See Newman [New05] for a general introduction to this concept. For our purpose let us assume that the distribution of jumps is symmetric, and that the distribution of jumps, likewise that of waiting times, either has finite second or first moment, respectively, or decays near infinity like a power with exponent ˛ or ˇ, respectively, 0<˛<2, 0<ˇ<1. Then we can state two lemmas. These lemmas and more general ones (e.g. with slowly varying decorations of the power laws (a) and (b)) can be distilled from the Gnedenko theorem on the domains of attraction of stable probability laws (see Gnedenko and Kolmogorov [GneKol54]), cf. [BiGoTe67, Ch. 9]. For wide generalizations (to several space dimensions and to anisotropy) see Meerschaert and Scheffler [MeeSch01]. They can also be modified to cover the special case of smooth densities w.x/ and .t/ and to the case of fully discrete random walks, see Gorenflo and Abdel-Rehim [GorAbdR04], Gorenflo and Vivoli [GorViv03]. For proofs see also Gorenflo and Mainardi [GorMai09]. Lemma 9.1 (for the jump distribution). Assume that W.x/ is increasing, W.1/ D 0; W.1/ D 1, and the symmetry W.x/ C W.x/ D 1 holds for all continuity points x of W.x/, and assume (a) or (b) are valid: Z C1 (a) 2 WD x2 dW.x/< 1, labelled as ˛ D 2 ; Z 1 1 (b) dW.x0/ b˛1x˛ for x !1; 0<˛<2; b>0: x Then, with D 2=2 in case (a) and D b=Œ .˛ C 1/ sin.˛=2/ in case (b) we have the asymptotics

1 Ow./ jj˛ for ! 0: Lemma 9.2 (for the waiting time distribution). Assume ˚.t/ is increasing, ˚.0/ D 0/, ˚.1/ D 1, and (A) or (B) is valid: Z 1 (A) WD t d˚.t/ < 1, labelled as ˇ D 1, 0 (B) 1 ˚.t/ cˇ1t ˇ for t !1; 0<ˇ<1;c>0: 9.5 The Fractional Diffusion Process 255

Then, with D in case (A) and D c=Œ .ˇ C 1/ sin.ˇ/ in case (B) we have the asymptotics

1 .s/Q sˇ for 0

1 Q .s/ 1 1 .s/Q 1 pQOh; .; s/D D : s 1 Owh./ Q .s/ s 1 Ow.h/ .s/Q (9.5.3)

Fixing now and s, both non-zero, replacing by h and s by s in the above lemmas, and sending h and to zero, we obtain by a trivial calculation the asymptotics

ˇsˇ1 pQO .; s/ (9.5.4) h; .hjj/˛ C .s/ˇ that we can rewrite in the form

sˇ1 h˛ pQO .; s/ with r.h;/ D : (9.5.5) h; r.h;/jj˛ C sˇ ˇ

Choosing r.h;/ Á 1 (it sufﬁces to choose r.h;/ ! 1)weget

sˇ1 pQO .; s/ ! pQO .; s/ D : (9.5.6) h; 0;0 jj˛ C sˇ

We will call the condition below the scaling relation

h˛ Á 1: (9.5.7) ˇ 256 9 Applications to Stochastic Models

Via D .=/h˛/1=ˇ we can eliminate the parameter , apply the inverse Laplace transform to (9.5.4), ﬁx and send h ! 0. So, by the continuity theorem (for the Fourier transform of a probability distribution, see Feller [Fel71]), we can identify

sˇ1 pQO .; s/ D 0;0 jj˛ C sˇ as the Fourier–Laplace solution uQO.; s/ of the space-time fractional Cauchy problem (for x 2 R, t 0)

ˇ ˛ t D u.x; t/ D xD0 u.x; t/ ; u.x; 0/ D ı.x/ ; (9.5.8) 0<˛Ä 2; 0<ˇ Ä 1: (9.5.9)

ˇ Here, for 0<ˇÄ 1, we denote by t D the regularized fractional differential operator, see Gorenﬂo and Mainardi [GorMai97], according to

ˇ ˇ t D g.t/ D t D Œg.t/ g.0/ (9.5.10) with the Riemann–Liouville fractional differential operator 8 Z ˆ 1 d t g.t0/ d < ;0<ˇ<1; ˇ .1 ˇ/ dt .t t 0/ˇ t D g.t/ WD 0 (9.5.11) :ˆ d g.t/ ; ˇ D 1: dt Hence, in longhand: 8 Z ˆ 1 d t g.t0/ d g.0/tˇ < ;0<ˇ<1 ˇ 0 ˇ D g.t/ D .1 ˇ/ dt 0 .t t / .1 ˇ/ t :ˆ d g.t/ ; ˇ D 1: dt (9.5.12)

If g0.t/ exists we can write Z 1 t g0.t 0/ D ˇ g.t/ D t 0 ; 0<ˇ<1; t 0 ˇ d .1 ˇ/ 0 .t t / and the regularized fractional derivative coincides with the form introduced by Caputo, see Caputo and Mainardi [CapMai71a], Gorenﬂo and Mainardi [GorMai97], Podlubny [Pod99], henceforth referred to as the Caputo derivative. ˇ Observe that in the special case ˇ D 1 the two fractional derivatives t D g.t/ and ˇ 0 t D g.t/ coincide, both then being equal to g .t/. The Riesz operator is a pseudo-differential operator according to

2˛ ˛ O R xD0 f Djj f./; 2 ; (9.5.13) 9.5 The Fractional Diffusion Process 257

(compare Samko, Kilbas and Marichev [SaKiMa93] and Rubin [Rub96]). It has the Fourier symbol jj˛. In the transform domain (9.5.8) means

sˇ uQO.; s/ sˇ1 Djj˛ uQO.; s/ hence

sˇ1 uQO.; s/ D ; (9.5.14) jj˛ C sˇ and looking back we see: uQO.; s/ D pQO0;0.; s/. Thus, under the scaling relation (9.5.7), the Fourier–Laplace solution of the CTRW integral equation converges to the Fourier–Laplace solution of the space-time fractional Cauchy problem (9.5.8) and (9.5.9), and we conclude that the sojourn probability of the CTRW converges weakly (or “in law”) to the solution of the Cauchy problem for the space-time fractional diffusion equation for every ﬁxed t>0. It is possible to present another way of passing to the diffusion limit, a way in which by decoupling the transitions in time and in space we circumvent doubts on the correctness of the transition. For a comprehensive study of integral representations of the solution to the Cauchy problem (9.5.8)and(9.5.9) we recommend the paper by Mainardi, Luchko and Pagnini [MaLuPa01].

The Fundamental Solution to the Space-Time Fractional Diffusion Equation

Let us note that the solution u.x; t/ of the Cauchy problem (9.5.8)and(9.5.9), known as the Green function or fundamental solution of the space-time fractional diffusion equation, is a probability density in the spatial variable x, evolving in time t. In the case ˛ D 2 and ˇ D 1 we recover the standard diffusion equation for which the fundamental solution is the Gaussian density with variance 2 D 2t. For completeness, let us consider the more general space-time fractional diffusion Cauchy problem (with skewness)

ˇ ˛ t D u.x; t/ D xD u.x; t/ ; u.x; 0/ D ı.x/ ; (9.5.15)

0<˛Ä 2; real ; j jÄmin.˛; 2 ˛/ ; 0 < ˇ Ä 1; (9.5.16) as comprehensively treated by Mainardi, Luchko and Pagnini [MaLuPa01]. Some- times, to point out the parameters, we may denote the fundamental solution by

u.x; t/ D G˛;ˇ.x; t/ : (9.5.17) 258 9 Applications to Stochastic Models

For our purposes let us conﬁne ourselves here to recalling the representation in the Laplace–Fourier domain of the (fundamental) solution as it results from an application of the Laplace and Fourier transforms to Eq. (9.5.8). Using ı./O Á 1 we have:

s ˇ uQO.; s/ s ˇ 1 Djj˛ i sign uQO.; s/ ; hence

b s ˇ 1 O e uQ.; s/ D G˛;ˇ.; s/ D : (9.5.18) s ˇ Cjj˛ i sign

For explicit expressions and plots of the fundamental solution of (9.5.8)and(9.5.9) in the space-time domain we refer the reader to [MaLuPa01]. There, starting from the fact that the Fourier transform of the fundamental solution can be written as a Mittag-Lefﬂer function with complex argument, Á b ˛ sign ˇ uO.; t/ D G˛;ˇ.; t/ D Eˇ jj i t ; (9.5.19)

a Mellin–Barnes integral representation of u.x; t/ D G˛;ˇ.x; t/ is derived, and is used to prove the non-negativity of the solution for values of the parameters f˛; ; ˇg in the range (9.5.9) and analyze the evolution in time of its moments. The representation of u.x; t/ in terms of Fox H-functions can be found in Mainardi, Pagnini and Saxena [MaPaSa05], see also Chapter 6 in the recent book [MaSaHa10] by Mathai, Saxena and Haubold.

9.5.3 Subordination in the Space-Time Fractional Diffusion Equation

Now we introduce the analytical and stochastic approaches to subordination in space-time fractional diffusion processes is the fundamental solution of the space-time fractional diffusion equation in the Laplace–Fourier domain given by (9.5.18). With an integration variable t that will play the role of operational time,weget the following instructive expression for (9.5.18): Z 1h Ái ˛ sign ˇ1 ˇ uQO.; s/ D exp tjj i s exp ts dt : (9.5.20) 0

We note that the ﬁrst factor in (9.5.20) Á ˛ sign fO˛; .; t/ WD exp tjj i (9.5.21) 9.5 The Fractional Diffusion Process 259 is the Fourier transform of a skewed stable density in x, evolving in operational time t, of a process x D y.t/ along the real axis x happening in operational time t, that we write as Á 1=˛ 1=˛ f˛; .x; t/ D t L˛ x=t : (9.5.22)

We can interpret the second factor in (9.5.20)

ˇ1 ˇ qQˇ.t;s/WD s exp.ts / (9.5.23) as the Laplace representation of the probability density in t evolving in t of a process t D t.t/, generating the operational time t from the physical time t, that is expressed via a fractional integral of a skewed Lévy density as

1=ˇ 1ˇ ˇ 1=ˇ ˇ ˇ qˇ.t;t/D t t J Lˇ .t=t / D t Mˇ.t=t /: (9.5.24)

We apply to (9.5.23) a second Laplace transformation with respect to t with parameter s to get

ˇ1 Q s qQˇ.s;s/D ˇ ; (9.5.25) s C s so, by inversion with respect to t Z 1 st ˇ qQˇ.s;t/D e qˇ.t;t/dt D Eˇ.st /; (9.5.26) tD0 and setting s D 0 we see that the density qˇ.t;t/is normalized: Z 1 qˇ.t;t/dt D Eˇ.0/ D 1: (9.5.27) tD0

Weighting the density of x D y.t/ with the density of t D t.t/ over 0 Ä t<1 yields the density u.x; t/ in x evolving with time t. In physical variables fx;tg, using Eqs. (9.5.20)–(9.5.24), we have the subordination integral formula Z 1 u.x; t/ D f˛; .x; t/qˇ.t;t/dt ; (9.5.28) tD0 where f˛; .x; t/ (density in x evolving in t) refers to the process x D y.t/ (t ! x) generating in “operational time” t the spatial position x,andqˇ.t;t/ (density in t evolving in t) refers to the process t D t.t/ (t ! t) generating from physical time t the “operational time” t. 260 9 Applications to Stochastic Models

Clearly f˛; .x; t/ characterizes a stochastic process describing a trajectory x D y.t/ in the .t;x/plane, that can be visualized as a particle travelling along space x, as operational time t is proceeding. Is there also a process t D t.t/, a particle moving along the positive t axis, happening in physical time t? Naturally we want t.t/ to be increasing, at least in the weak sense,

t2 >t1 H) t.t2/ t.t1/:

We answer this question in the afﬁrmative by inverting the stable process t D t.t/ whose probability density (in t, evolving in operational time t) is the extremely positively skewed stable density

1=ˇ ˇ 1=ˇ rˇ.t; t/ D t Lˇ .t=t /: (9.5.29)

In fact, recalling

ˇ rQˇ.s; t/ D exp.ts /; (9.5.30) there exists the stable process t D t.t/, weakly increasing, with density in t evolving in t given by (9.5.29). We call this process the leading process. Happily, we can invert this process. Inversion of a weakly increasing trajectory means that, in a graphical visualization, horizontal segments are converted to vertical segments and conversely jumps (as vertical segments) are converted to horizontal segments. Consider a ﬁxed sample trajectory t D t.t/ and its ﬁxed inversion t D t.t/. FixaninstantT of physical time and an instant T of operational time. Then, because t D t.t/ is increasing, we have the equivalence

t.T / Ä T ” T Ä t.T/; which, with the notation slightly changed by

0 0 t.T / ! t ;T ! t ;T! t; t.T/ ! t ; implies Z Z t 1 0 0 0 0 q.t;t/dt D rˇ.t ;t/ dt ; (9.5.31) 0 t for the probability density q.t;t/in t evolving in t. It follows that Z Z 1 1 @ 0 0 @ 0 0 q.t;t/D rˇ.t ;t/ dt D rˇ.t ;t/ dt : @t t t @t 9.5 The Fractional Diffusion Process 261

We continue in the s-Laplace domain assuming t>0, Z 1 0 0 0 qQ.s;t/D srQˇ.t ;s/ ı.t / dt : t

It sufﬁces to consider t>0,sothatwehaveı.t0/ D 0 in this integral. Observing from (9.5.30)

1 rQ .s; s / D ; ˇ ˇ (9.5.32) s C s we ﬁnd

ˇ1 0 ˇ rQˇ.t; s/ D ˇt Eˇ.st /; (9.5.33) so that Z 1 0ˇ1 0 0ˇ 0 ˇ q.sQ ;t/D sˇt Eˇ.st / dt D Eˇ.st /; (9.5.34) t and ﬁnally

ˇ ˇ q.t;t/D t Mˇ.t=t /: (9.5.35)

From (9.5.34) we also see that

sˇ1 q.sQ ;s/D D qQ .s ;s/; ˇ ˇ (9.5.36) s C s implying (9.5.23), see (9.5.25),

q.t;t/Á qˇ.t;t/; (9.5.37) so that the process t D t.t/ is indeed the inverse to the stable process t D t.t/ and has density qˇ.t;t/. Remark 9.3. For more details on the Gorenﬂo–Mainardi view of subordination, we draw the reader’s attention to the two complementary papers [GorMai12a, GorMai11]. The highlight of [GorMai12a] is an outline of the way how, by appropriate analytical manipulations from the subordination integral (9.5.28), a CTRW can be derived to produce snapshots of a particle trajectory. In the other paper [GorMai11], the authors show how, from a generic power law CTRW, by a properly scaled diffusion limit, the subordination integral can be found. 262 9 Applications to Stochastic Models

9.5.4 The Rescaling and Respeeding Concept Revisited: Universality of the Mittag-Lefﬂer Density

Now we use again the concept of rescaling and respeeding in order to obtain one more property of the Mittag-Lefﬂer waiting time density. First, we generalize the Kolmogorov–Feller equation (9.3.22) by introducing in the Laplace domain the auxiliary function

1 .s/Q .s/Q H.s/Q D D ; (9.5.38) s .s/Q .s/Q which is equivalent to h i H.s/Q sp.;s/QO 1 D ŒwO ./ 1 p.;s/QO : (9.5.39)

Then in the space-time domain we get the following generalized Kolmogorov– Feller equation Z Z t C1 0 @ 0 0 0 0 0 H.tt / 0 p.x;t / dt Dp.x;t/C w.xx /p.x ;t/dx ; (9.5.40) 0 @t 1 with p.x;0/ D ı.x/. Rescaling time means: with a positive scaling factor (intended to be small) we replace the waiting time T by T. This amounts to replacing the unit 1 of time by 1=,andif 1 then in the rescaled process there will occur very many jumps in a moderate span of time (instead of the original moderate number in a moderate span of time). The rescaled waiting time density and its corresponding Laplace transform are .t/ D .t=/=, Q .s/ D .s/Q .Furthermore:

1 Q .s/ 1 .s/Q 1 HQ .s/ D D ; hence Q .s/ D ; (9.5.41) s Q .s/ s .s/Q 1 C s HQ .s/ and (9.5.39) goes over into h i HQ .s/ spQO .; s/ 1 D ŒwO ./ 1 pQO .; s/ : (9.5.42)

Respeeding the process means multiplying the left-hand side (actually @ p.x;t0/ of Eq. (9.5.40) by a positive factor 1=a, or equivalently its right-hand @t0 side by a positive factor a. We honour the number a by the name respeeding factor. a>1means acceleration and a<1deceleration. In the Fourier–Laplace domain the rescaled and respeeded CTRW process then assumes the form, analogous to (9.5.39)and(9.5.42), 9.5 The Fractional Diffusion Process 263 h i Q O O H;a.s/ spQ;a.; s/ 1 D aŒwO ./ 1 pQ;a.; s/ ; (9.5.43) with

HQ .s/ 1 .s/Q HQ;a.s/ D D : a as.s/Q

What is the effect of such combined rescaling and respeeding?Weﬁnd

1 a .s/Q Q;a.s/ D D ; (9.5.44) 1 C sHQ;a.s/ 1 .1 a/ .s/Q and are now in a position to address the asymptotic universality of the Mittag-Lefﬂer waiting time density. Using Lemma 9.2 with s in place of s and taking

a D ˇ ; (9.5.45)

ﬁxing s as required by the continuity theorem of probability for Laplace transforms, the asymptotics .s/Q D 1 .s/ˇ/ C o..s/ˇ/ for ! 0 implies

ˇ ˇ ˇ 1 .s/ / C o..s/ / 1 ML Q ˇ .s/D ! DQ ; ; 1 .1 ˇ/ 1 .s/ˇ/ C o..s/ˇ/ 1 C sˇ ˇ (9.5.46) corresponding to the Mittag-Lefﬂer density

d ML.t/ D E .t ˇ/: ˇ dt ˇ

Observe that the parameter does not appear in the limit 1=.1 C sˇ/. We can make it reappear by choosing the respeeding factor ˇ in place of ˇ. In fact:

1 Q ˇ ! : ; 1 C sˇ Formula (9.5.46) says that the general density .t/ with power law asymptotics as in Lemma 9.2 is gradually deformed into the Mittag-Leffler waiting time ML density ˇ .t/. This means that with larger and larger unit of time (by sending ! 0) and stronger and stronger deceleration (by a D ˇ) as described our process becomes indistinguishable from one with Mittag-Leffler waiting time (the probability distribution of jumps remaining unchanged). Likewise a pure renewal process with asymptotic power law density becomes indistinguishable from the one with Mittag-Leffler waiting time (the fractional generalization of the Poisson process due to Laskin [Las03] and Mainardi, Gorenflo and Scalas [MaGoSc04a]). 264 9 Applications to Stochastic Models

9.6 Historical and Bibliographical Notes

A fractional generalization of the Poisson probability distribution was presented by Pillai in 1990 in his pioneering work [Pil90]. He introduced the probability distribution (which he called the Mittag-Leffler distribution) using the complete monotonicity of the Mittag-Leffler function. As has already been mentioned, the complete monotonicity of this function was proved by Pollard in 1948, see [Poll48] and [Fel49], as well as in more recent works by Schneider [Sch96], and Miller and Samko [MilSam97]. Nowadays the concept of complete monotonicity is widely investigated in the framework of the Bernstein functions (non-negative functions with a complete monotone first derivative), see the recent book by Schiling et al. [SchSoVo12] The concept of a geometrically infinitely divisible distribution was introduced in 1984 in [KlMaMe84]. Later in 1995 Pillai introduced [PilJay95](seealso [JayPil96]) a discrete analogue of such a distribution (the discrete Mittag-Leffler distribution). Another possible generalization of the Poisson distribution is that introduced by Lamperti in 1958 [Lam58] whose density has the expression

sin ˛ y˛1 f .y/ D ; X˛ y2˛ C 2y˛ cos ˛ C 1 see also [Jam10]. We recognize in it the spectral distribution of the Mittag-Lefﬂer ˛ function E˛.t / formerly derived in 1947 by Gross for linear viscoelasticity [Gro47] and then used by Caputo and Mainardi [CapMai71a,CapMai71b]. Lamperti considered a random variable equal to the ratio of two variables X˛ D S˛=S˛;0;0< ˛<1,whereS˛ is a positive stable random variable, with density f˛, and having Laplace transform

˛ E esS˛ D es ; and S˛ is a variable independent of S˛; ; >˛; whose laws follow a polynomially tilted stable distribution having density proportional to t f˛.t/. The concept of a renewal process has been developed as a stochastic model for describing the class of counting processes for which the times between successive events are independent identically distributed (i.i.d.) non-negative random variables, obeying a given probability law. These times are referred to as waiting times or inter-arrival times. The process of accumulation of waiting times is inverse to the counting number process, and is called the Erlang process in honour of the Danish mathematician and telecommunication engineer A.K. Erlang (see [BrHaJe48]). For more details see e.g. the classical treatises by Khintchine [Khi37, Khi60], Cox [Cox67], Gnedenko and Kovalenko [GneKov68], Feller [Fel71], and the book by Ross [Ros97], as well as the recent survey paper [MaiRog06] and the book [RogMai11] by Rogosin and Mainardi. The Mittag-Leffler function also appears in the solution of the fractional master equation. This equation characterizes the renewal processes with reward modelled 9.6 Historical and Bibliographical Notes 265 by the random walk model known as continuous time random walk (abbreviated CTRW). In this the waiting time is assumed to be a continuous random variable. The name CTRW became popular in physics in the 1960s after Montroll, Weiss and Scher (just to cite the pioneers) published a celebrated series of papers on random walks to model diffusion processes on lattices, see e.g. [Wei94] and the references therein. The basic role of the Mittag-Leffler waiting time probability density in time fractional continuous time random walk (CTRW) has become well-known via the fundamental paper of Hilfer and Anton [HilAnt95]. Earlier in the theory of thinning (rarefaction) of a renewal process under power law assumptions (see Gnedenko and Kovalenko’s book [GneKov68]), this density had been found as a limit density by a combination of thinning followed by rescaling the time and imposing a proper relation between the rescaling factor and the thinning parameter. In 1985 Balakrishnan [BalV85] defined a special class of anomalous random walk where the anomaly appears by growth of the second moment of the sojourn probability density like a power of time with exponent between 0 and 1. This paper appeared a few years before the fundamental paper by Schneider and Wyss [SchWys89], but did not attract much attention (probably because of its style of presentation). However, it should be mentioned that by a well-scaled passage to the limit from CTRW the space-time fractional diffusion equation in the form of an equivalent integro-differential equation was obtained in [BalV85]. Remarkably, Gnedenko and Kovalenko [GneKov68] and Balakrishnan [BalV85] ended their analysis by giving the solution only as a Laplace transform without inverting it. CTRWs are rather good and general phenomenological models for diffusion, including anomalous diffusion, provided that the resting time of the walker is much greater than the time it takes to make a jump. In fact, in the formalism, jumps are instantaneous. In more recent times, CTRWs have been applied to economics and finance by Hilfer [Hil84], by Gorenflo–Mainardi–Scalas and their co-workers [ScGoMa00,Mai-et-al00,Gor-et-al01,RaScMa02,Sca-et-al03], and, later, by Weiss and co-workers [MaMoWe03]. It should be noted, however, that the idea of combining a stochastic process for waiting times between two consecutive events and another stochastic process which associates a reward or a claim to each event dates back at least to the first half of the twentieth century with the so-called Cramér–Lundberg model for insurance risk, see for a review [EmKlMi01]. In a probabilistic framework, we now find it more appropriate to refer to all these processes as compound renewal processes. Serious studies of the fractional generalization of the Poisson process – replacing the exponential waiting time distribution by a distribution given via a Mittag-Leffer function with modified argument – began around the turn of the millennium, and since then many papers on its various aspects have appeared. There are in the literature many papers on this generalization where the authors have outlined a number of aspects and definitions, see e.g. Repin and Saichev [RepSai00], Wang et al. [WanWen03, WaWeZh06], Laskin [Las03, Las09], Mainardi et al. [MaGoSc04a], Uchaikin et al. [UcCaSi08], Beghin and Orsingher [BegOrs09], Cahoy et al. [CaUcWo10], Meerschaert et al. [MeNaVe11], Politi et al. [PoKaSc11], Kochubei [Koc11], so it would be impossible to list them all exhaustively. 266 9 Applications to Stochastic Models

However, in effect this generalization had already been used in 1995: Hilfer and Anton [HilAnt95] showed (using different terminology) that the fractional Kolmogorov–Feller equation (replacing the first order time derivative by a fractional derivative of order between 0 and 1) requires the underlying random walk to be subordinated to a renewal process with Mittag-Leffer waiting time. Gorenflo and Mainardi [GorMai08, Gor10] mention the asymptotic universality of the Mittag-Leffler waiting time density for the family of power law renewal processes. An alternative renewal process called the Wright process was investigated by Mainardi et al. [Mai-et-al00, MaGoVi05, MaGoVi07] as a process arising by discretization of the stable subordinator. This approach is based on the concept of the extremal Lévy stable density (Lévy stable processes are widely discussed in several books on probability theory, see, e.g., [Fel71, Sat99]). For the study of the Wright processes an essential role is played by the so-called M -Wright function (see, e.g., [Mai10]). A scaled version of this process has been used by Barkai [Bark02] to approximate the time-fractional diffusion process directly by a random walk subordinated to it (executing this scaled version in natural time), and he has found rather poor convergence in refinement. In Gorenflo et al. [GoMaVi07]the way of using this discretized stable subordinator has been modified. By appropriate discretization of the relevant spatial stable process we have then obtained a simulation method equivalent to the solution of a pair of Langevin equations, see Fogedby [Fog94] and Kleinhans and Friedrich [KleFri07]. For simulation of space- time fractional diffusion one then obtains a sequence of precise snapshots of a true particle trajectory, see for details Gorenflo et al. [GoMaVi07], and also Gorenflo and Mainardi [GorMai12a, GorMai12b]. There are several ways to generalize the classical diffusion equation by introducing space and/or time derivatives of fractional order. We mention here the seminal paper by Schneider and Wyss [SchWys89] for the time fractional diffusion equation and the influential paper by Saichev and Zaslavsky [SaiZas97] for diffusion in time as well as in space. In the recent literature several authors have stressed the viewpoint of subordination, and special attention is being paid to diffusion equations with distributed order of fractional temporal or/and spatial differentiation. The transition from CTRW to such generalized types of diffusion has been investigated by different methods, not only for its purely mathematical interest but also due to its applications in Physics, Chemistry, and other Applied Sciences, including Economics and Finance. As good reference texts on these topics, containing extended lists of relevant works, we refer to the review papers by Metzler and Klafter [MetKla00, MetKla04].

9.7 Exercises

9.7.1. A random variable X is said to be gamma distributed (or have gamma density) with parameters .˛; ˇ/, ˛>0;ˇ>0, if its density of probability has the form 9.7 Exercises 267

˛1 x x f.x/ D e ˇ ; for x 0; and f.x/ Á 0; x < 0: ˇ˛.˛/

Let X1, X2 be independently distributed gamma variables with parameters 1 .˛; 1/ and .˛ C 2 ;1/, respectively. Let U D X1X2. Show that the density f.u/ of this distribution is given by

22˛1 1 f.u/ D u˛1e2u 2 ; u 0I and f.x/ Á 0; x < 0: (9.7.1) .2˛/

9.7.2. Let X1, X2, X3 be independently distributed gamma variables with parame- 1 2 ters .˛; 1/, .˛ C 3 ;1/,and.˛ C 3 ;1/, respectively. Let U D X1X2X3. Show that the density f.u/ of this distribution can be represented via an H-function in the form Ä ˇ ˇ 27 1;0 ˇ f.u/ D H 27u ˇ ; u 0I and f.x/ Á 0; x < 0: .3˛/ 0;1 .3˛ 3; 3/ (9.7.2)

9.7.3. Let us consider the stochastic process x D x.t/; t 0; with x 0 (called in some sources the Mittag-Lefﬂer process) having the density ([Pil90, p. 190])

X1 .t C k/ f .x; t/ D .1/k x˛.tCk/: (9.7.3a) ˛ kŠ .t/ .˛.t C k/ C 1/ kD0

Show that this process obeys the following subordination formula

f˛.x; t/ D r˛.x; t/.t;t/dt; (9.7.3b) 0

where r˛.x; t/ is the distribution of the stable process with the Laplace transform equal to exp .t˛ / and

1 .t ;t/D t t1et : (9.7.3c) .t/

9.7.4. Consider the stochastic process x D x.t/; t 0; with x 0 having the density .t;t/with Laplace transform (a so-called -process)

t t log .1Cs/ .sQ ;t/D .1 C s/ D e : (9.7.4a)

Find the density ˇ.t; t/ of the inverse process t D t.t/ happening in t 0, running along t 0. 268 9 Applications to Stochastic Models

Answer.

s .1 C s /t d L1 ˇ.t; t/ D s : (9.7.4b) dt log .1 C s/

G 9.7.5. Represent the fundamental solution ˇ .x; t/ of the rightward time fractional drift equation ([GorMai12a]) Á @u Dˇu .x; t/ D .x; t/; 1

X1 .z/n M .z/ WD W .z/ D ; 0<<1: ;1 nŠ ŒnC .1 / nD0 (9.7.5b)

Answer. ( Á t ˇM x ;x>0; G ˇ t ˇ ˇ .x; t/ D (9.7.5c) 0; x < 0:

9.7.6. Rescaling and respeeding. Show that the Mittag-Lefﬂer waiting time density

d ML.t/ D E .t ˇ/; 0 < ˇ Ä 1; (9.7.6) ˇ dt ˇ

is invariant under combined rescaling and respeeding if a D ˇ. Answer. Use Eq. (9.5.44) Appendix A The Eulerian Functions

Here we consider the so-called Eulerian functions, namely the well-known Gamma function and Beta function together with some special functions that turn out to be related to them, such as the Psi function and the incomplete gamma functions. We recall not only the main properties and representations of these functions, but we also brieﬂy consider their applications in the evaluation of certain expressions relevant for the fractional calculus.

A.1 The Gamma Function

The Gamma function .z/ is the most widely used of all the special functions: it is usually discussed ﬁrst because it appears in almost every integral or series representation of other advanced mathematical functions. We take as its deﬁnition the integral formula Z 1 .z/ WD uz1 eu du ; Re z >0: (A.1.1) 0 This integral representation is the most common for , even if it is valid only in the right half-plane of C. The analytic continuation to the left half-plane can be done in different ways. As will be shown later, the domain of analyticity D of is

D D C nf0; 1; 2;:::;g : (A.1.2)

Using integration by parts, (A.1.1) shows that, at least for Re z >0, satisﬁes the simple difference equation

.z C 1/ D z .z/; (A.1.3)

© Springer-Verlag Berlin Heidelberg 2014 269 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 270 A The Eulerian Functions which can be iterated to yield

.z C n/ D z .z C 1/::: .z C n 1/ .z/; n2 N : (A.1.4)

The recurrence formulas (A.1.3 and A.1.4) can be extended to any z 2 D .In particular, since .1/ D 1,wegetfornon-negative integer values

.n C 1/ D nŠ ; n D 0; 1; 2; : : : : (A.1.5)

As a consequence can be used to deﬁne the Complex Factorial Function

zŠ WD .z C 1/ : (A.1.6)

By the substitution u D v2 in (A.1.1)wegettheGaussian Integral Representation Z 1 2 .z/ D 2 ev v2z1dv ; Re .z/>0; (A.1.7) 0 which can be used to obtain when z assumes positive semi-integer values. Starting from Â Ã Z C1 1 2 p D ev dv D 1:77245 ; (A.1.8) 2 1 we obtain for n 2 N, Â Ã Z Â Ã C1 1 2 1 .2n 1/ŠŠ p .2n/Š n C D v vn vD D :1 e d n 2n (A.1.9) 2 1 2 2 2 nŠ

A.1.1 Analytic Continuation

A common way to derive the domain of analyticity (A.1.2) is to carry out the analytic continuation by the mixed representation due to Mittag-Lefﬂer: Z X1 .1/n 1 .z/ D C eu uz1 du ; z 2 D : (A.1.10) nŠ.z C n/ nD0 1

This representation can be obtained from the so-called Prym’s decomposition, namely by splitting the integral in (A.1.1) into two integrals, one over the interval

1The double factorial mŠŠ means mŠŠ D 1 3 :::mif m is odd, and mŠŠ D 2 4 :::mif m is even. A.1 The Gamma Function 271

0 Ä u Ä 1 which is then developed as a series, the other over the interval 1 Ä u Ä 1, which, being uniformly convergent inside C, provides an entire function. The terms of the series (uniformly convergent inside D ) provide the principal parts of at the corresponding poles zn Dn. So we recognize that is analytic in the entire complex plane except at the points zn Dn (n D 0; 1; : : : ), which turn n out to be simple poles with residues Rn D .1/ =nŠ. The point at inﬁnity, being an accumulation point of poles, is an essential non-isolated singularity. Thus is a transcendental meromorphic function. A formal way to obtain the domain of analyticity D is to carry out the required analytical continuation via the Recurrence Formula

.z C n/ .z/ D ; (A.1.11) .z C n 1/ .z C n 2/:::.z C 1/ z that is obtained by iterating (A.1.3) written as .z/ D .z C 1/=z.Inthiswaywe can enter the left half-plane step by step. The numerator on the R.H.S. of (A.1.11) is analytic for Re z > n; hence, the L.H.S. is analytic for Re z > n except for simple poles at z D 0; 1;:::;.n C 2/; .n C 1/.Sincen can be arbitrarily large, we deduce the properties discussed above. Another way to interpret the analytic continuation of the Gamma function is provided by the Cauchy–Saalschütz representation, which is obtained by iterated integration by parts in the basic representation (A.1.1). If n 0 denotes any non- negative integer, we have Z Ä 1 1 1 .z/D uz1 eu 1 C u u2 CC.1/nC1 un du (A.1.12) 0 2 n in the strip .n C 1/ < Re z < n. To prove this representation the starting point is provided by the integral Z 1 uz1 Œeu 1 du ; 1

Integration by parts gives (the integrated terms vanish at both limits) Z Z 1 1 1 1 uz1 Œeu 1 du D uzeu du D .z C 1/ D .z/: 0 z 0 z

So, by iteration, we get (A.1.12).

A.1.2 The Graph of the Gamma Function on the Real Axis

Plots of .x/ (continuous line) and 1=.x/ (dashed line) for 4

6 Γ(x)

2 1/Γ(x) 0

−2

−4

−6 −4 −3 −2 −1 0 1 2 3 4 x

Fig. A.1 Plots of .x/ (continuous line)and1=.x/ (dashed line)

2.5 Γ(x)

2 1/Γ(x) 1.5

X: 1.462 Y: 0.8856 0.5

0 0 0.5 1 1.5 2 2.5 3 3.5 4 x

Fig. A.2 Plots of .x/ (continuous line)and1=.x/ (dashed line)

Hereafter we provide some analytical arguments that support the plots on the real axis. In fact, one can get an idea of the graph of the Gamma function on the real axis using the formulas

.x/ .x C 1/ D x.x/; .x 1/ D ; x 1 to be iterated starting from the interval 0

For x>0the integral representation (A.1.1) yields .x/ > 0 and 00.x/ > 0 since Z Z 1 1 .x/ D eu ux1 du ;00.x/ D eu ux1 .log u/2 du : 0 0

As a consequence, on the positive real axis .x/ turns out to be positive and convex so that it ﬁrst decreases and then increases, exhibiting a minimum value. Since .1/ D .2/ D 1, we must have a minimum at some x0, 1

A.1.3 The Reﬂection or Complementary Formula .z/.1 z/ D : (A.1.13) sin z

This formula, which shows the relationship between the function and the trigonometric sin function, is of great importance together with the recurrence formula (A.1.3). It can be proved in several ways; the simplest proof consists in proving (A.1.13)for0

A.1.4 The Multiplication Formulas

Gauss proved the following Multiplication Formula

nY1 k .nz/ D .2/.1n/=2 nnz1=2 .z C /; nD 2;3;::: ; (A.1.14) n kD0 which reduces, for n D 2,toLegendre’s Duplication Formula

1 1 .2z/ D p 22z1=2 .z/.z C /; (A.1.15) 2 2 274 A The Eulerian Functions and, for n D 3,totheTriplication Formula

1 1 2 .3z/ D 33z1=2 .z/.z C /.z C /: (A.1.16) 2 3 3

A.1.5 Pochhammer’s Symbols

Pochhammer’s symbols .z/n are deﬁned for any non-negative integer n as

.z C n/ .z/ WD z .z C 1/ .z C 2/:::.z C n 1/ D ;n2 N ; (A.1.17) n .z/ with .z/0 D 1. In particular, for z D 1=2 ; we obtain from (A.1.9) Â Ã 1 .n C 1=2/ .2n 1/ŠŠ WD D : n 2 n .1=2/ 2

We extend the above notation to negative integers, deﬁning

.z C 1/ .z/ WD z .z 1/ .z 2/:::.z n C 1/ D ;n2 N : (A.1.18) n .z n C 1/

A.1.6 Hankel Integral Representations

In 1864 Hankel provided a complex integral representation of the function 1=.z/ valid for unrestricted z;itreads: Z 1 1 et .z/ D t; z 2 C ; z d (A.1.19a) 2i Hat where Ha denotes the Hankel path deﬁned as a contour that begins at t D1ia (a>0), encircles the branch cut that lies along the negative real axis, and ends up at t D1Cib (b>0). Of course, the branch cut is present when z is non-integer because t z is a multivalued function; in this case the contour can be chosen as in Fig. A.3 left, where

C;above the cut; arg .t/ D ; below the cut:

When z is an integer, the contour can be taken to be simply a circle around the origin, described in the counterclockwise direction. A.1 The Gamma Function 275

Fig. A.3 The left Hankel contour Ha and the right Hankel contour HaC

An alternative representation is obtained assuming the branch cut along the positive real axis; in this case we get Z 1 1 et .z/ D t; z 2 C ; z d (A.1.19b) 2i HaC.t/ where HaC denotes the Hankel path deﬁned as a contour that begins at t DC1Cib (b>0), encircles the branch cut that lies along the positive real axis, and ends up at t DC1ia (a>0). When z is non-integer the contour can be chosen as in Fig. A.3 left, where

0; above the cut; arg .t/ D 2 ; below the cut:

When z is an integer, the contour can be taken to be simply a circle around the origin, described in the counterclockwise direction. We note that

i Ci Ha ! HaC if t ! t e ; and HaC ! Ha if t ! t e :

The advantage of the Hankel representations (A.1.19a)and(A.1.19b) compared with the integral representation (A.1.1) is that they converge for all complex z and not just for Rez >0. As a consequence 1= is a transcendental entire function (of maximum exponential type); the point at inﬁnity is an essential isolated singularity, which is an accumulation point of zeros (zn Dn; n D 0; 1; : : : ). Since 1= is entire, does not vanish in C. The formulas (A.1.19a)and(A.1.19b) are very useful for deriving integral representations in the complex plane for several special functions. Furthermore, using the reﬂection formula (A.1.13), we can get the integral representations of itself in terms of the Hankel paths (referred to as Hankel integral representations for ), which turn out to be valid in the whole domain of analyticity D . 276 A The Eulerian Functions

The required Hankel integral representations that provide the analytical continuation of turn out to be:

(a) Using the path Ha Z 1 t z1 .z/ D e t dt; z 2 D I (A.1.20a) 2i sin z Ha

(b) Using the path HaC Z 1 t z1 .z/ D e .t/ dt; z 2 D : (A.1.20b) 2i sin z HaC

A.1.7 Notable Integrals via the Gamma Function Z 1 .˛ C 1/ st t ˛ t D ; .s/ > 0 ; .˛/ > 1: e d ˛C1 Re Re (A.1.21) 0 s

This formula provides the Laplace transform of the power function t ˛ . Z 1 ˇ .1 C 1=ˇ/ at t D ; .a/>0; ˇ >0: e d 1=ˇ Re (A.1.22) 0 a p This integral for fixed a>0and ˇ D 2 attains the well-known value =a related to the Gauss integral.Forfixeda>0, the L.H.S. of (A.1.22) may be referred to as the generalized Gauss integral. The function I.ˇ/ WD .1 C 1=ˇ/ strongly decreases from infinity at ˇ D 0 to a positive minimum (less than the unity) attained around ˇ D 2 andthenslowly increases to the asymptotic value 1 as ˇ !1. The minimum value is attained at ˇ0 D 2:16638 : : : and I.ˇ0/ D 0:8856 : : : A more general formula is Z 1 1 .=/ 1 .1 C =/ zt t 1 t D D ; e d = = (A.1.23) 0 z z where Rez >0, >0,Re./ > 0. This formula includes (A.1.21)and(A.1.22); it reduces to (A.1.21)forz D s, D 1 and D ˛ C 1,andto(A.1.22)forz D a, D ˇ and D 1. A.1 The Gamma Function 277

A.1.8 Asymptotic Formulas Ä p 1 1 .z/ ' 2ez zz1=2 1 C C C ::: I (A.1.24) 12 z 288 z2 as z !1with jargzj <. This asymptotic expression is usually referred to as Stirling’s formula, originally given for nŠ. The accuracy of this formula is surprisingly good on the positive real axis and also for moderate values of z D x>0;as can be noted from the following exact formula, p xC xC1=2 xŠ D 2e. 12x / x I x>0; (A.1.25) where is a suitable number in .0; 1/. The two following asymptotic expressions provide a generalization of the Stirling formula. If a; b denote two positive constants, we have p .az C b/ ' 2eaz .az/azCb1=2 ; (A.1.26) as z !1with jargzj <,and Ä .z C a/ .a b/.a C b 1/ ' zab 1 C C ::: ; (A.1.27) .z C b/ 2z as z !1along any curve joining z D 0 and z D1provided z ¤a; a 1;:::, and z ¤b; b 1;::: :

A.1.9 Inﬁnite Products

An alternative approach to the Gamma function is via inﬁnite products, described by Euler in 1729 and Weierstrass in 1856. Let us start with the original formula given by Euler,

z 1 Y1 1 C 1 nŠ nz .z/ WD n D lim : (A.1.28) z 1 C z n!1 z.z C 1/:::.z C n/ nD1 n

The above limits exist for all z 2 D C : From Euler’s formula (A.1.28) it is possible to derive Weierstrass’ formula

1 h Á i 1 Y z D z e C z 1 C ez=n ; (A.1.29) .z/ n nD1 278 A The Eulerian Functions where C , called the Euler–Mascheroni constant,isgivenby 8 P < n 1 limn!1 kD1 k log n ; C D 0:5772157 : : : D : R (A.1.30) 0 1 u .1/ D 0 e log u du :

A.2 The Beta Function

A.2.1 Euler’s Integral Representation

The standard representation of the Beta function is Z 1 Re.p/ > 0 ; B.p; q/ D u p1 .1 u/ q1 du ; (A.2.1) 0 Re.q/ > 0 :

Note that, from a historical viewpoint, this representation is referred to as the Euler integral of the ﬁrst kind, while the integral representation (A.1.1)for is referred to as the Euler integral of the second kind. The Beta function is a complex function of two complex variables whose analyticity properties will be deduced later, as soon as the relation with the Gamma function has been established.

A.2.2 Symmetry

B.p; q/ D B.q; p/ : (A.2.2)

This property is a simple consequence of the deﬁnition (A.2.1).

A.2.3 Trigonometric Integral Representation Z =2 Re.p/ > 0 ; B.p; q/D2 .cos #/2p1 .sin #/2q1 d#; (A.2.3) 0 Re.q/ > 0 :

This noteworthy representation follows from (A.2.1) by setting u D .cos #/2. A.2 The Beta Function 279

A.2.4 Relation to the Gamma Function .p/.q/ B.p; q/ D : (A.2.4) .p C q/

This relation is of fundamental importance. Furthermore, it allows us to obtain the analytical continuation of the Beta function. The proof of (A.2.4) can easily be obtained by writing the product .p/.q/ as a double integral that is to be evaluated introducing polar coordinates. In this respect we must use the Gaussian representation (A.1.7) for the Gamma function and the trigonometric representation (A.2.3) for the Beta function. In fact, Z Z 1 1 2 2 .p/.q/ D 4 e.u Cv / u2p1 v2q1 du dv 0 0 Z Z 1 =2 2 D 4 e 2.pCq/1 d .cos #/2p1 .sin #/2q1 d# 0 0 D .p C q/B.p; q/ :

Henceforth, we shall exhibit other integral representations for B.p; q/, all valid for Rep>0;Req>0:

A.2.5 Other Integral Representations

Integral representations on Œ0; 1/ are 8 Z ˆ 1 xp1 ˆ dx; ˆ pCq ˆ 0 .1 C x/ ˆ <ˆ Z 1 xq1 B.p; q/ D dx; (A.2.5) ˆ pCq ˆ 0 .1 C x/ ˆ ˆ Z ˆ 1 p1 q1 :ˆ 1 x C x pCq dx: 2 0 .1 C x/

x The ﬁrst representation follows from (A.2.1) by setting u D 1Cx ; the other two are easily obtained by using the symmetry property of B.p; q/ : A further integral representation on Œ0; 1 is Z 1 yp1 C yq1 B.p; q/ D y: pCq d (A.2.6) 0 .1 C y/ 280 A The Eulerian Functions

This representation is obtained from the ﬁrst integral in (A.2.5)asasumofthetwo contributions Œ0; 1 and Œ1; 1/.

A.2.6 Notable Integrals via the Beta Function

The Beta function plays a fundamental role in the Laplace convolution of power functions. We recall that the Laplace convolution is the convolution between causal functions (i.e. vanishing for t<0), Z Z C1 t f.t/ g.t/ D f./g.t /d D f./g.t /d: 1 0

The convolution satisﬁes both the commutative and associative properties:

f.t/ g.t/ D g.t/ f.t/; f.t/ Œg.t/ h.t/ D Œf .t/ g.t/ h.t/ :

It is straightforward to prove the following identity, by setting in (A.2.1) u D =t ; Z t t p1 t q1 D p1 .t /q1 d D t pCq1 B.p; q/ : (A.2.7) 0

Introducing the causal Gel’fand–Shilov function

t 1 ˚ .t/ WD C ;2 C ; ./

(where the sufﬁx C just denotes the causality property of vanishing for t<0), we can write the previous result in the following interesting form:

˚p.t/ ˚q.t/ D ˚pCq .t/ : (A.2.8)

In fact, dividing the L.H.S. of (A.2.7)by.p/.q/, and using (A.2.4), we obtain (A.2.8). As a consequence of (A.2.8) we get the semigroup property of the Riemann–Liouville fractional integral operator. In the following we describe other relevant applications of the Beta function. The results (A.2.7–A.2.8) show that the convolution integral between two (causal) functions, which are absolutely integrable in any interval Œ0; t and bounded in every ﬁnite interval that does not include the origin, is not necessarily continuous at t D 0, even if a theorem ensures that this integral turns out to be continuous for any t>0, see e.g. [Doe74, pp. 47–48]. In fact, considering two arbitrary real numbers ˛; ˇ greater than 1,wehave

˛ ˇ ˛CˇC1 I˛;ˇ.t/ WD t t D B.˛ C 1; ˇ C 1/ t ; (A.2.9) A.2 The Beta Function 281 so that 8 < C1 if 2<˛C ˇ<1; lim I˛;ˇ.t/ D c.˛/ if ˛ C ˇ D1; (A.2.10) ! C : t 0 0 if ˛ C ˇ>1; where c.˛/ D B.˛ C 1; ˛/ D .˛ C 1/ .˛/ D = sin.˛/. We note that in the case ˛ C ˇ D1 the convolution integral attains for any t>0the constant value c.˛/ :In particular, for ˛ D ˇ D1=2 ; we obtain the minimum value for c.˛/,i.e. Z t d p p D : (A.2.11) 0 t The Beta function is also used to prove some basic identities for the Gamma function, like the complementary formula (A.1.13)andtheduplication formula (A.1.15). For the complementary formula it is sufﬁcient to prove it for a real argument in the interval .0; 1/, namely .˛/.1 ˛/ D ;0<˛<1: sin ˛ We note from (A.2.4–A.2.5)that Z 1 x˛1 .˛/.1 ˛/ D B.˛; 1 ˛/ D dx: 0 1 C x

Then it remains to use well-known formula Z 1 x˛1 dx D : 0 1 C x sin ˛

To prove the duplication formula we note that it is equivalent to

.1=2/ .2z/ D 22z1 .z/.z C 1=2/ ; and hence, after simple manipulations, to

B.z;1=2/D 22z1 B.z; z/: (A.2.12)

This identity is easily veriﬁed for Re.z/>0, using the trigonometric representation (A.2.3) for the Beta function and noting that Z Z Z =2 =2 =2 .cos #/˛ d# D .sin #/˛ d# D 2˛ .cos #/˛ .sin #/˛ d#; 0 0 0 with Re.˛/ > 1,sincesin2# D 2 sin # cos #: 282 A The Eulerian Functions

A.3 Historical and Bibliographical Notes

For the historical development of the Gamma function we refer the reader to the notable article [Dav59]. It is surprising that the notation and the name Gamma function were ﬁrst used by Legendre in 1814 whereas in 1729 Euler had represented his function via an inﬁnite product, see Eq. (A.1.28). As a matter of fact Legendre introduced the representation (A.1.1) as a generalization of Euler’s integral expression for nŠ, Z 1 nŠ D . log t/n dt: 0

Indeed, changing the variable t ! u Dlog t,weget Z 1 nŠ D eu un du D .n C 1/ : 0

Lastly, we mention the well-known Bohr–Mollerup theorem which states that the -function is the only function which satisﬁes the relation f.z C 1/ D zf.z/ where log f.z/ is convex and f.1/ D 1. The proof of this fact is presented, e.g., in the book by Artin [Art64].

A.4 Exercises

A.4.1. The Pochhammer symbol is deﬁned as follows (see, e.g. [Tem96, p. 72])

.a C n/ .a/ WD a.a C 1/:::.aC n 1/ D : n .a/

Verify that for all a 2 C;m;nD 0; 1; : : : the following identities hold 8 <ˆ 0; if n>m; (a) .m/n D ˆ : n mŠ .1/ .mn/Š ; if n Ä mI n (b) .a/n D .1/ .a n C1/nI 2n a aC1 (c) .a/2n D 2 2 n 2 n I 2nC1 a aC1 (d) .a/2nC1 D 2 2 nC1 2 n : Prove the following formulas for suitable values of parameters A.4.2. ([Rai71, p. 103])

.1 C a=2/ cos a.1 a/ D 2 : .1C a/ .1 a=2/ A.4 Exercises 283

A.4.3. ([Rai71, p. 103])

.1 C a b/ sin .b a=2/.b a=2/ D : .1C a=2 b/ sin .b a/.b a/

A.4.4. Prove the formula ([Tem96,p.72]) Z 1 Á x 1 z t z1e˛t dt D ˛z=x; Re ˛>0;Re x>0;Re z >0: 0 x x

A.4.5. Verify the formulas ([Tem96, p. 73]) p (a) n C 1 D .1/n 22n nŠ ;nD 0; 1; 2; : : : : 2 .2n/Š 1 np 2n .2n/Š (b) n C 2 D .1/ 2 nŠ ;nD 0;1;2;:::: A.4.6. Verify the alternative reﬂection formula for the Gamma function ([Tem96, p. 74]) Â Ã Â Ã 1 1 1 z C z D ; z 62 Z; 2 2 cos z 2

in particular Â Ã Â Ã 1 1 iy C iy D ;y2 R: 2 2 cosh y

A.4.7. Verify the generalized reﬂection formula for the Gamma function ([Tem96, p. 74])

.z/.1 z/ .z n/ D .1/n .n C 1 z/ .1/n D ; z 62 Z;nD 0;1;2;:::: sin z.n C 1 z/

A.4.8. By using the reﬂection formula show that s 2 j.iy/j ejyj; as y !˙1: jyj

Calculate the following integrals A.4.9. ([Tem96, p. 74])

Z 2 .cos t/x cos ty dt; Re x>1: 0 284 A The Eulerian Functions

.x C 1/ Answer. h i . 2xC1 xCy xy 2 C 1 2 C 1

A.4.10. ([Tem96, pp. 76–77]) Z 1 t z1 log j1 tj dt; 1

cot z Answer. . z

A.4.11. ([Tem96, pp. 76–77], [Sne56]) R 1 z1 (a) 0 t cos t dt; Re z >1: z Answer. .z/ cos . R 2 1 z1 (b) 0 t sin t dt; Re z >1: z Answer. .z/ sin . 2 A.4.12.([WhiWat52, p. 300]) Show that for a<0, a D C ˛, 2 N;˛ >0,

.x/.a/ X1 R D n C G .x/ ; .x C a/ x C n n nD1

where

.1/n.a 1/.a 2/:::.a n/ R C G.n/; n nŠ Á Á Á x x x G.x/ D 1 C 1 C ::: 1 C ; a 1 a 2 a G.x/ G.n/ G .x/ D : n x C n Appendix B The Basics of Entire Functions

B.1 Deﬁnition and Series Representations

A complex-valued function F W C ! C is called an entire function (or integral function) if it is analytic (C-differentiable) everywhere on the complex plane, i.e. if at each point z0 2 C the following limit exists

F.z/ F.z / lim 0 2 C: z!z0 z z0

Typical examples of entire functions are the polynomials, the exponential functions and also sums, products and compositions of these functions, thus trigonometric and hyperbolic functions. Among the special functions we point out the following entire functions: Airy functions Ai.z/; Bi.z/, Bessel functions of the first and second m;n kind J.z/; Y.z/,FoxH-functions Hp;q .z/ for certain values of parameters, the 1 reciprocal Gamma function .z/ , the generalized hypergeometric function pFq .z/, m;n Meijer’s G-functions Gp;q .z/, the Mittag-Leffler function E˛.z/ and its different generalizations, and the Wright function .zI ; ˇ/. According to Liouville’s theorem an entire function either has a singularity at infinity or it is a constant. Such a singularity can be either a pole (as is the case for a polynomial), or an essential singularity. In the latter case we speak of transcendental entire functions. All of the above-mentioned special functions are transcendental. Every entire function can be represented in the form of a power series

X1 k F.z/ D ck z ; (B.1.1) kD0 converging everywhere on C. Thus, according to the Cauchy–Hadamard formula, the coefﬁcients of the series (B.1.1) satisfy the following condition (the necessary

© Springer-Verlag Berlin Heidelberg 2014 285 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 286 B The Basics of Entire Functions and sufficient condition for the sum of a power series to represent an entire function):

1 lim jckj k D 0: (B.1.2) k!1

The absolute value of the coefﬁcients of an entire function necessarily decreases to zero (although not monotonically, in general). One can classify the corresponding function in terms of the speed of this decrease (see below in Sect. B.2). Thus jckj! 0 for z !1is a necessary but not sufﬁcient condition for convergence of a power series.

B.2 Growth of Entire Functions: Order, Type and Indicator Function

The global behaviour of entire functions of finite order is characterized by their order and type. Recall (see e.g. [Lev56]) that the order of an entire function F.z/ is defined as an infimum of those k for which we have the inequality

rk MF .r/ WD max jF.z/j < e ; 8r>r.k/: jzjDr

Equivalently

log log M .r/ WD D lim sup F : (B.2.1) F r!1 log r

A more delicate characteristic of an entire function is its type. Recall (see e.g. [Lev56]) that the type of an entire function F.z/ of finite order is defined as the infimum of those A>0for which the inequality

Ar MF .r/ < e ; 8r>r.k/; holds. Equivalently

log M .r/ WD D lim sup F : (B.2.2) F r!1 r

For an entire function F.z/ represented in the form of the series

X1 k F.z/ D ckz kD0 B.3 Weierstrass Canonical Representation: Distribution of Zeros 287 its order and type can by found by the following formulae

klog k D lim supk!1 1 ; (B.2.3) log jc j k Á 1 1 p k . e / D lim supk!1 k jckj : (B.2.4)

The asymptotic behaviour of an entire function is usually studied via its restriction to rays. In order to describe this we introduce the so-called indicator function of an entire function of order :

log jF.rei /j h. / D lim sup : (B.2.5) r!1 r For instance, the exponential function ez D expz has order D 1, type D 1 and indicator function h. / D cos ; Ä Ä .

B.3 Weierstrass Canonical Representation: Distribution of Zeros

The Weierstrass canonical representation generalizes the representation of complex polynomials in the form of a product of prime factors. By the fundamental theorem of algebra any complex polynomial of order n can be split into exactly n linear factors

Yn Pn.z/ D a .z ak/ with a 6D 0: (B.3.1) kD1

Entire functions can have infinitely many zeroes. In this case the finite product in (B.3.1) has to be replaced by an infinite product. Then the main question is to take an infinite product in such form that it can represent an entire function (so, is convergent on the whole complex plane). Another problem is that there are some entire functions which have very few zeroes (or have no zeroes at all, such as the exponential function). These two ideas were taken into account by Weierstrass. He took prime factors (or Weierstrass elementary factors) in the form 8 <ˆ .1 z/; if n D 0I En.z/ D ˆ n o (B.3.2) : z z2 zn .1 z/exp 1 C 2 C :::C n ; otherwise:

Usingsuchfactorsheproved 288 B The Basics of Entire Functions Theorem B.3.1 (Weierstrass). Let zj be a sequence of complex numbers j 2N0 (0 Djz0j < jz1jÄjz2jÄ:::), satisfying the following conditions:

(i) zj !1as j !1; (ii) There exists a sequence of positive integers pj j 2N such that ˇ ˇ X1 ˇ ˇ1Cpj ˇ z ˇ ˇ ˇ < 1: z j D1 j Then there exists an entire function which has zeroes only at the points zj , j 2N0 in particular, the following one Â Ã Y1 z F.z/ D zk E : (B.3.2a) pj z j D1 j

The following theorem is in a sense the converse of the above. Theorem B.3.2 (Weierstrass). Let F.z/ be an entire function and let zj be j 2N0 the zeros of F.z/, then there exists a sequence pj j 2N and an entire function g.z/ such that the following representation holds Â Ã Y1 z F.z/ D C zkeg.z/ E ; (B.3.3) pj z j D1 j where C is a constant and k 2 N0 is the multiplicity of the zero of F at the origin. For entire functions of finite order the Weierstrass theorems have a more exact form. Theorem B.3.3 (Hadamard). Let F.z/ be an entire function of finite order and let zj be the zeros of F.z/, listed with multiplicity, then the rank p of F.z/ is j 2N0 defined as the least positive integer such that ˇ ˇ X1 ˇ ˇ1Cp ˇ 1 ˇ ˇ ˇ < 1: (B.3.4) z j D1 j

Then the canonical Weierstrass product is given by Â Ã Y1 z F.z/ D C zkeg.z/ E ; (B.3.5) p z j D1 j B.4 Entire Functions of Completely Regular Growth 289 where g.z/ is a polynomial of degree q Ä . The genus of F.z/ deﬁned as max fp; qg is then also ﬁnite and

Ä : (B.3.6)

B.4 Entire Functions of Completely Regular Growth

Recall (see, e.g., [GoLeOs91, Lev56, Ron92]) that a ray arg z D is a ray of completely regular growth (CRG) for an entire function F of order if the following weak limit exists

log jF.rei /j lim ; (B.4.1) r!1 r where the term “weak limit” (lim in (B.4.1)) means that r tends to inﬁnity omitting the values of a set E RC which satisﬁes the condition

mes E \ .0; r/ lim D 0; r!1 r i.e. is relatively small. If all rays 2 Œ0; 2/ are rays of CRG for an entire function F (with the same exceptional set E D E ; 8 ), then such a function is called an entire function of completely regular growth. The main characterization of entire functions of completely regular growth is the following: An entire function F.z/ of order is a function of completely regular growth if and only if its set of zeroes zj has j 2N0 (a) In the case of a non-integer – an angular density:

n.r; ; / 4 .; / WD lim I (B.4.1a) r!1 r (b) In the case of an integer – an angular density and a ﬁnite angular symmetry: 8 9 < = 1 X 1 a WD lim q C ; (B.4.1b) r!1 : ; zj 0

B.5 Historical and Bibliographical Notes

The period from 1850 to 1950 was the “golden century of the theory of entire functions”. This theory was one of the central subjects of complex analysis, it was rapidly developed in connection with several deep problems in mathematics as well as due to the usefulness of the analytic machinery for the solution of a wide range of applied problems. Here we mention only a few results of that time which have great influence on contemporary mathematics and which are related in some way to the subject of our book. Many of these results were influenced by Weierstrass (see, e.g., [Wei66], and the modern description of Weierstrass’s results in [AblFok97, Rud74], and [Kra04]). He introduced the notion of single-valued analytic functions (eindeutig analytis- che Funktionen), established the method of uniform convergence for families of complex functions which he successfully applied to the proof of his celebrated factorization theorem, introduced and applied the notion of (single-valued) analytic elements which became the core of the analytic continuation method, and developed the theory of elliptic functions illustrated by a special collection of these function now bearing his name and used by him and his successors to create and develop the analytic theory of complex differential equations. Mittag-Leffler was influenced by the genius of Weierstrass during his stay in Berlin (see Chap. 2 of this book). Among Mittag-Leffler’s results we point out the Mittag-Leffler factorization theorem for meromorphic functions which he obtained by developing the ideas of Weierstrass. The final form of this theorem was published in 1884 (see [Mit84]). The evolution of the hypotheses of the Mittag-Leffler theorem is worth studying; there were several varying publications of the theorem between 1876 and 1884, the majority of which were by Mittag-Leffler himself, and there is a noticeable evolution of the ideas which is marked by changes in notation from the original Weierstrassian style and the simplification of proofs. In particular, the later versions of Mittag-Leffler’s theorem differ markedly from those of his initial papers. Specifically, in 1882 we see a rather abrupt appearance of Cantor’s recently developed theory of transfinite sets in statements and proofs of Mittag-Leffler’s theorem, despite the generally negative reception of Cantor’s work by prominent mathematicians of that period. Mittag-Leffler was interested in the solution of the analytic continuation problem as applied to the study of the convergence of divergent series (we discuss the corresponding results in detail in Chap. 2). For this purpose he introduced a new entire function (now called the Mittag-Leffler function) which serves as the simplest generalization of the exponential function and also helped him to get a criterion for analytic continuation generalizing Borel’s result. As for the general theory of entire functions, we point out several results which constitute special directions in mathematics (see, e.g., [Boa54, Rud74]). First of all, from the results of Borel–Picard appeared the modern theory of value distributions for entire functions (see, e.g., [Lev56]) and Nevanlinna’s theory of value distributions for meromorphic functions (see, [Nev25, Nev36]). B.5 Historical and Bibliographical Notes 291

Levin and Pfluger discovered that the asymptotics of the distribution of zeros of entire functions and the growth of these functions at infinity are closely related for a large class of entire functions, namely the functions of completely regular growth. The theory of entire functions of completely regular growth of one variable, developed in the late 1930s independently by Levin and Pfluger, soon found applications in both mathematics and physics. Later, the theory was extended to functions in the half-plane, subharmonic functions in space, and entire functions of several variables. The monograph [Ron92] describes this theory and presents recent developments based on the concept of weak convergence. This enables a unified approach and provides a comparatively simple presentation of the classical Levin– Pfluger theory (see [GoLeOs91, Lev56]). This theory has also been generalized to analytic functions in angle domains (see [Gov94]). Rolf Nevanlinna’s most important mathematical achievement is the value distribution theory of meromorphic functions. The roots of the theory go back to the result of Picard in 1879, showing that a non-constant complex-valued function which is analytic in the entire complex plane assumes all complex values except at most one. In the early 1920s Rolf Nevanlinna, partly in collaboration with his brother Frithiof, extended the theory to cover meromorphic functions, i.e. functions analytic in the plane except for isolated points in which the Laurent series of the function has a finite number of terms with a negative power of the variable. Nevanlinna’s value distribution theory, or Nevanlinna theory, is crystallized in its two Main Theorems. Qualitatively, the first one states that if a value is assumed less frequently than average, then the function comes close to that value more often than average. The Second Main Theorem, more difficult than the first one, says that there are relatively few values which the function assumes less often than average. One more classical topic related to the subject of this book is the theory of complex differential equations. Many of the considered special functions (the Mittag-Leffler function and its generalizations are among them) satisfy certain complex differential equations. Thus, for the study of the properties of some functions it is natural to construct the corresponding differential equation and apply the general results of the theory. From another perspective, the Mittag-Leffler function and its generalizations serve as solutions of a new type of equation (in particular integral and differential equations of fractional order). The theory of the latter can be compared with the theory of classical complex differential equations. Among the results on complex differential equations we mention their relations to Nevanlinna Theory (see, e.g., [Lai93]) and those results which are related to the Riemann–Hilbert problem (see, [AnoBol94, Bol90]). In this appendix we collect a number of notions and results on entire functions since for certain values of parameters the Mittag-Leffler function and its generalization are entire functions. Thus we use in the main text approaches describing the asymptotics of entire functions, distribution of zeros, series and integral representations and other analytic properties. 292 B The Basics of Entire Functions

B.6 Exercises

Series. B.6.1. Represent the following functions in the form of Taylor series [Vo l 7 0, p. 65]. (a) cosh z, (b) sinh z, (c) sin2 z,(d)cosh2 z. Answers:

X1 z2n X1 z2nC1 .a/ I .b/ I .2n/Š .2n C 1/Š nD0 nD0

X1 22n1z2n 1 X1 22n1z2n .c/ .1/nC1 I .d/ C : .2n/Š 2 .2n/Š nD0 nD1

B.6.2. Prove the following formulas [Evg69, p. 134] (a)

X1 2.2 22/ .2 4n2/ cos. arcsin z/ D 1 .1/nz2nC2I .2n C 2/Š nD0

(b)

X1 .2 12/ .2 .2n 1/2/ sin. arcsin z/ D z .1/nz2nC1I .2n C 1/Š nD1 (c)

X1 22n1..n 1/Š/2 .arcsin z/2 D z2nI .2n/Š nD1 (d)

p 1 ln.z C 1 C z2/ X 22n..n/Š/2 p D .1/n z2nC1: 2 .2n C 1/Š 1 C z nD0

Inﬁnite products. B.6.3. Construct an inﬁnite product (Weierstrass product) for a function having the following collection of zeros [Mark66, p. 253]

(a) zk D k of multiplicity k; k 2 NI (b) zk D k of multiplicity j k j;k2 Z: B.6 Exercises 293

B.6.4. Prove the following formulas [LavSha65, p. 433] (a) Â Ã Y1 z2 sin z D z 1 I n22 nD1

(b) Â Ã z Y1 4z2 expz 1 D zexp 1 C I 2 4n22 nD1 (c) Â Ã Y1 4z2 cos z D 1 : .2n 1/22 nD1

B.6.5. Prove the following formulas [Evg69, pp. 266–267] (a) Â Ã z Y1 z2 cos D 1 I 2 .2n C 1/2 nD1 (b) Â Ã Y1 z z sin.z a/ D sin aexpz cot a 1 exp I a C n a C n nD1

(c)

1 Ä h i 2 Y z C a z a cos z cos a D .z2 a2/ 1 . /2 1 . /2 I 2 2n 2n nD1

(d)

1 h i z C a Y z a expz expa D exp .z a/ 1 C . /2 I 2 2n nD1

(e) Â Ã Y1 z4 cosh z cos z D z2 1 C I 44n4 nD1 294 B The Basics of Entire Functions

(f) Â Ã z2 Y1 z2 sin z z cos z D 1 I tan D >0I 3 2 n n nD1 n

(g) Â Ã sin z Y1 z z2 D 1 C I z.1 z/ n C n2 nD1

(h)

1 Á 1 Y z z D zexpC z 1 C exp ; .z/ n n nD1

where C is Euler’s constant; B.6.6. Prove the following formulas [GunKuz58, pp. 200–202] (a) Â Ã Y1 z2 sinh z D z 1 C I n22 nD1

(b) Â Ã Y1 .a b/2z2 expaz expbz D .a b/zexp1=2.a C b/z 1 C I 4n22 nD1

(d) Â Ã Y1 .z 1/4 expz2Cexp.2z 1/ D 2exp1=2.z2 C 2z 1/ 1 C : 2.2n 1/2 nD1 B.6 Exercises 295

B.6.7. Calculate the values of the following inﬁnite products [Vo l 7 0, p. 115] Y1 Â Ã Á 1 Q1 2 .a/ 1 D 1=2; .b/ n 4 D 1=4; n2 nD3 n21 nD2 Â Ã Y1 3 Á n 1 Q1 .c/ D 2=3; .d/ 1 C 1 D 2; n3 C 1 nD1 n.nC2/ nD2 Y1 Â Ã Á 2 Q1 nC1 .e/ 1 D 1=3; .f/ 1 C . 1/ D 1; n.n C 1/ nD1 n nD2 Á 2 2 2 Q p q 1 2n 2n .g/p p p D ;.h/ D C D : 2 p 2 n 1 2n 1 2n 1 2 2 C 2 2 C 2 C 2 B.6.8. Find domains of convergence for the following inﬁnite products [Vo l 7 0 , pp. 116–117]

Â Ã Â Ã Y1 Y Y1 1 zn z2 .a/ .1 zn/; .b/ 1 C ;.c/ 1 ; nD1 2n n2 nD1 nD1 Â Ã Â Ã Y1 Y 1 1 1 2 .d/ 1 .1 /nzn ;.e/ 1 C .1 C /n zn ; n nD1 n nD1 1 Á Y z .f/ 1 C e.z=n/: n nD1

Answers:

.a/ j z j<1; .b/ j z j<1; .c/ j z j< 1; .d/ j z j>1; .e/ j z j<1=e;.f/ j z j< 1:

Characteristics of entire functions. B.6.9. Find the order and type of the following entire functions [Vo l 7 0, p. 122]

.a/ expazn;a>0;n2 NI .b/ znexp3zI .c/ z2exp2z exp3zI .d/ exp5z 3exp2z3I .e/ exp.2 i/z2I .f/ sin zI p .g/ cosh zI .h/ expz cos zI .i/ cos z: 296 B The Basics of Entire Functions

Answers:

.a/ D n; D a.b/ D 1; D 3.c/ D 1; D 3 p .d/ D 3; D 2.e/ D 2; D 5.f/ D 1; D 1 p .g/ D 1; D 1.h/ D 1; D 2.i/ D 1=2; D 1:

B.6.10. Calculate the order and type of the following entire functions, represented in the form of seriesP [Vo l 7 0, p. 124] P 1 z n 1 ln n n=a n (a) f.z/ D nD1 n ,(b)f.z/ D nD1 n z ;a >0, P P 1 1 n=a n 1 n2 n (c) f.z/ D nD2 n ln n z ;a >0,(d)f.z/ D nD0 e z , P P p 1 zn 1 cosh n n (e) f.z/ D D C ;a >0 (f) f.z/ D D z , n 1 nn1 a n 1 nŠ

1 X .1/nz2n .g/ zJ .z/ D ;.>1/ where J is the Bessel function. nŠ .n C C 1/ nD0

Answers:

.a/ D 1; D 1=e .b/ D a; D1.c/ D 0; D 0 .d/ D 0; .e/ D 0; .f/ D 1; D 1 .g/ D 1; D 2

B.6.11. Find the order and indicator function of the following entire functions [Vo l 7 0, p. 125] (a) expz,(b)exp.z/ C z2,(c)sinz z z .zn/ (d) cosp , (e) cosh , (f) exp , sin z p (g) z Answers:

cos ; =2 Ä <=2 .a/ D 1; h./ D cos .b/ D 1; h./ D 0; =2 Ä <3=2; .c/ D 1; h./ Dj sin j;.d/ D 1; h./ Dj sin j; .e/ D 1; h./ Dj cos j;.f/ D n; h./ D cos n; .g/ D 1=2; j sin =2 j :

Zeros of entire functions. B.6.12. Find all zeros and their multiplicities for the following entire functions [Vo l 7 0, p. 70] 2 3 .z22/2 sin z (a) .1 expz/.z 4/ ,(b)1 cos z,(c) z 3 sin3 z 3 (d) sin z,(e)z , (f) sin .z /, (g) cos3 z,(h)cosz3. B.6 Exercises 297

Answers: (a) z D˙2; 3rd orderI z D 2ki.k D 0; ˙1; / simple; (b) z D 2k.k D 0; ˙1; / 2nd order; (c) z D˙; 3rd orderI z D k.k D˙2;˙3; / simple; (d) z D k.k D 0; ˙1; / 3rd order; (e) z D 0;p 2nd order; zpD k.k D˙p 1; ˙2;/ 3rd order; (f) z D 3 k;z D 1=2 3 k.1 ˙ i 3/; .k D˙1; ˙2;/ simple; (g) z D .2k C 1/ ;.k D 0; ˙1; / 3rd order; p 2 p p z D 3 .2k C 1/=2; z D 1=2 3 .2k C 1/=2.1 ˙ i 3/; (h) . .k D 0; ˙1; ˙2;/ simple B.6.13. Find the solutions of the following equations [Evg69,p.74] 4i 5 3i (a) sin z D 3 ,(b)sinz D 3 , (c) cos z D 4 , 3Ci i 1 (d) cos z D 4 ,(e)sinhz D 2 , (f) cosh z D 2 . Answers: (a) z D i.1/kln 3 C k;k D 0; ˙1; , (b) z D˙iln 3 C =2 C 2k; k D 0; ˙1; ; (c) z D˙.iln 2 C =4/ C 2k; k D 0; ˙1; , (d) z D˙.i=2ln 2 C =4/ C 2k; k D 0; ˙1; ; k i (e) z D .1/ 6 C k;k D 0; ˙1; ; i (f) z D˙3 C 2k; k D 0; ˙1; : Appendix C Integral Transforms

In this appendix we give an outline of the properties of some integral transforms. The main focus is on the properties which are often useful in treating applied problems. It is not our intention to present a complete theory of these transforms. In applications, however, it is advantageous to have at our disposal an arsenal of formal manipulations that should be used with a critical mind. Among the thousands of books on the subject we refer to a few in which the theory is developed with different degrees of rigour.

C.1 Fourier Type Transforms

The most general deﬁnition of the Fourier transform is

ZC1 .Ff /./ D F./ D A eiBtf.t/dt; A;B 2 R;A6D 0; B 6D 0: 1

In this book we use the following definition which is commonly used in Probability Theory and Stochastic Modelling. Definition C.1. The Fourier transform of a function f W R ! R.C/ is denoted by Ff D F./, 2 R, and defined by the integral

ZC1 .Ff /./ D F./ D eitf.t/dt; (C.1.1) 1

© Springer-Verlag Berlin Heidelberg 2014 299 R. Gorenflo et al., Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 300 C Integral Transforms where F is called the Fourier transform operator or the Fourier transformation.1 The Fourier image Ff D F./is also denoted by f./O . From the theory of harmonic oscillations comes the following terminology: the pre-image (original) of the Fourier transform is usually called the signal or amplitude function depending on the time-variable t, while the Fourier image is called the spectral function depending on the frequency variable . The simplest class of functions for which the Fourier integral transform exists is the so-called class of rapidly decreasing functions, i.e. real- or complex-valued functions defined for all x 2 R and infinitely differentiable everywhere, such that each derivative tends to zero as jxj!C1faster that any positive power of x:

lim jxjN f .n/.x/ D 0 (C.1.2) jxj!C1 for each positive integer N and n. A more general sufﬁcient condition for a function f to have a Fourier transform is that f is absolutely integrable on R (see, e.g., [Doe74, p. 154]), i.e. belongs to 2 L1.R/ 8 9 < ZC1 = R R R C L1. / WD :f W ! . / W jf.x/jdx

Theorem C.1. Let the function f be absolutely integrable, i.e.

ZC1 jf.t/jdt<1: (C.1.3) 1

Then at every point t0,wheref is of bounded variation in some (arbitrarily small) neighbourhood of t0, the following inversion formula for the Fourier transform holds:

1Among other deﬁnitions of the Fourier transform we mention the symmetric form of the Fourier C1R transform .Ff /./ D p1 eitf.t/dt used in Functional Analysis, and .F'/.f / D 2 1 C1R ei.2f/t'.t/dt which is commonly used in Signal Processing. In the last deﬁnition the 1 variable t is time and f is the frequency of a signal. The function Ff is called the spectrum of the signal f.t/. 2 More precisely the space L1.R/ consists of equivalence classes with respect to the equivalence C1R relation: f g ” jf.x/ g.x/jdx D 0. 1 C.1 Fourier Type Transforms 301

ZC1 f.t C 0/ C f.t 0/ 1 0 0 D eitf./d;O (C.1.4) 2 2 1 where the integral is understood in the sense of the Cauchy principal value. If t0 is a point of continuity for f , then the value of the right-hand side of (C.1.4) coincides with the value f.t0/.

Moreover, if f 2 L1.R/ then its Fourier transform Ff D F./is a (uniformly) continuous and bounded function in 2 R, and lim F./ D 0. jj!C1 Theorem C.1 determines the inverse Fourier operator

ZC1 1 F 1F .t/ D eitf./d:O 2 1

In particular, if a function is locally integrable and has a compact support (i.e. vanishes outside some interval), then its Fourier transform exists. In this case, the Fourier transform F./ possesses an analytic continuation into the whole complex plane 2 C. In order to understand the meaning of the Fourier transform let us recall the Dirichlet condition. A real- or complex-valued function defined on the whole real line R is said to satisfy the Dirichlet condition on R if: (a) f.t/has in R no more than a finite number of finite discontinuity points (jump points) and has no infinite discontinuity points; (b) f.t/has in R no more than a finite number of maximum and minimum points. If f.t/satisfies the Dirichlet condition in R and is absolutely integrable, then the following Fourier integral formula holds:

ZC1 ZC1 f.t C 0/ C f.t 0/ 1 D eitd f./eid (C.1.5) 2 2 1 1 at any ﬁnite discontinuity point t 2 .1; C1/. This result is also known as the Fourier integral theorem. In particular, if f is continuous at t, then the Fourier integral formula can be written as

ZC1 ZC1 1 f.t/ D eitd f./eid: (C.1.6) 2 1 1 302 C Integral Transforms

Let us recall some basic properties of the Fourier transform (for more detailed information we refer to the treatise by Titchmarsh (see [Tit86])). Á Ff ./ D f.O /: (C.1.7)

Z1 If f.t/ D f.t/; then .Ff/./D 2 f.t/cos t dt: (C.1.8)

0 Z1 If f.t/ Df.t/; then .Ff/./D2i f.t/sin t dt: (C.1.9)

0 Ff.a1t C b/ ./ D aeiabf.a/;a>0;O (C.1.10) Â Ã 1 b Feibtf.at/ ./ D fO ;a>0; (C.1.11) a a d nf./O .Ft nf.t//./D i n ;n2 N; (C.1.12) dn Ff .n/.t/ ./ D i nnf./;nO 2 N; (C.1.13)

.F.f g/.t// ./ D f./O Og./; (C.1.14) provided that all Fourier images on the left- and right-hand sides exist, where .f g/ .t/ is the Fourier convolution,i.e.

ZC1 .f g/.t/ D f.t /g./d: (C.1.15) 1

Sufﬁcient conditions for the fulﬁllment of equality (C.1.14) read that this equality holds if both functions f and g are integrable and square integrable on the real line: \ f; g 2 L1.R/ L2.R/: (C.1.16)

These conditions coincide with the conditions which guarantee the fulﬁllment of the Cauchy–Schwarz inequality:

ZC1 ZC1 ZC1 jf.t/ g.t/jdt Ä jf.t/j2dt jg.t/j2dt: (C.1.17) 1 1 1

Analogous conditions give us the so-called Parseval identity for the Fourier transform. C.1 Fourier Type Transforms 303

Theorem C.2. Let f be integrable and square integrable on the real line, i.e. \ f 2 L1.R/ L2.R/:

Then for all real x the following formula holds:

ZC1 ZC1 1 f.t/f.t x/dt D eixf./O f./O d: (C.1.18) 2 1 1

Moreover for x D 0 this yields the Parseval formula

ZC1 ZC1 1 jf.t/j2dt D jf./O j2d: (C.1.19) 2 1 1

The Mittag-Lefﬂer function is one of the most important special functions related to the Fractional Calculus. Thus we recall two composition formulas for the Fourier transform and fractional integrals/derivatives.

'./O FI ˛ ' ./ D ;0

˛ where I˙ are fractional integrals of the Liouville type:

Zx ZC1 1 '.t/dt 1 '.t/dt I ˛ ' .x/ D ; I ˛' .x/ D : C .˛/ .x t/1˛ .˛/ .t x/1˛ 1 x

Formulas (C.1.20) are valid for any function ' 2 L1.1; C1/. Corresponding formulas for fractional derivatives of the Liouville type

Zx 1 dn '.t/dt D˛ ' .x/ D ; C .n ˛/ dxn .x t/˛nC1 1 ZC1 .1/n dn '.t/dt D˛ ' .x/ D ;nD ŒRe ˛ C 1; .n ˛/ dxn .t x/1˛ x have the form FD˛ ˛ ˙' ./ D .i/ './;O Re ˛>0: (C.1.21) 304 C Integral Transforms

Formulas (C.1.21) are valid, in particular, for all functions having derivatives up to the order n D ŒRe ˛ C 1, and rapidly decreasing at inﬁnity together with all derivatives (see, e.g. [SaKiMa93]). In (C.1.20)and(C.1.21) the values of the function .i/˛ are calculated according to the formula

˛ ˛ log jxj ˛i sgn x .ix/ D e 2 :

We mention two formulas for cosine- and sine-integral transforms of the Riemann–Liouville fractional integral (see, e.g., [SaKiMa93, p. 140]) h i ˛ ˛ F I ˛ ' ./ D ˛ cos .F '/./ sin .F '/./ ;>0; (C.1.22) c 0C 2 c 2 s h i ˛ ˛ F I ˛ ' ./ D ˛ sin .F '/./ C cos .F '/./ ;>0: (C.1.23) s 0C 2 c 2 s The above results are considered in the classical case and in the distributional sense (see, e.g., [Bre65]). We use them in the text only in the ﬁrst sense.

C.2 The Laplace Transform

The classical Laplace transform is deﬁned by the following integral formula

Z1 .Lf/.s/D estf.t/dt; (C.2.1)

0 provided that the function f (the Laplace original) is absolutely integrable on the semi-axis .0; C1/. In this case the image of the Laplace transform (also called the Laplace image), i.e. the function

F.s/ D .Lf/.s/ (C.2.2)

(sometimes denoted F.s/ D f.s/Q ) is deﬁned and analytic in the half-plane Re s>0. It may happen that the Laplace image can be analytically continued to the left of the imaginary axes Re s D 0 in a bigger domain, i.e. there exist a non-positive real number s (called the Laplace abscissa of convergence) such that F.s/ D f.s/Q is analytic in the half-plane Re s s . Then the following inverse Laplace transform can be introduced Z 1 L1F .t/ D estF.s/ds; (C.2.3) 2i Lic C.2 The Laplace Transform 305 where Lic D .c i1;cC i1/, c> s , and the integral is usually understood in the sense of the Cauchy principal value, i.e.

Z cZCiT estF.s/ds D lim estF.s/ds: (C.2.4) T !C1 Lic ciT

If the integral (C.2.2) converges absolutely on the line Re s D c,thenatany continuity point t0 of the original f the integral (C.2.3) gives the value of f at this point, i.e. Z 1 est0 f.s/Q ds D f.t /: (C.2.5) 2i 0 Lic

Thus under these conditions operators L and L1 constitute an inverse pair of operators. Correspondingly, the functions f and F D fQ constitute a Laplace transform pair. To describe this fact the following notation is used Z 1 st f.t/ f.s/Q D e f.t/dt; Re s> c ; (C.2.6) 0 where c is the abscissa of the convergence. Here the sign denotes the juxtaposition of a function (depending on t 2 RC) with its Laplace transform (depending on s 2 C). In the following the conjugate variables ft;sg may be different, e.g., fr; sg and the abscissa of the convergence may sometimes be omitted. Furthermore, throughout our analysis, we assume that the Laplace transforms obtained by our formal manipulations are invertible by using the standard Bromwich formula. Among the rules for the Laplace transform pairs we recall the following one, which turns out to be useful for our purposes, Z 1 1=2 1 2 f.sQ / p r =.4t/ f.r/ r : e d 1=2 (C.2.7) t 0 s

Since an examination of convergence conditions is not always possible (see, e.g., [Wid46]) sometimes the terminology “Laplace transform pair” is used for pairs f; fQ not necessarily satisfying the equality (C.2.5) at certain points. There are several properties of the Laplace transform which make it very useful in the study of a wide class of differential and integral equations. Let us recall a few main properties of the Laplace transform (in the form of Laplace transform pairs) (more information can be found, e.g., [BatErd54a, Wid46, Doe74]).

eatf.t/ f.sQ C a/; a > 0I (C.2.8) 306 C Integral Transforms

dnfQ t nf.t/ .1/n .s/; n 2 NI (C.2.9) dt n Z1 Z1 Z1 n t f.t/ dsn dsn1 ::: f.sQ 1/ds1;n2 NI (C.2.10)

s sn s2 f .n/.t/ snf.s/Q sn1f.0/ sn2f 0.0/ ::: f .n1/.0/; n 2 NI (C.2.11)

Zt Ztn Zt2 n dtn dtn1 ::: f.t1/dt1 s f.s/;nQ 2 NI (C.2.12) 0 0 0 Â Ã Â Ã d n d n t f.t/ s f.s/;nQ 2 NI (C.2.13) dt ds Â Ã Â Ã d n d n t f.t/ s f.s/;nQ 2 NI (C.2.14) dt ds

.f1 f2/.t/ fQ1.s/ fQ2.s/; (C.2.15) where

.f1 f2/.t/D f1./f2.t /d (C.2.16) 0 is the so-called Laplace convolution. Among simple sufﬁcient existence conditions for the Laplace transform we point out the following: The Laplace transform exists in the half-plane Re s>a(a>0) provided that the original f is locally integrable on RC D .0; C1/ and has an exponential growth of order a at inﬁnity, i.e. there exists a positive constant K>0and positive t0 >0 such that

at jf.t/j

From the formula (C.2.15) follows immediately the Laplace transform of the Riemann–Liouville fractional integral L ˛ ˛ L I0C' .s/ D s . '/ .s/; Re ˛>0: (C.2.17)

The Laplace transform of the Riemann–Liouville fractional derivative is given by the formula (see, e.g., [OldSpa74, p. 134]) ˇ Xn ˇ L ˛ ˛ L ˛k ˛ ˇ D0C' .s/ D s . '/.s/ s D0C' .t/ ;n 1

Xn1 L C ˛ ˛ L ˛k1 D0C' .s/ D s . '/.s/ s '.0C/; n 1

˛ Formula (C.2.19) is simpliﬁed in the case of the Marchaud fractional derivative DC ˛ and Grünwald–Letnikov fractional derivative GLD0C: L ˛ ˛ L L ˛ ˛ L DC' .s/ D s . '/ .s/; GLD0C' .s/ D s . '/ .s/: (C.2.20)

C.3 The Mellin Transform

Let Z C1 s1 M ff.r/I sgDf .s/ D f.r/r dr; 1 < Re .s/ < 2 (C.3.1) 0 be the Mellin transform of a sufﬁciently well-behaved function f.r/,andlet Z 1 Ci1 M1 ff .s/I rgDf.r/D f .s/ rs ds (C.3.2) 2i i1 be the inverse Mellin transform, where r>0, D Re .s/, 1 <<2. For the existence of the Mellin transform and the validity of the inversion formula we need to recall the following theorems, adapted from Marichev’s treatise [Mari83], see THEOREMS 11, 12, on page 39. Theorem C.3. Let f.r/ 2 Lc. ; E/ ; 0 < < E < 1, be continuous in the intervals .0; ; ŒE; 1/, and let jf.r/jÄMr1 for 0E,whereM is a constant. Then for the existence of a strip in the s-plane s1 c in which f.r/r belongs to L .0; 1/ it is sufﬁcient that 1 <2 : When this condition holds, the Mellin transform f .s/ exists and is analytic in the vertical strip 1 <D Re.s/ < 2 : Theorem C.4. If f.t/is piecewise differentiable, and f.r/r1 2 Lc .0; 1/,then the formula (C.3.2) holds true at all points where f.r/ is continuous and the (complex) integral in it must be understood in the sense of the Cauchy principal value. We refer to specialized treatises and/or handbooks, see, e.g. [BatErd54a,Mari83, PrBrMa-V3], for more details and tables on the Mellin transform. Here, for our convenience we recall the main rules that are useful when adapting the formulae from the handbooks and which are also relevant in the following. 308 C Integral Transforms

M Denoting by $ the juxtaposition of a function f.r/ with its Mellin transform f .s/, the main rules are:

M f.ar/ $ as f .s/ ; a > 0 ; (C.3.3)

M ra f.r/ $ f .s C a/ ; (C.3.4)

M 1 f.rp / $ f .s=p/ ; p ¤ 0; (C.3.5) jpj Z1 1 M h.r/ D f . / g.r= / d $ h.s/ D f .s/ g.s/ : (C.3.6)

The Mellin convolution formula (C.3.6) is useful in treating integrals of Fourier type for x Djxj >0: Z 1 1 I .x/ D f./ cos . x/ d; (C.3.7) c Z0 1 1 Is.x/ D f./ sin . x/ d; (C.3.8) 0 when the Mellin transform f .s/ of f./ is known. In fact we recognize that the integrals Ic.x/ and Is.x/ can be interpreted as Mellin convolutions (C.3.6) between f./and the functions gc ./ ; gs./, respectively, with r D 1=jxj ; D ;where Â Ã Á 1 1 M .1 s/ s g ./ WD cos $ sin WD g.s/; 0 < Re.s/ < 1 ; c jxj jxj 2 c (C.3.9) Â Ã Á 1 1 M .1 s/ s g ./ WD sin $ cos WD g.s/; 0 < Re.s/ < 2 : s jxj jxj 2 s (C.3.10)

The Mellin transform pairs (C.3.9)and(C.3.10) have been adapted from the tables in [Mari83]byusing(C.3.3)–(C.3.5) and the duplication and reﬂection formulae for the Gamma function. Finally, the inverse Mellin transform representation (C.3.2) provides the required integrals as Z Ci1 Á 1 1 s s Ic.x/ D f .s/ .1 s/ sin x ds; x >0; 0< <1; x 2i i1 2 (C.3.11) C.3 The Mellin Transform 309 Z Ci1 Á 1 1 s s Is.x/ D f .s/ .1 s/ cos x ds; x >0; 0< <2: x 2i i1 2 (C.3.12)

First, we present some known properties of the Mellin integral transform.

k 1 Theorem C.5. Let x 2 f.x/ 2 L2.0; C1/. Then the following four statements hold: 10. The functions Z a M.sI a/ D f.x/xs1dx; Re s D k; (C.3.13) 1=a converge in the mean when a !C1on the line .k i1;kC i1/, that is, there exists a function M.s/ 2 L2.k i1;kC i1/ such that Z kCi1 lim jM.s/ M.sI a/j2 jdsjD0: (C.3.14) a!C1 ki1

20. The functions Z 1 kCia f.x;a/ D M.s/xs ds; 0 < x < C1 (C.3.15) 2i kia converge in the mean on the semi-axis with the weight function x2k1 to the function f.x/when a !C1, that is, Z C1 lim jf.x/ f.x;a/j2x2k1 dx D 0; (C.3.16) a!C1 0 moreover, almost everywhere on the semi-axis .0; C1/ we have Z 1 d kCi1 x1s f.x/ D M.s/ ds: (C.3.17) 2i dx ki1 1 s

30. The Parseval identity Z Z C1 1 C1 jf.x/j2x2k1dx D jM.k C it/j2 dt (C.3.18) 0 2 1 is valid. 0 4 . Conversely, for any function M.s/ 2 L2.ki1;kCi1/ the functions (C.3.15) converge in the mean when a !C1in the sense of (C.3.16) to some function f.x/ 2 L.0; C1/ which can be represented in the form (C.3.17). The functions 310 C Integral Transforms

(C.3.13) converge in the mean in the sense of (C.3.14) to the function M.s/ when a !C1and, moreover, the Parseval identity (C.3.18) is valid. We present a number of important formulas for the Mellin transform.

.M x˛f.x//.p/D f .p C ˛/; (C.3.19) Â Ã .1 x/˛1H.1 x/ .p/ M .p/ D ; (C.3.20) .˛/ .pC ˛/ where H.x/ is the Heaviside function

1; 0 Ä x

Z1 1 W ˛f.x/ D .t x/˛1f.t/dt; 0 < Re ˛ < 1; x > 0: (C.3.22) .˛/ x

The Mellin transform of the Riemann–Liouville fractional integral is given by the formula

.1 ˛ p/ M I ˛ f.x/ .p/ D f .p C ˛/; Re .˛ C p/ < 1: (C.3.23) 0C .1 p/

The Mellin transform of the Riemann–Liouville fractional derivative is represented in the form

.1C ˛ p/ M D˛ f.x/ .p/ D f .p ˛/ 0C .1 p/

Xn1 .1C k p/ xD1 C D˛k1f .x/xpk1 ; (C.3.24) .1 p/ 0C xD0 kD0 and the formula for the Mellin transform of the Caputo derivative has the form

.1C ˛ p/ M CD˛ f.x/ .p/ D f .p ˛/: (C.3.25) 0C .1 p/ C.4 Simple Examples and Tables of Transforms of Basic Elementary and. . . 311

C.4 Simple Examples and Tables of Transforms of Basic Elementary and Special Functions

Table of selected values of the Fourier integral transform.

.Ff /./ D F./ D f./O C1R n/n f(t) D eitf.t/dt Conditions 1 2a 1. exp.ajtj/ .a2 C2/ a>0 4ai 2. texp.ajtj/ 2 C 2 2 a>0 p.a / Á 2 2 3. exp.at / a exp 4a a>0 1 4. t2Ca2 a exp.ajj/ a>0 t i 5. (t2Ca2 2a exp.ajj/ a>0 c; a Ä t Ä b; ˚ « 6. ic eib eia a 0 0; otherwise sin at 7. t 2H .a jj/ a>0 C C ˛ cos . 2 .˛ 1// ˛ 2 ;˛ 6D 0; 8. jtj 2 .˛ C 1/ jj1C˛ ˛ 6D1 2k; k 2 N0 cos . ˛ / ˛ 2 N 9. jtj sgn t 2i .˛ C 1/ jj1C˛ sgn ˛ 6D2k; k 2 i 10. expft.a i!/gH.t/ !CCia a>0 j j H.ap t / 11. 2J0.a/ a>0 a2t2 aCi 12. exp.at/H.t/ a2Ch2 i a>0 1 .i/n 1 N 13. tn i .n1/Š sign n 2 2ı./" # ˇ ˇ 2 .2; 2/; .1; 1/ ˇ 1 14. E˛.jtj/ 2 21 2 ˛>1 .˛ C 1;" 2˛/ # ˇ .2; 2/; .1; 1/ ˇ 15. E .jtj/ 2ı./ 2 ˇ 1 ˛>1;ˇ2 C ˛;ˇ .ˇ/ 2 2 1 .˛ C ˇ; 2˛/ 2

Table of selected values of the Laplace integral transform.

C1R n/n f.t/ .Lf /.s/ D F.s/ D f.s/Q D estf.t/dt Conditions 0 a .aC1/ 1. t saC1 a>1 at sa 2. e cos bt .sa/2 Cb2 ˛1 1 ˛ s Re ˛ E˛.at / s˛ a Re ˛>0;s>jaj at b 3. e sin bt .sa/2 Cb2 s2b2 4. t cos bt .s2Cb2/2 (continued) 312 C Integral Transforms

.Lf /.s/ D F.s/ C1R n/n f.t/ D f.s/Q D estf.t/dt Conditions 0 5. t sin bt 2bs p.s2Cb2/2 p 6. p1 exp a exp 2 as t t s C C Z ˛1 .˛/ 1 a; b 2 ;c 2 n 0 ; 7. t 2F1.a; bI cI t/ ˛ 3F1.˛;a;bI cI / s s Re ˛>0I Re s>0 C C Z ˛1 .˛/ 1 a 2 ;c 2 n 0 ; 8. t ˚.aI cI t/ ˛ 2F1.˛; aI bI / s s Re ˛>0I Re s>0 .˛/.˛c1/ C C Z .acC˛C1/ a 2 ;c 2 n 0 ; ˛1 9. t .aI cI t/ 2F1.˛; ˛ c C 1I Re ˛>0I Re c<1C Re ˛I a c C ˛ C 1I 1 1 / j1 sj <1 p s 10. exp a2t erf a t p a p s.sa2/ 11. exp a2t erfc a t p pa s. sCa/ ˛1 1 ˛ s Re ˛ 12. E˛.at / s˛ a Re ˛>0;s>jaj ˛1 1 ˛ s Re ˛ 13. E˛.at / s˛ a Re ˛>0;s>jaj

ˇ1 ˛ s˛ ˇ Re ˛>0;Re ˇ>0; 14. t E˛;ˇ.at / s˛ a 1 s>jaj Re ˛ s

m˛Cˇ1 .m/ mŠs˛ ˇ Re ˛>0;Re ˇ>0; 15. t E˛;ˇ .at/ .s˛ a/mC1 1 m 2 N;s >jaj Re ˛ " # ˇ Re ˛>0;Re ˇ>0; ˇ .1; 1/; . ; / 1 1 a ˇ 16. t E˛;ˇ.at / s 21 s˛ Re >0;Re >0; .ˇ; ˛/ 1 s>jaj Re ˛ Re ˛>0;Re ˇ>0; ˇ1 ˛ ˇ ˛ 17. t E˛;ˇ.at / s .1 as / Re >0; 1 s>jaj Re ˛ " # ˇ Re ˛>0;Re ˇ>0; ˇ .ı; 1/; .1; 1/ ı 1 1 ˇ 18. Eˇ; .t/ s 21 s Re >0;Re ı>0; .ˇ; / 1 " # Re >0;s>jaj Re ˛ ˇ ˇ 1 1 ˇ . ; 1/; .1; 1/ Re ˛j >0;Re ˇj >0; 19. E ..˛j ;ˇj /1;mI t/ s 2m s 1 .˛j ;ˇj /1;m Re >0;s>jaj Re ˛ C.5 Historical and Bibliographical Notes 313

Table of selected values of the Mellin integral transform.

.Mf /.p/ D F.p/ D C1R n/n f.t/ f .p/ D f.t/tp1dt Conditions 0 t .p/ 1. e p Re >0;Re p>0 t2 .p=2/ 2. e 2p=2 Re >0;Re p>0 .p/. p/ 3. .t C 1/ . / 0

.p/.ap/ .bp/ .c/ a; b; c 2 C;c 62 Z0 8. 2F1.a; bI cIt/ .cp/ .a/ .b/ 0

C.5 Historical and Bibliographical Notes

Here we present a few historical remarks on the early development of the integral transform method. For an extended historical exposition we refer to [DebBha07, Ch. 1]. The Fourier integral theorem was originally stated in J. Fourier’s famous treatise entitled La Théorie Analytique da la Chaleur (1822), and its deep significance was recognized by mathematicians and mathematical physicists. This theorem is one of the most monumental results of modern mathematical analysis and has widespread physical and engineering applications. Fourier’s treatise provided the modern mathematical theory of heat conduction, Fourier series, and Fourier integrals with applications. He gave a series of examples before stating that an arbitrary function defined on a finite interval can be expanded in terms of a trigonometric series which is now universally known as the Fourier series. In an attempt to extend his new ideas 314 C Integral Transforms to functions defined on an infinite interval, Fourier discovered an integral transform and its inversion formula which are now well-known as the Fourier transform and the inverse Fourier transform. However, this celebrated idea of Fourier was known to P.S. Laplace and A.L. Cauchy as some of their earlier work involved this transformation. On the other hand, S.D. Poisson also independently used the method of transform in his research on the propagation of water waves. It was the work of Cauchy that contained the exponential form of the Fourier Integral Theorem. Cauchy also introduced the functions of the operator D:

ZC1 C1Z 1 .D/f.x/ D .i/ei.xy/f.y/dyd; 2 1 1 which led to the modern form of operator theory. The birth of the operational method in its popularization as a powerful method for solving differential equations is probably due to O. Heaviside (see, e.g., [Hea93]). He developed this technique as a purely algebraic one, using and developing the classical ideas of Fourier, Laplace and Cauchy. An extended use of the machinery of complex analytic functions in this theory was proposed by T.J. Bromwich (see, e.g., [Bro09, Bro26]). In this direction several types of integral transform appeared. In particular, elaborating the ideas of B. Riemann, Mellin introduced a new type of integral transform which was later given his name. It turned out that (see [Sla66, Mari83]) the Mellin transform is highly suitable for the study of properties of a wide class of special functions, namely G-andH-functions [MaSaHa10]. In [Mari83], a calculation method for integrals with a ratio of products of Gamma functions was developed. This method is based on the properties of the Mellin transform. Many results were obtained by different authors due to a combination of the complex analytic approach and results from the theory of special functions which rapidly developed in the ﬁrst part of the twentieth century. These results led to the theory of operational calculus in its modern form. We mention here several treaties on integral transform theory [Dav02,DebBha07,DitPru65,Doe74,Mik59,Sne74,Tit86,Wid46]. Applications of the integral transform method are presented in many books on integral and differential equations (see, e.g., [AnKoVa93, Boa83, PolMan08]), in particular those related to the fractional calculus [Die10, KiSrTr06, SaKiMa93]. Tables of integral transforms (see [BatErd54a,GraRyz00,Mari83,Obe74]) constitute a very useful source for applications.

C.6 Exercises

C.6.1. Evaluate the Fourier transform of the functions ([DebBha07, p. 119]) n o n o at2 2 t 2 .a/f .t/ D texp 2 ; a > 0 .b/f .t/ D t exp 2 : C.6 Exercises 315

C.6.2. Solve the following integral equations with respect to the unknown function y.t/ ([DebBha07, p. 121])

ZC1 .a/ .x t/y.t/dt D g.x/; 1 ZC1 .b/ exp at2 y.x t/dt D exp ax2 ;a>b>0: 1

Hint. Use the Fourier transform. C.6.3. Use the Fourier transform to solve the boundary value problem ([DebBha07, p. 128]) 2 uxx C uyy D xexp x ; 1

in the class of continuously differentiable functions such that

lim u.x; y/ D 0; 8x; 1

Answer.

ZC1 Â Ã p sin tx t 2 u.x; y/ D 2 Œ1 exp .ty/ exp dt: t 4 0

C.6.4. Find the Laplace transform of the functions ([DebBha07, p. 173]):

.a/2t C a sin at;a>0; .b/.1 2t/exp f2tg ; .c/H.t a/exp ft ag ;a>0;.d/.t a/kH.t a/;a>0;k2 N:

C.6.5. Evaluate the inverse Laplace transform of the functions ([DebBha07, p. 174])

1 1 .a/ .sa/.sb/2 ;a;b>0I .b/ s2.sa/2 ;a>0I 1 s .c/ s2.s2Ca2/ ;a>0I .d/ .s2Ca2/.s2Cb2/ ;a;b>0: 316 C Integral Transforms

C.6.6. Using the change of variables, s D c C i!, show that the inverse Laplace transformation is a Fourier transformation, that is ([DebBha07, p. 179]),

Á ZC1 ect f.t/ D L1f.s/Q .t/ D f.cQ C i!/ei!td!: 2 1

C.6.7. Calculate the Mellin transform of the functions ([DebBha07, p. 365])

.a/f.t/D H.a t/; a > 0I .b/f.t/ D t ˛eˇt ;˛;ˇ>0I

1 2 .c/f.t/D 1Ct 2 I .d/f.t/ D J0 .t/:

C.6.8. Prove that ([DebBha07, p. 365]) Â Ã 1 .p/.n p/ M .p/ D ;a>0I .1 C at/n ap.n/ Á p 1 a np 1 M n 2 . t Jn.at// .p/ D p ;a>0;n> : 2 2 n 2 C 1 2

C.6.9. Prove the following relations of the Mellin transform to the Laplace and the Fourier transforms ([DebBha07, p. 370]): .Mf.t//.p/D Lf.et / .p/I .Mf.t//.aC i!/ D Ff.et /eat .!/:

C.6.10. Calculate the Laplace transform of the Bessel function of the ﬁrst kind J.t/ for Re >1, where ([KiSrTr06, p. 36, (1.7.34)])

X1 .1/k.z=2/2kC J .z/ D .z 2 C n .1;0I 2 C/: kŠ . C k C 1/ kD0

Answer.

L h 1 i . J.t// .s/ D p p .Re s>0/: s2 C 1 s C s2 C 1

C.6.11. Calculate the Laplace transform of the Bessel function of the second kind Y.t/ for jRe j <1;6D 0, where ([KiSrTr06, p. 36, (1.7.35)])

cos ./J .z/ J .z/ Y .z/ D . 2 C n Z/: sin C.6 Exercises 317

Answer. h i p 2 cot ./ csc ./ s C s2 C 1 L h i . Y.t// .s/ D p p .Re s>0/: s2 C 1 s C s2 C 1

C.6.12. Prove the following relation (˛; ˇ; 2 C; Re ˛>0)([KiSrTr06, p. 50, (1.10.10)]) Â Ä Â Ã Ã ˇ ˇ n ˛ˇ ˇ ˇ ˛nCˇ1 @ ˛ nŠs ˇ ˇ L t E˛;ˇ.t / .s/ D .ˇ ˇ <1/: @ .s˛ /nC1 s˛

z ˛1 ˛ C C C.6.13. Let e˛ D z E˛;˛.z /.z 2 I Re ˛>0I 2 / be the so-called ˛- Exponential function. Prove the following relation for it ([KiSrTr06, p. 52, (1.10.27)]): Â ÄÂ Ã Ã ˇ ˇ n ˇ ˇ @ t nŠ ˇ ˇ L e .s/ D .n 2 NI Re s>0I ˇ ˇ <1/: @ ˛ .s˛ /nC1 s˛

C.6.14. Calculate the Laplace transform of the Meyer G-function ([MaSaHa10, p. 52, (2.29)]) Ä ˇ ˇ 1 m;n ˇ a1;:::;ap t Gp;q at b1;:::;bq

2 C; >0;Re C min Re bj >0; 1Äj Äm c p C q jarg aj < ;c D m C n >0I Re s>0: 2 2 C.6.15. Calculate for all s 2 C,Res>0, the Laplace transform of the Fox H- function ([MaSaHa10, p. 50, (2.19)]) Ä ˇ ˇ 1 m;n ˇ .a1;˛1/;:::;.ap;˛p / t Hp;q at .b1;ˇ1/;:::;.bq;ˇq / ; a 2 C; >0; ˛ ˛>0;jarg aj < or ˛ D 0 and Re ı<1 2 Re b Re C min j >0; for ˛>0or ˛ D 0; 0; 1Äj Äm ˇj 318 C Integral Transforms

and Ä Re b Re ı C 1=2 Re C min j C >0; for ˛ D 0; < 0; 1Äj Äm ˇj

where

Xn Xp Xm Xq ˛ D ˛j ˛j C ˇj ˇj I j D1 j DnC1 j D1 j DmC1 q p q p X X p q X X ı D b a C I D ˇ ˛ : j j 2 j j j D1 j D1 j D1 j D1

C.6.16. Calculate the Laplace and Mellin transforms of the classical Wright function .˛;ˇI t/, where ([KiSrTr06, p. 55, (1.11.6–7)])

X1 zk .˛;ˇI z/ D : kŠ .˛k C ˇ/ kD0

Answers. Â Ã 1 1 .L .˛;ˇI t//.s/ D E .˛ > 1I ˇ 2 CI Re s>0/I s ˛;ˇ s .s/ .M .˛;ˇI t//.s/ D .˛ > 1I ˇ 2 CI Re s>0/: .ˇ ˛s/

C.6.17. Find the Mellin transform of the Gauss hypergeometric function 2F1.a; bI cIt/ (a; b; c 2 C)([MaSaHa10, p. 46, Ex. 2.1]). Answer.

.s/.a s/ .b s/ .c/ .M F .a; bI cIt//.s/ D 2 1 .c s/.a/.b/ .min fRe a; Re bg > Re s>0/:

C.6.18. Calculate the Mellin transform of the Meyer G-function ([MaSaHa10, p. 48, (2.9)]) Ä ˇ ˇ m;n ˇ a1;:::;ap Gp;q at b1;:::;bq c p C q a 2 C; jarg aj ;c D m C n >0I 2 2

s 2 C; min Re bj < Re s<1 max Re aj : 1Äj Äm 1Äj Än Appendix D The Mellin–Barnes Integral

D.1 Deﬁnition: Contour of Integration

In the modern theory of special functions it has become customary to call to an integral of the type (see, e.g., [ParKam01]) Z 1 I.z/ D f.s/zs ds; (D.1.1) 2i L a Mellin–Barnes integral. Here the density function f.s/ is usually a solution to a certain ordinary differential equation with polynomial coefficients. Thus this integral is similar to the Mellin transform applied to special types of originals (see, e.g., [Mari83]). The most crucial part of this definition is the choice of the contour of integration. The contour L is usually either a loop in the complex s plane, a vertical line indented to avoid certain poles of the integrand, or a curve midway between these two, in the sense of avoiding certain poles of the integrand and tending to infinity in certain fixed directions (see, e.g., [ParKam01]). A short introduction to the theory of Mellin–Barnes integrals is given in [Bat-1, pp. 49–50]. In order to be more precise we consider a density function f.s/of the type

A.s/B.s/ f.s/ D ; (D.1.2) C.s/D.s/ where A; B; C; D are products of Gamma functions depending on parameters. Such integrals appear, in particular, in the representation of the solution to the hypergeometric equation

d2u du z.1 z/ C Œc .b C a 1/z abu D 0; (D.1.3) dz2 dz

© Springer-Verlag Berlin Heidelberg 2014 319 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 320 D The Mellin–Barnes Integral i.e. in the representation of the Gauss hypergeometric function

X1 .a/ .b/ F.a;bI cI z/ D n n zn; .c/ nD0 n and in the representation of the solution to a more general equation, namely, the generalized hypergeometric equation 2 3 p Â Ã q Â Ã Y d d Y d 4z z C a z z C b 1 5 u.z/ D 0; (D.1.4) dz j dz dz k j D0 kD0 i.e. in the representation of the generalized hypergeometric function

X1 .a / :::.a / F .a ;:::;a I b ;:::;b I z/ D 1 n p n zn: p q 1 p 1 q .b / :::.b / nD0 1 n q n

In [Mei36] more general classes of transcendental functions were introduced via a generalization of the Gauss hypergeometric function presented in the form of a series (commonly known now as Meijer G-functions). Later this deﬁnition was replaced by the Mellin–Barnes representation of the G-function Ä ˇ Z ˇ 1 m;n ˇ a1;:::;ap Gm;n s Gp;q z ˇ D p;q .s/z ds; (D.1.5) b1;:::;bq 2i L where L is a suitably chosen path, z 6D 0, zs WD expŒs.ln jzjCiarg z/ with a single valued branch of arg z, and the integrand is deﬁned as Q Q m .b s/ n .1 a C s/ Gm;n Q kD1 k j DQ1 j p;q .s/ D q p : (D.1.6) kDmC1 .1 bk C s/ j DnC1 .aj s/

In (D.1.6) the empty product is assumed to be equal to 1 (the empty product convention), the parameters m; n; p; q satisfy the relation 0 Ä n Ä q, 0 Ä n Ä p, and the complex numbers aj , bk are such that no pole of .bk s/;k D 1;:::;m, coincides with a pole of .1 aj C s/;j D 1;:::;n. Let

q p 1 X X ı D m C n .p C q/; D b a : 2 k j kD1 j D1

The contour of integration L in (D.1.5) can be of the following three types (see [Mari83], [Kir94, p. 313]): D.1 Deﬁnition: Contour of Integration 321

(i) L D Li1, which starts at i1 and terminates at Ci1, leaving to the right all poles of -functions .bk s/;k D 1;:::;m, and leaving to the left all poles of -functions .1 aj C s/;j D 1;:::;n.Integral(D.1.5)converges for ı>0, jarg zj <ı.Ifjarg zjDı, ı 0, then the integral converges absolutely when p D q,Re<1,andwhenp 6D q if .q p/Res> 1 Re C 1 2 .q p/ as Ims !˙1. (ii) L D LC1, which starts at '1 C i1,terminatesat'2 C i1, 1 <'1 < '2 < C1, encircles once in the negative direction all poles of the -functions .bk s/;k D 1;:::;m, but no pole of the -functions .1aj Cs/;j D 1; :::;n. Integral (D5) converges if q 1 and either pqor p D q and jzj >1.

Since the generalized hypergeometric function pFq .a1;:::;apI b1;:::;bqI z/ can be considered as a special case of the Meijer G-functions, the function pFq.a1;:::;apI b1;:::;bq I z/ also possesses a representation via a Mellin–Barnes integral

Qq Ä ˇ .bk/ ˇ kD1 1;p ˇ .1 a1/;:::;.1 ap/ pFq.a1;:::;apI b1;:::;bqI z/ D Qp Gp;qC1 z ˇ ; 0; .1 b1/;:::;.1 bq/ .aj / j D1 (D.1.7) where Ä ˇ ˇ 1;p ˇ .1 a1/;:::;.1 ap/ Gp;qC1 z ˇ 0; .1 b1/;:::;.1 bq/

CZi1 1 .a1 C s/:::.ap C s/ .s/ D .z/s ds; (D.1.8) 2i .b1 C s/:::.bq C s/ i1

aj 6D 0; 1;:::I j D 1;:::;pIjarg .1 iz/j <:

Although the Meijer G-functions are quite general in nature, there still exist examples of special functions, such as the Mittag-Lefﬂer and the Wright functions, which do not form particular cases of them. A more general class which includes those functions can be obtained by introducing the Fox H-functions [Fox61], whose representation in terms of the Mellin–Barnes integral is a straightforward generalization of that for the G-functions. To introduce it one needs to add to the sets of the complex parameters aj and bk the new sets of positive numbers ˛j and ˇk 322 D The Mellin–Barnes Integral with j D 1;:::;p, k D 1;:::;q, and to replace in the integral of (D.1.5)thekernel p;q Gm;n.s/ by the new one Q Q m .b ˇ s/ n .1 a C ˛ s/ Hm;n Q kD1 k k j DQ1 j j p;q .s/ D q p : (D.1.9) kDmC1 .1 bk C ˇks/ j DnC1 .aj ˛j s/

The Fox H-functions are then deﬁned in the form " ˇ # ˇ Z ˇ .a ;˛ /p 1 p;q p;q ˇ j j j D1 Hm;n s Hm;n.z/ D Hm;n.z/ z ˇ q D p;q .s/z ds: (D.1.10) .bk;ˇk/ 2i kD1 L

A representation of the type (D.1.10) is usually called a Mellin–Barnes integral representation. The convergence questions for these integrals are completely discussed in [ParKam01, Sect. 2.4]. For further information we refer the reader to the treatises on Fox H-functions by Mathai, Saxena and Haubold [MaSaHa10], Srivastava, Gupta and Goyal [SrGuGo82], Kilbas and Saigo [KilSai04]andthe references therein.

D.2 Asymptotic Methods for the Mellin–Barnes Integral

The asymptotics of the Mellin–Barnes integral is based on the following two general lemmas on the expansion of quotients of Gamma functions as inverse factorial expansions (see [Wri40b, Wri40c, Bra62]). Let us consider the quotient of products of Gamma functions

Qp .˛j s C aj / j D1 P.s/ D Qq : (D.2.1) .ˇk s C bk/ kD1

Deﬁne the parameters 8 ˆ Qp Qq ˆ ˛j ˇk ˆ h D ˛j ˇk ; ˆ j D1 kD1 < Pp Pq 1 0 # D aj bk C 2 .q p C 1/; # D 1 #; (D.2.2) ˆ j D1 ˆ kD1 ˆ Pq Pp :ˆ D ˇk ˛j : kD1 j D1

The following lemma presents the inverse factorial expansions for the functions P.s/ (see [Wri40c, Bra62]). D.2 Asymptotic Methods for the Mellin–Barnes Integral 323

Lemma D.1 ([ParKam01,p.39]). Let M be a positive integer and suppose >0. Then there exist numbers Ar (0 Ä r Ä M 1), independent of s and M , such that the function P.s/ in (D.2.1) possesses the inverse factorial expansion given by ( ) MX1 A .s/ P.s/ D .h/s r C M ; (D.2.3) .sC #0 C r/ .sC #0 C M/ rD1 where the parameters h; and #0 are deﬁned in (D.2.2). In particular, the coefﬁcient A0 has the value

Yp Yq 1=2.pqC1/ 1=2# aj 1=2 1=2bk A0 D .2/ ˛j ˇk : (D.2.4) j D1 kD1

The remainder function M .s/ is analytic in s except at the points s D.aj Ct/= ˛j , t D 0; 1; 2; : : : (1 Ä j Ä p), where P.s/ has poles, and is such that

M .s/ D O.1/ for jsj!1uniformly in jargsjÄ ", ">0. Let now Q.s/ be another type of quotient of products of Gamma functions:

Qq .1 bk C ˇks/ kD1 Q.s/ D Qp : (D.2.5) .1 aj C ˛j s/ j D1

Let the parameters h; #; #0; be deﬁned in the same manner as in (D.2.2). The following lemma presents the corresponding inverse factorial expansions for the functions Q.s/ (see [Wri40c, Bra62]). Lemma D.2 ([ParKam01,p.39]). Let M be a positive integer and suppose >0. Then there exist numbers Ar (0 Ä r Ä M 1), independent of s and M , such that the function Q.s/ in (D.2.5) possesses the inverse factorial expansion given by ( ) .h/s MX1 Q.s/ D .1/r A .sC # r/C .s/ .s C # M/ ; .2/pqC1 r M rD1 (D.2.6) where the parameters h; and #0 are deﬁned in (D.2.2). The remainder function M .s/ is analytic in s except at the points s D .bk 1 t/=ˇk, t D 0; 1; 2; : : : (1 Ä k Ä q), where Q.s/ has poles, and is such that

M .s/ D O.1/ for jsj!1uniformly in jargsjÄ ", ">0. 324 D The Mellin–Barnes Integral

An algebraic method for determining the coefﬁcients Ar in (D.2.3)and(D.2.6) is presented in [ParKam01, pp. 46–49].

D.3 Historical and Bibliographical Notes

As a historical note, we point out that “Mellin–Barnes integrals” are named after the two authors (namely Hj. Mellin and E.W. Barnes) who in the early 1910s developed the theory of these integrals, using them for a complete integration of the hypergeometric differential equation. However, these integrals were first used in 1888 by Pincherle, see, e.g., [MaiPag03]. Recent treatises on Mellin–Barnes integrals are [Mari83]and[ParKam01]. In the classical treatise on Bessel functions by Watson [Wat66, p. 190], we read “By using integrals of a type introduced by Pincherle and Mellin, Barnes has obtainedrepresentationsofBesselfunctions...”. Salvatore Pincherle (1853–1936) was Professor of Mathematics at the University of Bologna from 1880 to 1928. He retired from the University just after the International Congress of Mathematicians which he had organized in Bologna, following the invitation received at the previous Congress held in Toronto in 1924. He wrote several treatises and lecture notes on Algebra, Geometry, and Real and Complex Analysis. His main book related to his scientific activity is entitled “Le Operazioni Distributive e loro Applicazioni all’Analisi”; it was written in collaboration with his assistant, Dr. Ugo Amaldi, and was published in 1901 by Zanichelli, Bologna. Pincherle can be considered one of the most prominent founders of Functional Analysis, and was described as such by J. Hadamard in his review lecture “Le développement et le rôle scientifique du Calcul fonctionnel”, given at the Congress of Bologna (1928). A description of Pincherle’s scientific works requested from him by Mittag-Leffler, who was the Editor of the prestigious journal Acta Mathematica, appeared (in French) in [Pin25]. A collection of selected papers (38 from 247 notes plus 24 treatises) was edited by Unione Matematica Italiana (UMI) on the occasion of the centenary of his birth, and published by Cremonese, Roma 1954. S. Pincherle was the first President of UMI, from 1922 to 1936. Here we point out that S. Pincherle’s 1888 paper (in Italian) on the Generalized Hypergeometric Functions led him to introduce what was later called the Mellin– Barnes integral to represent the solution of a generalized hypergeometric differential equation investigated by Goursat in 1883. Pincherle’s priority was explicitly recognized by Mellin and Barnes themselves, as reported below. In 1907 Barnes, see p. 63 in [Barn07b], wrote: “The idea of employing contour integrals involving gamma functions of the variable in the subject of integration appears to be due to Pincherle, whose suggestive paper was the starting point of the investigations of Mellin (1985) [Mel95] though the type of contour and its use can be traced back to Riemann.” In 1910 Mellin, see p. 326ff in [Mel10], devoted a section (§10: Proof of Theorems of Pincherle) to revisit the original work of Pincherle; in D.4 Exercises 325 particular, he wrote “Before we prove this theorem, which is a special case of a more general theorem of Mr. Pincherle, we want to describe more closely the lines L over which the integration is preferably to be carried out.” [free translation from German]. The Mellin–Barnes integrals are the essential tools for treating the two classes of higher transcendental functions known as G-and H-functions, introduced by Meijer [Mei46]andFox[Fox61] respectively, so Pincherle can be considered their precursor. For an exhaustive treatment of the Mellin–Barnes integrals we refer to the recent monograph by Paris and Kaminski [ParKam01].

D.4 Exercises

D.4.1. The exponential function can be represented in terms of the Mellin–Barnes integral ([ParKam01, p. 67])

cZCi1 1 ez D .s/zs ds.c>0/: 2i ci1

Find the domain of convergence of the integral in the right-hand side of this representation. D.4.2. Prove the following Mellin–Barnes integral representation of the hypergeometric function ([ParKam01, p. 68]) Z .a/.b/ 1 .sC a/ .s C b/ F .a; bI cI z/ D .s/ .z/s ds; .c/ 2 1 2i .sC c/ Li1

where the contour of integration Li1 is a vertical line, which starts at i1 ends at Ci1 and separates the poles of the Gamma function .s/ from those of the Gamma functions .sC a/, .s C b/ for a and b 6D 0; 1; 2;:::. Find the domain of convergence of the integral in the above representation. If the vertical line Li1 is replaced by the contour L1, then show that the domain of convergence coincides with an open unit disk. D.4.3. Find the domain of convergence of the following Mellin–Barnes integral ([ParKam01,p.68]): Z 1 .a s=2/.s C b/.c C s=4/ .z/s ds; 2i .d C s=4/ .s C 1/ LC1 326 D The Mellin–Barnes Integral

where the parameters a; b; c and and the contour of integration LC1 are such that the poles of .as=2/ are separated from those of .sCb/ and .cC s=4/. D.4.4. Prove that the following Mellin–Barnes integral representation ([ParKam01, p. 109])

cZCi1 .a/ 1 .s/ .s C a/ F .aI bI cI z/ D .z/sds .b/1 1 2i .sC b/ ci1

is valid for all ﬁnite values of c provided that the contour of integration can be deformed to separate the poles of .s/ and .sC a/ (which is always possible when a is not a negative integer or zero). Here 1F1.aI bI cI z/ is the conﬂuent hypergeometric function

X1 .a/ zn F .aI bI cI z/ D n ;.jzj < 1/: 1 1 .b/ nŠ nD n

D.4.5. For the incomplete gamma function ([ParKam01, p. 113])

Zz .a;z/ D t a1et dt.Re a>0/

prove the representation

cZCi1 1 .s/ .a;z/ D .a/C zsCads 2i s C a ci1

where c

Z1 .a;z/ D t a1et dt z

prove the representation

cZCi1 1 .sC a/ .a;z/ D zsds 2i s ci1

for all z; jarg zj=2. Appendix E Elements of Fractional Calculus

E.1 Introduction to the Riemann–Liouville Fractional Calculus

As is customary, let us take as our starting point for the development of the so-called Riemann–Liouville fractional calculus the repeated integral Z Z Z x xn1 x1 n N IaC .x/ WD ::: .x0/ dx0 ::: dxn1 ;aÄ x1 and b ÄC1. The function .x/ is assumed to be well-behaved; for this it sufﬁces that .x/ is locally integrable in the interval Œa; b/, meaning in particular that a possible singular behaviour at x D a does not destroy integrability. It is well known that the above formula provides an n-fold primitive n.x/ of .x/, precisely that primitive which vanishes at x D a jointly with its derivatives of order 1;2;:::n 1: We can re-write this n-fold repeated integral by a convolution-type formula (often attributed to Cauchy) as, Z x n 1 n1 IaC .x/ D .x / ./d; aÄ x

In a natural way we are now led to extend the formula (E.1.2) from positive integer values of the index n to arbitrary positive values ˛;thereby using the relation .n 1/Š D .n/:So, using the Gamma function, we deﬁne the fractional integral of order ˛ as Z x ˛ 1 ˛1 IaC .x/ WD .x / ./d; a0: (E.1.3) .˛/ a

© Springer-Verlag Berlin Heidelberg 2014 327 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 328 E Elements of Fractional Calculus

n N We remark that the values IaC .x/ with n 2 are always ﬁnite for a Ä x0are ﬁnite for a0we obtain the dual form of the fractional integral (E.1.3), i.e. Z b ˛ 1 ˛1 Ib .x/ WD . x/ ./d; a0: (E.1.5) .˛/ x

˛ Now it may happen that the limit (if it exists) of Ib.x/ for x ! b ,thatwe ˛ denote by Ib .b /, is inﬁnite. ˛ ˛ We refer to the fractional integrals IaC and Ib as progressive or right-handed and regressive or left-handed, respectively. Let us point out the fundamental property of the fractional integrals, namely the additive index law (semigroup property,) according to which

˛ ˇ ˛Cˇ ˛ ˇ ˛Cˇ IaC IaC D IaC ;Ib Ib D Ib ;˛;ˇ 0; (E.1.6)

0 0 I where, for complementation, we have deﬁned IaC D Ib WD (Identity operator) 0 0 which means IaC .x/ D Ib .x/ D .x/. The proof of (E.1.6) is based on Dirichlet’s formula concerning the change of the order of integration and the use of the Beta function in terms of the Gamma function. We note that the fractional integrals (E.1.3)and(E.1.5) contain a weakly singular kernel only when the order is less than one. We can now introduce the concept of fractional derivative based on the fundamental property of the common derivative of integer order n

dn Dn .x/ D .x/ D .n/.x/ ; a < x < b dxn E.1 Introduction to the Riemann–Liouville Fractional Calculus 329 to be the left inverse operator of the progressive repeated integral of the same n order n, IaC .x/. In fact, it is straightforward to recognize that the derivative of any integer order n D 0; 1; 2; : : : satisﬁes the following composition rules with n n respect to the repeated integrals of the same order n, IaC .x/ and Ib .x/ W ( n n D IaC .x/ D .x/; P C a

In view of the above properties we can define the progressive/regressive fractional derivatives of order ˛>0provided that we first introduce the positive integer m such that m 1<˛Ä m. Then we define the progressive/regressive fractional ˛ ˛ derivative of order ˛, DaC=Db,astheleft inverse of the corresponding fractional ˛ ˛ integral IaC=Ib. As a consequence of the previous formulas (E.1.2)–(E.1.8)the fractional derivatives of order ˛ turn out to be defined as follows ( D˛ .x/ WD Dm I m˛ .x/; aC aC a

In fact, for the progressive/regressive derivatives we get ( D˛ I ˛ D Dm I m˛ I ˛ D Dm I m D I ; aC aC aC aC aC (E.1.10) ˛ ˛ m m m˛ ˛ m m m I Db Ib D .1/ D Ib Ib D .1/ D Ib D :

0 0 I For complementation we also deﬁne DaC D Db WD (Identity operator). When the order is not an integer (m 1<˛

We stress the fact that the “proper” fractional derivatives (namely when ˛ is non-integer) are non-local operators being expressed by ordinary derivatives of convolution-type integrals with a weakly singular kernel, as is evident from (E.1.11) 330 E Elements of Fractional Calculus and (E.1.12). Furthermore, they do not necessarily obey the analogue of the “semigroup” property of the fractional integrals: in this respect the initial point a 2 R (or the end point b 2 R) plays a “disturbing” role. We also note that when the order ˛ tends to both the end points of the interval .m 1; m/ we recover the common derivatives of integer order (unless of the sign for the regressive derivative).

E.2 The Liouville–Weyl Fractional Calculus

We can also deﬁne the fractional integrals over unbounded intervals and, as left inverses, the corresponding fractional derivatives. If the function .x/ is locally integrable in 1 0: (E.2.1) .˛/ 1

Analogously, if the function .x/ is locally integrable in 1 Ä a0: (E.2.2) .˛/ x

Note the kernel . x/˛1 for (E.2.2). The names of Liouville and Weyl are here adopted for the fractional integrals (E.2.1)and(E.2.2), respectively, following a standard terminology, see e.g. Butzer and Westphal [ButWes00],basedonhistorical reasons.1 We note that a sufﬁcient condition for the integrals in (E.2.1)and(E.2.2)to converge is that

.x/D O.jxj˛ /; >0;x!1; respectively. Integrable functions satisfying these properties are sometimes referred to as functions of Liouville class (for x !1), and of Weyl class (for x !C1), respectively, see Miller and Ross [MilRos93]. For example, power functions jxjı

1In fact, Liouville considered in a series of papers from 1832 to 1837 the integrals of progressive type (E.2.1), see, e.g., [Lio32a, Lio32a, Lio37]. On the other hand, H. Weyl [Wey17] arrived at the regressive integrals of type (E.2.2) indirectly by deﬁning fractional integrals suitable for periodic functions. E.2 The Liouville–Weyl Fractional Calculus 331 with ı>˛>0and exponential functions ecx with c>0are of Liouville class (for x !1). For these functions we obtain 8 ˆ ˛ ı .ı˛/ ıC˛ < IC jxj D .ı/ jxj ; ˆ ı >˛ >0; x <0; (E.2.3) : ˛ ı .ıC˛/ ı˛ DC jxj D .ı/ jxj ; and 8 ˛ cx ˛ cx < IC e D c e ; R : c>0;x2 : (E.2.4) ˛ cx ˛ cx DC e D c e ;

For the Liouville and Weyl fractional integrals we can also state the corresponding semigroup property

˛ ˇ ˛Cˇ ˛ ˇ ˛Cˇ IC IC D IC ;I I D I ;˛;ˇ 0; (E.2.5)

0 0 I where, for complementation, we have defined IC D I WD (Identity operator). For more details on Liouville–Weyl fractional integrals we refer to Miller [Mil75], Samko, KIlbas and Marichev [SaKiMa93] and Miller and Ross [MilRos93]. For the definition of the Liouville–Weyl fractional derivatives of order ˛ we follow the scheme adopted in the previous section for bounded intervals. Having introduced the positive integer m so that m 1<˛Ä m we define 8 ˛ m m˛ < DC .x/ WD D IC .x/; 1

0 0 I with DC D D D : In fact we easily recognize using (E.2.5)and(E.2.6)the fundamental property

˛ ˛ I m ˛ ˛ DC IC D D .1/ D I : (E.2.7)

The explicit expressions for the “proper” Liouville and Weyl fractional derivatives (m 1<˛

Because of the unbounded intervals of integration, fractional integrals and derivatives of Liouville and Weyl type can be (successfully) handled via the Fourier transform and the related theory of pseudo-differential operators, that, as we shall see, simplifies their treatment. For this purpose, let us now recall our notations and the relevant results concerning the Fourier transform. Let Z C1 ./O D F f.x/I g D e Cix .x/dx; 2 R ; (E.2.10) 1 be the Fourier transform of a sufficiently well-behaved function .x/,andlet n o Z 1 C1 .x/ D F 1 ./O I x D e ix ./O d; x2 R ; (E.2.11) 2 1 denote the inverse Fourier transform.2 In this framework we also consider the class of pseudo-differential operators of which the ordinary repeated integrals and derivatives are special cases. A pseudo- differential operator A, acting with respect to the variable x 2 R, is defined through its Fourier representation, namely Z C1 e ix A.x/dx D A./O ./;O (E.2.12) 1 where A./O is referred to as the symbol of A. An often applicable practical rule is A./O D A e ix e Cix ;2 R : (E.2.13)

If B is another pseudo-differential operator, then we have

AB./b D A./O B./:O (E.2.14)

F For the sake of convenience we adopt the notation $ to denote the juxtaposition of a function with its Fourier transform and that of a pseudo-differential operator with its symbol, i.e.

F F .x/$ ./;O A$ A:O (E.2.15)

2In the ordinary theory of the Fourier transform the integral in (E.2.10) is assumed to be a “Lebesgue integral” whereas the one in (E.2.11) can be the “principal value” of a “generalized integral”. In fact, .x/ 2 L1.R/, necessary for writing (E.2.10), is not sufﬁcient to ensure ./O 2 L1.R/. However, we allow for an extended use of the Fourier transform which includes Dirac-type generalized functions: then the above integrals must be properly interpreted in the framework of the theory of distributions. E.2 The Liouville–Weyl Fractional Calculus 333

We now consider the pseudo-differential operators represented by the Liouville– Weyl fractional integrals and derivatives. Of course we assume that the integrals in their deﬁnitions are in a proper sense, in order to ensure that the resulting functions of x can be Fourier transformable in the ordinary or generalized sense. The symbols of the fractional Liouville–Weyl integrals and derivatives can easily be derived according to ( Ic˛ D .i/˛ Djj˛ e ˙i.sgn/˛=2 ; ˙ (E.2.16) b˛ C˛ C˛ i.sgn/˛=2 D˙ D .i/ Djj e :

Based on an idea of Marchaud, see e.g. Marchaud [Marc27], Samko, Kilbas and Marichev [SaKiMa93], Hilfer [Hil97], we now give purely integral expressions for ˛ D˙ which are alternative to the integro-differential expressions (E.2.8)and(E.2.9). We limit ourselves to the case 0<˛<1:Let us ﬁrst consider from Eq. (E.2.8) the progressive derivative

d D˛ D I 1˛ ; 0<˛<1: (E.2.17) C dx C We have, see Hilfer [Hil97],

d D˛ .x/ D I 1˛ .x/ C dx C Z 1 d x D .x /˛ ./d .1 ˛/ dx 1 Z 1 d 1 D ˛ .x /d .1 ˛/ dx 0 Z Ä Z ˛ 1 1 d D 0.x / ; 1C˛ d .1 ˛/ 0 so that, interchanging the order of integration Z ˛ 1 .x/ .x / D˛ .x/ D ; 0<˛<1: C 1C˛ d (E.2.18) .1 ˛/ 0

Here 0 denotes the ﬁrst derivative of with respect to its argument. The coefﬁcient in front of the integral in (E.2.18) can be re-written, using known formulas for the Gamma function, as

˛ 1 sin ˛ D D .1C ˛/ : (E.2.19) .1 ˛/ .˛/ 334 E Elements of Fractional Calculus

Similarly we get for the regressive derivative

d D˛ D I 1˛ ;0<˛<1; (E.2.20) dx Z ˛ 1 .x/ .x C / D˛ .x/ D ; 0<˛<1: 1C˛ d (E.2.21) .1 ˛/ 0

Similar results can be given for ˛ 2 .m 1; m/ ; m 2 N :

E.3 The Abel–Riemann Fractional Calculus

RC In this section we consider sufﬁciently well-behaved functions .t/ (t 2 0 ) with Laplace transform deﬁned as Z 1 st Q .s/ D L f .t/I sg D e .t/ dt; Re.s/ > a ; (E.3.1) 0 where a denotes the abscissa of convergence. The inverse Laplace transform is then given as Z ˚ « 1 .t/ D L1 Q .s/I t D e st Q .s/ ds; t >0; (E.3.2) 2i Br

3 where Br is a Bromwich path, namely f i1; C i1g with >a . It may be convenient to consider .t/ as a “causal” function in R, i.e. vanishing L for all t<0:For the sake of convenience we adopt the notation $ to denote the juxtaposition of a function with its Laplace transform, with its symbol, i.e.

L .t/ $ Q .s/ : (E.3.3)

E.3.1 The Abel–Riemann Fractional Integrals and Derivatives

We ﬁrst deﬁne the Abel–Riemann (A-R) fractional integral and derivative of any order ˛>0for a generic (well-behaved) function .t/ with t 2 RC :

3 C In the ordinary theory of the Laplace transform the condition .t/ 2 Lloc.R / is necessarily required, and the Bromwich integral is intended in the “principal value” sense. However, as mentioned in the previous footnote, by interpreting these integrals in the framework of the theory of distributions we include Dirac-type generalized functions. E.3 The Abel–Riemann Fractional Calculus 335

For the A-R fractional integral (of order ˛)wehave Z 1 t J ˛ .t/ WD .t /˛1 ./ d; t>0 ˛>0: (E.3.4) .˛/ 0

For complementation we put J 0 WD I (Identity operator), as can be justiﬁed by passing to the limit ˛ ! 0: The A-R integrals possess the semigroup property

J ˛ J ˇ D J ˛Cˇ ; for all ˛; ˇ 0: (E.3.5)

The A-R fractional derivative (of order ˛>0)isdeﬁnedastheleft-inverse operator of the corresponding A-R fractional integral (of order ˛>0), i.e.

D˛ J ˛ D I : (E.3.6)

Therefore, introducing the positive integer m such that m 1<˛Ä m and noting that .Dm J m˛/J˛ D Dm .J m˛ J ˛/ D Dm J m D I ; we deﬁne

D˛ WD Dm J m˛ ;m 1<˛Ä m; (E.3.7) i.e. ( R 1 dm t ./ m ˛C1m d; m 1<˛

For complementation we put D0 WD I : For ˛ ! m we thus recover the standard derivative of order m but the integral formula loses its meaning for ˛ D m: By using the properties of the Eulerian Beta and Gamma functions it is easy to show the effect of our operators J ˛ and D˛ on the power functions: we have 8 ˆ J ˛t D .C1/ t C˛ ; <ˆ .C1C˛/ D˛ t D .C1/ t ˛ ; (E.3.9) ˆ .C1˛/ :ˆ t>0;˛ 0; >1:

These properties are of course a natural generalization of those known when the order is a positive integer. Note the remarkable fact that the fractional derivative D˛ .t/ is not zero for the constant function .t/ Á 1 if ˛ 62 N : In fact, the second formula in (E.3.9) with D 0 teaches us that

t ˛ D˛ 1 D ;˛ 0; t >0: (E.3.10) .1 ˛/ 336 E Elements of Fractional Calculus

This, of course, is identically 0 for ˛ 2 N, due to the poles of the Gamma function at the points 0; 1; 2;:::.

E.4 The Caputo Fractional Calculus

E.4.1 The Caputo Fractional Derivative

An alternative definition of the fractional derivative was introduced in the late sixties by Caputo [Cap67, Cap69] and was soon adopted in physics to deal with long-memory visco-elastic processes by Caputo and Mainardi, see [CapMai71a, CapMai71b], and, for a more recent review, Mainardi [Mai97]. ˛ In this case the fractional derivative, denoted by D ; is defined by exchanging the operators J m˛ and Dm in the classical definition (E.3.7), i.e.

˛ m˛ m D WD J D ;m 1<˛Ä m: (E.4.1)

In the literature, following its appearance in the book by Podlubny [Pod99], this derivative is known simply as the Caputo derivative. Based on (E.4.1)wehave 8 R < 1 t .m/./ C d; m 1<˛

For m1<˛

˛ m m˛ m˛ m ˛ D .t/ D D J .t/ ¤ J D .t/ D D .t/ ; (E.4.3) unless the function .t/ along with its ﬁrst m 1 derivatives vanishes at t D 0C.In fact, assuming that the passage of the m-th derivative under the integral is legitimate, one recognizes that, for m 1<˛0

mX1 t k˛ D˛ .t/ D D˛ .t/ C .k/.0C/: (E.4.4) .k ˛ C 1/ kD0

As noted by Samko, Kilbas and Marichev [SaKiMa93] and Butzer and Westphal [ButWes00] the identity (E.4.4) was considered by Liouville himself (but not used as an alternative deﬁnition of the fractional derivative). E.4 The Caputo Fractional Calculus 337

Recalling the fractional derivative of the power functions we can rewrite (E.4.4) in the equivalent form ! mX1 t k D˛ .t/ .k/.0C/ D D˛ .t/ : (E.4.5) kŠ kD0

The subtraction of the Taylor polynomial of degree m1 at t D 0C from .t/ yields a sort of regularization of the fractional derivative. In particular, as is easily shown, according to this deﬁnition we retrieve the property that the fractional derivative of a constant is zero,

˛ D1 Á 0; ˛ >0: (E.4.6)

As a consequence of (E.4.5) we can interpret the Caputo derivative as a sort of regularization of the R-L derivative as soon as the values k .0C/ are ﬁnite; in this sense such a fractional derivative was independently introduced in 1968 by Dzherbashyan and Nersesian [DzhNer68], as pointed out in interesting papers by Kochubei, see [Koc89, Koc90]. In this respect the regularized fractional derivative is sometimes referred to as the Caputo–Dzherbashyan derivative. We now explore the most relevant differences between the two fractional derivatives (E.3.7)and(E.4.1). We agree to call (E.4.1)theCaputo fractional derivative to distinguish it from the standard A-R fractional derivative. We observe, again by looking at the second equation in (E.3.9), that D˛t ˛k Á 0; t > 0 for ˛>0,andk D 1;2;:::;m. We thus recognize the following statements about functions which for t>0 admit the same fractional derivative of order ˛;with m 1<˛Ä m, m 2 N ;

Xm ˛ ˛ ˛j D .t/ D D .t/ ” .t/ D .t/ C cj t ; (E.4.7) j D1 Xm ˛ ˛ mj D .t/ D D .t/ ” .t/ D .t/ C cj t ; (E.4.8) j D1 where the coefﬁcients cj are arbitrary constants. For the two deﬁnitions we also note a difference with respect to the formal limit as ˛ ! .m 1/C; from (E.3.7)and(E.4.1) we obtain respectively,

D˛ .t/ ! Dm J .t/D Dm1 .t/ I (E.4.9)

˛ m m1 .m1/ C D .t/ ! JD .t/ D D .t/ .0 /: (E.4.10)

We now consider the Laplace transform of the two fractional derivatives. For the A-R fractional derivative D˛ the Laplace transform, assumed to exist, requires the knowledge of the (bounded) initial values of the fractional integral J m˛ and of its 338 E Elements of Fractional Calculus integer derivatives of order k D 1;2;:::;m 1:The corresponding rule reads, in our notation,

m1 L X D˛ .t/ $ s˛ Q .s/ Dk J .m˛/ .0C/sm1k ;m1<˛Ä m: (E.4.11) kD0

For the Caputo fractional derivative the Laplace transform technique requires the knowledge of the (bounded) initial values of the function and of its integer derivatives of order k D 1;2;:::;m 1;in analogy with the case when ˛ D m:In ˛ ˛ ˛ m˛ m m m fact, noting that J D D J J D D J D ,wehave

mX1 t k J ˛ D˛ .t/ D .t/ .k/.0C/ ; (E.4.12) kŠ kD0 so we easily prove the following rule for the Laplace transform,

m1 L X ˛ ˛ Q .k/ C ˛1k D .t/ $ s .s/ .0 /s ;m 1<˛Ä m: (E.4.13) kD0

Indeed the result (E.4.13), first stated by Caputo [Cap69], appears as the “natural” generalization of the corresponding well-known result for ˛ D m: Gorenflo and Mainardi [GorMai97] have pointed out the major utility of the Caputo fractional derivative in the treatment of differential equations of fractional order for physical applications. In fact, in physical problems, the initial conditions are usually expressed in terms of a given number of bounded values assumed by the field variable and its derivatives of integer order, despite the fact that the governing evolution equation may be a generic integro-differential equation and therefore, in particular, a fractional differential equation. Nowadays all the books on Fractional Calculus after Podlubny book [Pod99] consider Caputo derivative. This derivative becomes very popular because of its importance for the theory and applications. Several applications have also been treated by Caputo himself from the seventies up to the present, see e.g. [Cap74, Cap79, Cap85, Cap89, Cap96, Cap00, Cap01].

E.5 The Riesz–Feller Fractional Calculus

The purpose of this section is to combine the Liouville–Weyl fractional integrals and derivatives in order to obtain the pseudo-differential operators considered around the 1950s by Marcel Riesz [Rie49] and William Feller [Fel52]. In particular the Riesz– Feller fractional derivatives will be used later to generalize the standard diffusion equation by replacing the second-order space derivative. In so doing we shall E.5 The Riesz–Feller Fractional Calculus 339 generate all the (symmetric and non-symmetric) Lévy stable probability densities according to our parametrization.

E.5.1 The Riesz Fractional Integrals and Derivatives

The Liouville–Weyl fractional integrals can be combined to give rise to the Riesz fractional integral (usually called the Riesz potential)oforder˛,deﬁnedas Z ˛ ˛ C1 ˛ IC.x/ C I.x/ 1 ˛1 I0 .x/ D D jx j ./d; 2 cos.˛=2/ 2.˛/ cos.˛=2/ 1 (E.5.1) for any positive ˛ with the exclusion of odd integers for which cos.˛=2/ vanishes. The symbol of the Riesz potential turns out to be

c˛ ˛ R I0 Djj ;2 ;˛>0;˛¤ 1; 3; 5 ::: : (E.5.2)

In fact, recalling the symbols of the Liouville–Weyl fractional integrals, see Eq. (E.2.16), we obtain Ä 1 1 .Ci/˛ C .i/˛ 2 cos.˛=2/ Ic˛ C Ic˛ D C D D : C .i/˛ .Ci/˛ jj˛ jj˛

We note that, in contrast to the Liouville fractional integral, the Riesz potential has the semigroup property only in restricted ranges, e.g.

˛ ˇ ˛Cˇ I0 I0 D I0 for 0<˛ <1; 0<ˇ <1; ˛C ˇ<1: (E.5.3)

From the Riesz potential we can deﬁne by analytic continuation the Riesz ˛ fractional derivative D0 , including also the singular case ˛ D 1;by formally setting ˛ ˛ D0 WD I0 , i.e., in terms of symbols,

c˛ ˛ D0 WD jj : (E.5.4)

We note that the minus sign has been used in order to recover for ˛ D 2 the standard second derivative. Indeed, noting that

jj˛ D.2/˛=2 ; (E.5.5) we recognize that the Riesz fractional derivative of order ˛ is the opposite of the d2 ˛=2-power of the positive deﬁnite operator dx2 Â Ã d 2 ˛=2 D˛ D : (E.5.6) 0 dx2 340 E Elements of Fractional Calculus

We also note that the two Liouville fractional derivatives are related to the ˛-power d of the first order differential operator D D dx : We note that it was Bochner [Boc49] who first introduced fractional powers of the Laplacian to generalize the diffusion equation. Restricting our attention to the range 0<˛Ä 2 the explicit expression for the Riesz fractional derivative turns out to be 8 ˛ ˛ <ˆ DC .x/CD .x/ 2 cos.˛=2/ ;if˛¤ 1; D˛ .x/ D (E.5.7) 0 :ˆ DH.x/; if˛D 1; where H denotes the Hilbert transform operator defined by Z Z 1 C1 ./ 1 C1 .x / H.x/WD d D d; (E.5.8) 1 x 1 the integral understood in the Cauchy principal value sense. Incidentally, we note that H 1 DH. By using the practical rule (E.2.13) we can derive the symbol of H, namely

HO D i sgn2 R : (E.5.9)

The expressions in (E.5.7) can easily be veriﬁed by manipulating with symbols of “good” operators as below

( ˛ ˛ .i/ C.Ci/ Djj˛ ;if˛¤ 1; c˛ b˛ ˛ 2 cos.˛=2/ D0 DI0 Djj D Ci isgn D sgn Djj ;if˛D 1:

In particular, from (E.5.7) we recognize that Â Ã 1 1 d2 d2 d2 d D2 D D2 C D2 D C D ; but D1 ¤ : 0 2 C 2 dx2 dx2 dx2 0 dx

˛ We also recognize that the symbol of D0 (0<˛Ä 2) is just the cumulative function (logarithm of the characteristic function) of a symmetric Lévy stable probability distribution function, see e.g. Feller [Fel71], Sato [Sat99]. We introduce the “illuminating” notation introduced by Zaslavsky, see e.g. Saichev and Zaslavsky [SaiZas97] to denote our Liouville and Riesz fractional derivatives d˛ d˛ D˛ D ;D˛ D ;0<˛Ä 2: (E.5.10) ˙ d.˙x/˛ 0 djxj˛ E.5 The Riesz–Feller Fractional Calculus 341

Recalling from (E.5.7) the fractional derivative in Riesz’s sense

D˛ .x/ C D˛ .x/ D˛ .x/ WD C ; 0<˛<1;1<˛<2; 0 2 cos.˛=2/ and using (E.2.18)and(E.2.21) we get for it the following regularized representation, valid also for ˛ D 1; Z sin .˛=2/ 1 .x C / 2.x/ C .x / D˛ .x/ D .1C ˛/ ; 0 1C˛ d 0 0<˛<2: (E.5.11) We note that Eq. (E.5.11) has recently been derived by Gorenﬂo and Mainardi, see [GorMai01], and improves the corresponding formula in the book by Samko, Kilbas and Marichev [SaKiMa93] which is not valid for ˛ D 1:

E.5.2 The Feller Fractional Integrals and Derivatives

A generalization of the Riesz fractional integral and derivative was proposed by Feller [Fel52] in a pioneering paper, recalled by Samko, Kilbas and Marichev [SaKiMa93], but only recently revised and used by Gorenﬂo and Mainardi [GorMai98]. Feller’s intention was indeed to generalize the second order space derivative appearing in the standard diffusion equation by a pseudo-differential operator whose symbol is the cumulative function (logarithm of the characteristic function) of a general Lévy stable probability distribution function according to his parametrization. Let us now show how to obtain the Feller derivative by inversion of a properly generalized Riesz potential, later called the Feller potential by Samko, Kilbas and ˛ Marichev [SaKiMa93]. Using our notation we deﬁne the Feller potential I by its symbol obtained from the Riesz potential by a suitable “rotation” by a properly restricted angle =2,i.e.

˛; if 0<˛<1; Ic˛./ Djj˛ e i.sgn / =2 ; j jÄ (E.5.12) 2 ˛;if 1<˛Ä 2; with ; 2 R. As for the Riesz potential, the case ˛ D 1 is omitted here. The ˛ integral representation of I turns out to be

˛ ˛ ˛ I .x/ D c.˛; / IC .x/ C cC.˛; / I .x/; (E.5.13) where, if 0<˛<2; ˛¤ 1;

sin Œ.˛ /=2 sin Œ.˛ C /=2 cC.˛; / D ;c.˛; / D ; (E.5.14) sin .˛/ sin.˛/ 342 E Elements of Fractional Calculus and, by passing to the limit (with D 0)

cC.2; 0/ D c.2; 0/ D1=2 : (E.5.15)

In the particular case D 0 we get

1 cC.˛; 0/ D c.˛; 0/ D ; (E.5.16) 2 cos .˛=2/ and thus, from (E.5.13)and(E.5.16) we recover the Riesz potential (E.3.1). Like the Riesz potential, the Feller potential also has the (range-restricted) semigroup property, e.g.

˛ ˇ ˛Cˇ I I D I for 0<˛<1;0<ˇ<1;˛C ˇ<1: (E.5.17)

From the Feller potential we can deﬁne by analytical continuation the Feller ˛ fractional derivative D , including also the singular case ˛ D 1;by setting

˛ ˛ D WD I ; so ( ˛; if 0<˛Ä 1; c˛ ˛ Ci.sgn / =2 D ./ Djj e ; j jÄ (E.5.18) 2 ˛;if 1<˛Ä 2:

˛ Since for D the case ˛ D 1 is included, the condition for in (E.5.18) can be shortened to

j jÄmin f˛; 2 ˛g ;0<˛Ä 2:

We note that the allowed region for the parameters ˛ and turns out to be a diamond in the plane f˛; g with vertices at the points .0; 0/, .1; 1/, .2; 0/, .1; 1/,see Fig. E.1. We call it the Feller–Takayasu diamond, in honour of Takayasu, who in his 1990 book [Tak90] ﬁrst gave the diamond representation for the Lévy-stable distributions. ˛ The representation of D .x/ can be obtained from the previous considerations. We have 8 < ˛ ˛ cC.˛; / D C c.˛; / D .x/; if ˛ ¤ 1; ˛ C D .x/ D (E.5.19) : 1 cos. =2/ D0 C sin. =2/ D .x/; if ˛ D 1:

For ˛ ¤ 1 it is sufﬁcient to note that c.˛; / D c˙.˛; / : For ˛ D 1 we need 1 O to recall the symbols of the operators D and D0 DDH , namely D D .i/ and c1 D0 Djj ; and note that E.5 The Riesz–Feller Fractional Calculus 343

Fig. E.1 The Feller–Takayasu diamond

c1 Ci.sgn / =2 D i Djj e Djj cos. =2/ .i/ sin. =2/ c1 O D cos. =2/ D0 C sin. =2/ D:

We note that in the extremal cases of ˛ D 1 we get

d D1 D˙D D˙ : (E.5.20) ˙1 dx We also note that the representation by hyper-singular integrals for 0<˛<2 (now excluding the cases f˛ D 1; ¤ 0g) can be obtained by using (E.2.18)and (E.2.21) in the first equation of (E.5.19). We get Z .1C ˛/ 1 .x C / .x/ D˛ .x/ D Œ.˛ C /=2 sin 1C˛ d 0 Z (E.5.21) 1 .x / .x/ C Œ.˛ /=2 ; sin 1C˛ d 0 which reduces to (E.5.11)for D 0: For later use we find it convenient to return to the “weight” coefficients c˙.˛; / in order to outline some properties along with some particular expressions, which can be easily obtained from (E.4.4) with the restrictions on given in (E.4.2). We obtain ( 0;if 0<˛<1; c˙ (E.5.22) Ä 0;if 1<˛Ä 2; 344 E Elements of Fractional Calculus and ( cos . =2/ >0;if 0<˛<1; cC C c D (E.5.23) cos .˛=2/ <0;if 1<˛Ä 2:

In the extremal cases we ﬁnd ( cC D 1; c D 0;if D˛; 0<˛<1; (E.5.24) cC D 0; c D 1;if DC˛; ( cC D 0; c D1;if D.2 ˛/ ; 1<˛<2; (E.5.25) cC D1; c D 0;if DC.2 ˛/ :

In view of the relation of the Feller operators in the framework of stable probability density functions, we agree to call the skewness parameter. We must note that in his original paper Feller [Fel52] used a skewness parameter ı different from our I the potential introduced by Feller is such that

˛ 2 Ic˛ D jj e i.sgn /ı ;ıD ; D ˛ı : (E.5.26) ı 2 ˛ In their book, Uchaikin and Zolotarev [UchZol99] have adopted Feller’s convention, but use the letter in place of Feller’s ı.

E.6 The Grünwald–Letnikov Fractional Calculus

Grünwald [Gru67] and Letnikov [Let68a, Let68b] independently generalized the classical deﬁnition of derivatives of integer order as limits of difference quotients to the case of fractional derivatives. Let us give an outline of their approach in a way appropriate for later use in the construction of discrete random walk models for fractional diffusion processes. For more detailed presentations, proofs of convergence, suitable function spaces etc., we refer the reader to Butzer and Westphal [ButWes00], Podlubny [Pod99], Miller and Ross [MilRos93], Samko, Kilbas and Marichev [SaKiMa93]. It is noteworthy that Oldham and Spanier in their pioneering book [OldSpa74] also made extensive use of this approach. We need one essential discrete operator, namely the shift operator Eh,whichfor h 2 R in its application to a function .x/ deﬁned in R has the effect

Eh .x/ D .x C h/ : (E.6.1)

Obviously the operators Eh,forh 2 R ; possess the group property

h1 h2 h2 h1 h1Ch2 E E D E E D E ;h1;h2 2 R ; (E.6.2) E.6 The Grünwald–Letnikov Fractional Calculus 345 and it is because of this property that we put h in the place of an exponent rather than in the place of an index. In fact, we can formally write

X1 1 Eh D e hD D hn Dn ; nŠ nD0 as a shorthand notation for the Taylor expansion of a function analytic at the point x. In the sequel, if not explicitly said otherwise, we will assume

h>0: (E.6.3)

We are now going to describe in some detail how to approximate our fractional ˛ ˛ derivatives, introduced earlier, such as DaC ;DC ;::: for ˛>0:The corresponding formulas with the minus sign in the index can then be obtained by consideration of symmetry. It is convenient to introduce the backward difference operator h as

h h D I E ; (E.6.4) whose process can, via the binomial theorem, readily be expressed in terms of the k operators Ekh D Eh :

Xn n n D .1/k Ekh ;n2 N : (E.6.5) h k kD0

0 I For complementation we also deﬁne h D : It is well-known that for sufﬁciently smooth functions and positive integer n

n n n C h h .x C ch/ D D .x/ C O.h/; h ! 0 : (E.6.6)

Here c stands for an arbitrary ﬁxed real number (independent of h). The choice c D 0 or c D n leads to completely one-sided (backward or forward, respectively) approximation, whereas the choice c D n=2 leads to approximation of second order accuracy, i.e. O.h2/ in place of O.h/:

E.6.1 The Grünwald–Letnikov Approximation in the Riemann–Liouville Fractional Calculus

The Grünwald–Letnikov approach to fractional differentiation consists in replacing in (E.6.5)and(E.6.6) the positive integer n by an arbitrary positive number ˛. Then, instead of the binomial formula, we have to use the binomial series, and we get 346 E Elements of Fractional Calculus

(formally) ! X1 ˛ ˛ D .1/k Ekh ;˛>0; (E.6.7) h k kD0 with the generalized binomial coefﬁcients ! ˛ ˛.˛ 1/:::.˛ k C 1/ .˛C 1/ D D : k kŠ .nC 1/ .˛ n C 1/

If now .x/ is a sufﬁciently well-behaved function (deﬁned in R) we have the limit relation

˛ ˛ ˛ C h h .x C ch/ ! DC .x/; h ! 0 ;˛>0: (E.6.8)

Here we usually take c D 0, but in general a fixed real number c is allowed if is smooth enough (compare Vu Kim Tuan and Gorenflo [VuGor95]). ˛ ˛ For sufficient conditions of convergence and also for convergence h h ! ˛ DC in norms of appropriate Banach spaces instead of pointwise convergence the readers may consult the above-mentioned references. From [VuGor95] one can take that in the case c D 0 convergence is pointwise (as written in (E.6.8)) and of order O.h/ if 2 C Œ˛C2.R/ and all derivatives of up to the order Œ˛ C 3 belong to L1.R/. In our later theory of random walk models for space-fractional diffusion such precise conditions are irrelevant because a posteriori we can prove their convergence. The Grünwald–Letnikov approximations will there serve us as a heuristic guide to constructional models. In fact, we will need these approximations just for 0<˛Ä 2,for0<˛<1with c D 0; for 1<˛Ä 2 with c D 1; in (E.6.9). (The case ˛ D 1 being singular in these special random walk models.) Before proceeding further, let us write in longhand notation (for the special choice c D 0) the formula (E.6.9) in the form ! X1 ˛ h˛ .1/k .x kh/ ! D˛ .x/; h ! 0C ;˛>0; (E.6.9) k C kD0 and the corresponding formula ! X1 ˛ h˛ .1/k .x C kh/ ! D˛ .x/; h ! 0C ;˛>0: (E.6.10) k kD0 N ˛ If ˛ D n 2 ,then k D 0 for k>˛D n;and we have finite sums instead of infinite series. E.6 The Grünwald–Letnikov Fractional Calculus 347

The Grünwald–Letnikov approach to the Riemann–Liouville derivative ˛ DaC .x/ where x>a>1 consists in extending .x/ by .x/ D 0 for x

˛ ˛ ˛ C GLDaC .x/ D lim h h .x/; h ! 0 ;x>a: (E.6.11) h!0C

If .x/ is smooth enough (in particular at x D a where we require .a/ D 0)we have

˛ ˛ GLDaC .x/ D DaC .x/; (E.6.12) either pointwise or in a suitable function space. Taking x a as a multiple of h;we can write the Grünwald–Letnikov derivative as a ﬁnite sum ! .xa/=h X ˛ D˛ .x/ ' h˛ ˛ .x/ D h˛ .1/k .x kh/: (E.6.13) aC h k kD0

One can show that [see Samko, Kilbas and Marichev [SaKiMa93] Eq. (20.45) and compare our formula (E.5.9)]

. C 1/ D˛ x D x˛ D D˛ x D D˛ x ; GL 0C . C 1 ˛/ 0C with x>0and >1; ˛ >0: We refrain from writing down the analogous formulas for approximation of the ˛ regressive derivative Db .x/: Let us ﬁnally point out that by formally replacing ˛ by ˛ we obtain an ˛ approximation of the operators of fractional integration as IaC with ˛>0;namely ! .xa/=h X ˛ I ˛ .x/ ' h˛ .1/k .x kh/; (E.6.14) 0C k kD0 where ! ˛ .˛/ .1/k .x kh/ D k >0: (E.6.15) k kŠ

Here we have used the convention

.˛/ kY1 k D D ˛.˛ 1/:::.˛ k C 1/ : kŠ j D0

Remark concerning notation. 348 E Elements of Fractional Calculus

In numerical analysis it is customary to write rh (instead of h) for the backward h difference operator I E andtouseh for the forward difference operator Eh I; as in Gorenﬂo [Gor97]. In our present work, however, we follow the notation employed by most workers in fractional calculus. In the modern notation the Grünwald–Letnikov fractional derivative has the form

Zt mX1 x.k/.0/t ˛Ck 1 D˛ x.t/ D C .t /m˛1x.m/./d; GL 0C .˛ C k C 1/ .m ˛/ kD0 0 (E.6.16) where m 1 Ä ˛

E.6.2 The Grünwald–Letnikov Approximation in the Riesz–Feller Fractional Calculus

The Grünwald–Letnikov approximation for the fractional Liouville–Weyl deriva- ˛ ˛ ˛ ˛ tives DC ;D gives us an approximation to the Riesz–Feller derivatives D0 ;D ; with 0<˛<2, according to Eqs. (E.2.7)and(E.2.19) (respectively), provided we disregard the singular case ˛ D 1.

E.7 Historical and Bibliographical Notes

Fractional calculus is the ﬁeld of mathematical analysis which deals with the investigation and application of integrals and derivatives of arbitrary order. The term fractional is a misnomer, but it has been retained following the prevailing use. The fractional calculus may be considered an old and yet novel topic. It is an old topic since, starting from some speculations of G.W. Leibniz (1695, 1697) and L. Euler (1730), it has been developed up to the present day. In fact the idea of generalizing the notion of derivative to non-integer order, in particular to the order 1/2, is contained in the correspondence of Leibniz with Bernoulli, L’Hôpital and Wallis. Euler took the ﬁrst step by observing that the result of the evaluation of the derivative of the power function has a meaning for non-integer order thanks to his Gamma function. A list of mathematicians who have provided important contributions up to the middle of the twentieth century includes P.S. Laplace (1812), J.B.J. Fourier E.7 Historical and Bibliographical Notes 349

(1822), N.H. Abel (1823–1826), J. Liouville (1832–1837), B. Riemann (1847), H. Holmgren (1865–1867), A.K. Grünwald (1867–1872), A.V. Letnikov (1868– 1872), N.Ya. Sonine (1872–1884), H. Laurent (1884), P.A. Nekrassov (1888), A. Krug (1890), J. Hadamard (1892), O. Heaviside (1892–1912), S. Pincherle (1902), G.H. Hardy and J.E. Littlewood (1917–1928), H. Weyl (1917), P. Lévy (1923), A. Marchaud (1927), H.T. Davis (1924–1936), A. Zygmund (1935–1945), E.R. Love (1938–1996), A. Erdélyi (1939–1965), H. Kober (1940), D.V. Widder (1941), M. Riesz (1949), and W. Feller (1952). In [LetChe11] A.V. Letnikov’s main results on fractional calculus are presented, including his dissertations and a long discussion between A.V. Letnikov and N.Ya. Sonine on the foundations of fractional calculus. The modern development of the ideas by Letnikov is given and their applications to underground dynamics and population dynamics are presented. However, it may be considered a novel topic as well, since it has only been the subject of specialized conferences and treatises in the last 30 years. The merit is due to B. Ross for organizing the First Conference on Fractional Calculus and its Applications at the University of New Haven in June 1974 and editing the proceedings [Ros75]. For the first monograph the merit is ascribed to K.B. Oldham and J. Spanier [OldSpa74], who, after a joint collaboration starting in 1968, published a book devoted to fractional calculus in 1974. Nowadays, to our knowledge, the list of texts in book form with a title explicitly devoted to fractional calculus (and its applications) includes around ten titles, namely Oldham and Spanier (1974) [OldSpa74], McBride (1979) [McB79], Samko, Kilbas and Marichev (1987–1993) [SaKiMa93], Nishimoto (1991) [Nis91], Miller and Ross (1993) [MilRos93], Kiryakova (1994) [Kir94], Rubin (1996) [Rub96], Podlubny (1999) [Pod99], and Kilbas, Strivastava and Trujillo (2006) [KiSrTr06]. Furthermore, we draw the reader’s attention to the treatises by Davis (1936) [Dav36], Erdélyi (1953–1954) [Bat-1, Bat-2, Bat-3], Gel’fand and Shilov (1959– 1964) [GelShi64], Djrbashian (or Dzherbashian) [Dzh66], Caputo [Cap69], Babenko [Bab86], Gorenflo and Vessella [GorVes91], West, Bologna and Grigolini (2003) [WeBoGr03], Zaslavsky (2005) [Zas05], Magin (2006) [Mag06], which contain a detailed analysis of some mathematical aspects and/or physical applications of fractional calculus. For more details on the historical development of the fractional calculus we refer the interested reader to Ross’ bibliography in [OldSpa74] and to the historical notes generally available in the above quoted texts. In recent years considerable interest in fractional calculus has been stimulated by the applications that it finds in different fields of science, including numerical analysis, economics and finance, engineering, physics, biology, etc. For economics and finance we quote the collection of articles on the topic of Fractional Differencing and Long Memory Processes, edited by Baillie and King (1996), which appeared as a special issue in the Journal of Econometrics [BaiKin96]. For engineering and physics we mention the book edited by Carpinteri and Mainardi [CarMai97], entitled Fractals and Fractional Calculus in Continuum Mechanics, which contains lecture notes of a CISM Course devoted to some 350 E Elements of Fractional Calculus applications of related techniques in mechanics, and the book edited by Hilfer (2000) [Hil00], entitled Applications of Fractional Calculus in Physics,which provides an introduction to fractional calculus for physicists, and collects review articles written by some of the leading experts. In the above books we recommend the introductory surveys on fractional calculus by Gorenflo and Mainardi [GorMai97] and by Butzer and Westphal [ButWes00], respectively. In addition to some books containing proceedings of international conferences and workshops on related topics, see e.g. [McBRoa85,Nis90,RuDiKi94,RuDiKi96], we mention regular journals devoted to fractional calculus, i.e. Journal of Fractional Calculus (Descartes Press, Tokyo), started in 1992, with Editor-in-Chief Prof. Nishimoto and Fractional Calculus and Applied Analysis from 1998, with Editor- in-Chief Prof. Kiryakova (Diogenes Press, Sofia). For information on this journal, please visit the WEB site www.diogenes.bg/fcaa. Furthermore, WEB sites devoted to fractional calculus have also appeared, of which we call attention to www. fracalmo.org, whose name comes from FRActional CALculus MOdelling, and the related WEB links. Appendix F Higher Transcendental Functions

F.1 Hypergeometric Functions

F.1.1 Classical Gauss Hypergeometric Functions Â Ã a; b The hypergeometric function F.aI bI cI z/ D F I z D F .aI bI cI z/ is c 2 1 deﬁned by the Gauss series

P1 .a/ .b/ ab a.a C 1/b.b C 1/ F.aI bI cI z/ D n n zn D 1 C z C z2 C ::: nD0 .c/nnŠ c c.c C 1/2Š .c/ X1 .aC n/ .b C n/ D zn .a/.b/ .cC n/nŠ nD0 (F.1.1) on the disk jzj <1, and by analytic continuation elsewhere. In general, F.aI bI cI z/ does not exist when c D 0; 1; 2;:::. The branch obtained by introducing a cut from 1 to C1 on the real z-axis and by ﬁxing the value F.0/ D 1 is the principal branch (or principal value) of F.aI bI cI z/. For all values of c another type of classical hypergeometric function is deﬁned as

X1 .a/ .b/ 1 F.aI bI cI z/ D n n zn D F.aI bI cI z/; jzj <1; (F.1.2) .cC n/nŠ .c/ nD0 again with analytic continuation for other values of z, and with the principal branch deﬁned in a similar way. In (F.1.2) it is supposed by deﬁnition that if for an index n

© Springer-Verlag Berlin Heidelberg 2014 351 R. Gorenﬂo et al., Mittag-Lefﬂer Functions, Related Topics and Applications, Springer Monographs in Mathematics, DOI 10.1007/978-3-662-43930-2 352 F Higher Transcendental Functions we have c C n as a non-positive integer, then the corresponding term in the series is identically equal to zero (e.g. if c D1, then we omit the term with n D 0 and n D 1). This agreement is natural, because 1= .m/ D 0 for m D 0; 1; 2;:::. On the circle jzjD1, the Gauss series: (a) converges absolutely when Re .c a b/ > 0; (b) converges conditionally when 1

The principal branch of F.aI bI cI z0/ is an entire function of parameters a; b, and c for any fixed values z0; jz0j <1(see [Bat-1, p. 68]). The same is true of other branches, since the series in (F.1.2)convergesforafixedz0; jz0j <1, in any bounded domain of the complex a; b; c-space. As a multi-valued function of z, F.aI bI cI z/ is analytic everywhere except for possible branch points at z D 1,andC1.The same properties hold for F.aI bI cI z/, except that as a function of c, F.aI bI cI z/ in general has poles at c D 0; 1; 2;:::. Because of the analytic properties with respect to a; b,andc, it is usually legitimate to take limits in formulas involving functions that are undefined for certain values of the parameters. For special values of parameters the hypergeometric function coincides with elementary functions

ln.1 z/ .a/ F.1I 1I 2I z/ D ; z Â Ã 1 1 C z .b/ F.1=2I 1I 3=2I z2/ D ln ; 2z 1 z

arctan z .c/ F.1=2I 1I 3=2Iz2/ D ; z

arcsin z .d/ F.1=2I 1=2I 3=2I z2/ D ; z p Á ln z C 1 C z2 .e/ F.1=2I 1=2I 3=2Iz2/ D ; z

.f / F.aI bI bI z/ D .1 z/a;

1 .g/ F.aI 1=2 C aI 1=2I z2/ D .1 C z/2a C .1 z/2a 2 (see also the additional formulas below in Sect. F.5,cf.[NIST]). F.1 Hypergeometric Functions 353

We have the following asymptotic formulas for the hypergeometric function near the branch point z D 1

F.aI bI a C bI z/ .aC b/ .i/ lim D I z!10 ln .1 z/ .a/.b/ Â Ã .c/.c a b/ .ii/ lim .1 z/aCbc F.aI bI cI z/ z!10 .c a/ .c b/ .c/.aC b c/ D ; where Re .c a b/ D 0; c 6D a C bI .a/.b/ .iii/ lim .1 z/aCbc F.aI bI cI z/ z!10 .c/.aC b c/ D ; where Re .c a b/ < 0: .a/.b/

All these formulas can be immediately veriﬁed, they follow from the deﬁnition (F.1.1).

F.1.2 Euler Integral Representation: Mellin–Barnes Integral Representation

If Re c>Re b>0, the hypergeometric function satisﬁes the Euler integral representation (cf., [Bat-1, p. 59])

Z1 .c/ t b1.1 t/cb1 F.aI bI cI z/ D dt: (F.1.3) .b/.c b/ .1 zt/a 0

Here the right-hand side is a single-valued analytic function of z in the domain jarg .1 z/j <. Therefore, this formula determines an analytic continuation of the hypergeometric function F.aI bI cI z/ into this domain too. In order to prove (F.1.3) in the unit disk jzj <1it is sufﬁcient to expand the function .1 zt/a into a binomial series and calculate the series term-by-term by using standard formulas for the Beta function. The Euler integral representation can be rewritten as

.1ZC/ i.c/ei.bc/ t b1.1 t/cb1 F.aI bI cI z/ D dt; .b/.c b/2sin .c b/ .1 zt/a 0 Re b>0;jarg .1 z/j <;c b 6D 1;2;:::I 354 F Higher Transcendental Functions

Z1 i.c/eib t b1.1 t/cb1 F.aI bI cI z/ D dt; .b/.c b/2sin b .1 zt/a .0C/ Re c>Re b>0;jarg .z/j <;b6D 1;2;:::I F.aI bI cI z/ Z i.c/eic t b1.1 t/cb1 D dt; .b/.c b/4sin b sin .c b/ .1 zt/a .1C;0C;1;0/ jarg .z/j <I b; 1 c;b c 6D 1;2;::::

In each case we suppose that the path of integration starts and ends at corresponding points on the Riemann surface of the function t b1.1t/cb1.1zt/a with real t, 0 Ä t Ä 1,wheret b , .1 t/cb mean the principal branches of these functions and .1 zt/a is deﬁned in such a way that .1 zt/a ! 1 whenever z ! 0. The second type of integral representation is the so called Mellin–Barnes representation (see Appendix D)

CZi1 .a/.b/ 1 .aC s/ .b C s/ .s/ F.aI bI cI z/ D .z/s ds; .c/ 2i .cC s/ i1 (F.1.4) where jarg .z/j <and the integration contour separates the poles of the functions .aC s/ and .b C s/ from those of the function .s/,and.z/s assumes its principal values.

F.1.3 Basic Properties of Hypergeometric Functions

In the domain of deﬁnition the hypergeometric function satisﬁes some differential relations

d ab dz F.aI bI cI z/ D c F.aC 1I b C 1I c C 1I z/;

n d .a/n.b/n n F.aI bI cI z/ D F.aC nI b C nI c C nI z/; dz .c/n Á (F.1.5) n c1 cn1 d z F.aI bI cI z/ D .cn/nz F.a nI b nI c nI z/; dzn .1z/cab .1z/cCnab Á n d a1 aCn1 z dz z z F.aI bI cI z/ D .a/nz F.aC nI bI cI z/: F.1 Hypergeometric Functions 355 Á Á n n d d Here the operator z dz z is deﬁned by the operator identity z dz z ./ D n dn n z dzn z ./; n D 1;2;:::.

F.1.4 The Hypergeometric Differential Equation

If the second order homogeneous differential equation has at most three singular points, we can assume that these are the points 0; 1; 1. If all these points are regular (see [Bol90]), then the equation can be reduced to the form

d2w dw z.z 1/ C Œc .a C b C 1/z abw D 0: (F.1.6) dz2 dz

This is the hypergeometric differential equation. It has regular singularities at z D 0; 1; 1, with corresponding exponent pairs f0; 1 cg, f0; c a bg, fa; bg, respectively. Here we use the standard terminology for the complex ordinary differential equation (see, e.g., [NIST, §2.7(i)])

d2w dw C f.z/ C g.z/w D 0: dz2 dz

A point z0 is an ordinary point for this equation if the coefficients f and g are analytic in a neighbourhood of z0. In this case all solutions of the equation are analytic in a neighbourhood of z0. All other points z0 are called singular points (or simply, singularities) for the differential equation. If both .z z0/f .z/ and 2 .z z0/ g.z/ are analytic in a neighbourhood of z0, then this point is a regular singularity. All other singularities are called irregular. An irregular singularity z0 is l 2l of rank l 1 if l is the least integer such that .z z0/ f.z/ and .z z0/ g.z/ are analytic in a neighbourhood of z0. The most common type of irregular singularity for special functions has rank 1 and is located at infinity. In this case the coefficients f , g have series representations

X1 f X1 g f.z/ D s ;g.z/ D s ; zs zs sD0 sD0 where at least one of f0;g0;g1 is non-zero. Regular singularities are characterized by a pair of exponents (or indices) that are roots ˛1;˛2 of the indicial equation

Q.˛/ Á ˛.˛ 1/ C f0˛ C g0 D 0;

2 where f0 D limz!z0 .z z0/f .z/ and g0 D limz!z0 .z z0/ g.z/. Provided that .˛1 ˛2/ 62 Z the differential equation has two linear independent solutions 356 F Higher Transcendental Functions

X1 ˛j s wj D .z z0/ as;j .z z0/ ;jD 1; 2: sD0

In the case of the hypergeometric differential equation, when none of c;c a b; a b is an integer, we have the pair f1.z/, f2.z/ of fundamental solutions. They are also numerically satisfactory ([NIST, §2.7(iv)]) in a neighbourhood of the corresponding singularity.

F.1.5 Kummer’s and Tricomi’s Conﬂuent Hypergeometric Functions

Kummer’s differential equation

d2w dw z C .c z/ aw D 0 (F.1.7) dz2 dz has a regular singularity at the origin with indices 0 and 1 b, and an irregular singularity at infinity of rank one. It can be regarded as the limiting form of the hypergeometric differential equation (F.1.6) that is obtained on replacing z by z=b, letting b !1, and subsequently replacing the symbol c by b. In effect, the regular singularities of the hypergeometric differential equation at b and 1 coalesce into an irregular singularity at 1. Equation (F.1.7) is called the confluent hypergeometric equation, and its solutions are called confluent hypergeometric functions. Two standard solutions to Eq. (F.1.7) are the following:

X1 .a/ a a.a C 1/ M.aI cI z/ D n zn D 1 C z C z2 C ::: (F.1.8) .c/ nŠ c c.c C 1/2Š nD0 n and

X1 .a/ 1 a a.a C 1/ M.aI cI z/ D n zn D C z C z2 C :::: .cC n/nŠ .c/ .cC 1/ .cC 2/2Š nD0 (F.1.9)

The ﬁrst of these functions M.aI cI z/ does not exist if c is a non-positive integer. For all other values of parameters the following identity holds:

M.aI cI z/ D .c/M.aI cI z/:

The series (F.1.8)and(F.1.9) converge for all z 2 C. M.aI cI z/ is an entire function in z and a, and is meromorphic in c. M.aI cI z/ is an entire function in z;a, and c. F.1 Hypergeometric Functions 357

The function M.aI cI z/ is known as the Kummer conﬂuent hypergeometric function. Sometimes the notation

M.aI cI z/ D 1F1.aI cI z/ is used, which is also Humbert’s symbol [Bat-1, p. 248]

M.aI cI z/ D ˚.aI cI z/:

Another standard solution to the conﬂuent hypergeometric equation (F.1.7)isthe function U.aI cI z/ which is determined uniquely by the property

U.aI cI z/ za; z !1; jarg zj < ı: (F.1.10)

Here ı is an arbitrary small positive constant. In general, U.aI cI z/ has a branch point at z D 0. The principal branch corresponds to the principal value of za in (F.9), and has a cut in the z-plane along the interval .1I 0. The function U.aI cI z/ was introduced by Tricomi (see, e.g., [Bat-1, p. 257]). Sometimes it is called the Tricomi conﬂuent hypergeometric function. It is related to the Erdélyi function 2F0.aI cI z/

a 2F0.aI cI1=z/ D z U.aI a c C 1I z/ and the Kummer conﬂuent hypergeometric function

.1 c/ .c 1/ U.aI cI z/ D M.aI cI z/ C z1aM.a c C 1I 2 cI z/: .a c C 1/ .a/

The following notation (Humbert’s symbol) is used too (see, e.g., [Bat-1,p.255]):

U.aI cI z/ D .aI cI z/:

Integral Representations

Two main types of integral representations for conﬂuent hypergeometric functions can be mentioned here. First of all these are representations analogous to the Euler integral representation of the classical hypergeometric function. The integral representation of Kummer’s conﬂuent hypergeometric function

Z1 .c/ M.aI cI z/ D etzt a1.1 t/ca1dt; Re c>Re a>0; .a/.c a/ 0 (F.1.11) can be immediately veriﬁed by expanding the exponential function etz. 358 F Higher Transcendental Functions

The formula

Z1 1 etzt a1.1 C t/ca1dt; Re a>0; (F.1.12) .a/ 0 gives the solution of the differential equation (F.1.7) in the right half-plane Re z >0 and coincides in this domain with U.aI cI z/. The analytic continuation of (F.1.12) yields the integral representation of Tricomi’s conﬂuent hypergeometric function U.aI cI z/. Another type of integral representation uses Mellin–Barnes integrals. For conﬂu- ent hypergeometric functions this type of integral representation has the form

ZCi1 1 .c/ .s/ .a C s/ M.aI cI z/ D .z/sds; (F.1.13) 2i .a/ .cC s/ i1 jarg .z/j <=2; Re a<<0;c6D 0; 1; 2; : : : I

ZCi1 1 .s/ .a C s/ .1 c s/ U.aI cI z/ D .z/s ds; (F.1.14) 2i .cC s/ .a c C 1/ i1 jarg .z/j <3=2; Re a<

The conditions on the parameters in (F.1.13)andin(F.1.14) can be relaxed by suitable deformation of the contour of integration. Thus, (F.1.13) is valid with any whenever a is not a non-negative integer, provided that the contour of integration separates the poles of .s/ from the poles of .aC s/. Similarly, (F.1.14) is valid with any whenever neither a nor a c C 1 is a non-negative integer, provided that the contour of integration separates the poles of .s/ .1 c s/ from the poles of .aC s/. The conditions on arg z cannot be relaxed.

Asymptotics

As z !1then (see, e.g., [NIST, p. 328])

ezzac X1 .1 a/ .c a/ M.aI cI z/ n n zn .a/ nŠ nD0 1 e˙iaza X .a/ .a c C 1/ C n n .z/n; (F.1.15) .c a/ nŠ nD0 =2 C ı<˙arg z <3=2 ı; a 6D 0; 1; 2;:::I c a 6D 0; 1; 2;:::I F.1 Hypergeometric Functions 359

X1 .a/ .a c C 1/ U.aI cI z/ za n n .z/n; (F.1.16) nŠ nD0 jarg zj <3=2 ı; with ı being sufﬁciently small positive:

Other asymptotic formulas with respect to large values of the variable z and/or with respect to large values of the parameters a and c can be found in [NIST, pp. 330–331].

Relation to Elementary and Other Special Functions

In the following special cases the conﬂuent hypergeometric functions coincide with elementary functions:

M.aI aI z/ D ezI ez M.1I 2I 2z/ D sinh zI z M.0I cI z/ D U.0I cI z/ D 1I U.aI a C 1I z/ D zaI with incomplete gamma function .a;z/ (in the case when a c is an integer or a is a positive integer):

M.aI a C 1Iz/ D ezM.1I a C 1I z/ D aza.a;z/I with the error functions erf and erfc: p M.1=2I 3=2Iz2/ D erf.z/I 2z p 2 U.1=2I 1=2I z2/ D ez erfc.z/I with the orthogonal polynomials:

– With the Hermite polynomials H.z/:

nŠ M.nI 1=2I z2/ D .1/n H .z/I .2n/Š 2n nŠ M.nI 3=2I z2/ D .1/n H .z/I .2n C 1/Š2z 2nC1 2n U.1=2 n=2I 3=2I z2/ D H .z/I z n 360 F Higher Transcendental Functions

.˛/ – With the Laguerre polynomials L .z/:

n n .˛/ U.nI ˛ C 1I z/ D .1/ .˛ C 1/nM.nI ˛ C 1I z/ D .1/ nŠLLn .z/I

with the modiﬁed Bessel functions I;K (in the case when c D 2b):

z M. C 1=2I 2 C 1I z/ D .C 1/e .z=2/ I.z/I

1 z U. C 1=2I 2 C 1I z/ D p e .2z/ K.z/:

F.1.6 Generalized Hypergeometric Functions and Their Properties Â Ã a Generalized hypergeometric functions F (or F I z D F .a; bI z/ D p q p q b p q pFq.a1;:::;apI b1;:::;bq I z/) are deﬁned by the following series (where none of the parameters bj is a non-positive integer): Â Ã X1 n a1;a2;:::ap .a1/n.a2/n :::.ap/n z pFq I z D : (F.1.17) b1;b2;:::bq .b / .b / :::.b / nŠ nD0 1 n 2 n q n This series converges for all z whenever p0, convergent except at z D 1 if 1 q C1. However, when one or more of the top parameters aj is a non-positive integer the series terminates and the generalized hypergeometric function is a polynomial in z. Note that if m D maxfa1;:::;aq g is a positive integer, then the following identity holds: Â Ã Â Ã m pCqC1 m; a .a/m.z/ m; 1 m b .1/ pC1Fq I z D qC1Fp I ; b .b/m 1 m a z which can be used to interchange a and b. F.2 Wright Functions 361

If p Ä q D 1 and z is not a branch point of the generalized hypergeometric function pFq , then the function Â Ã Â Ã X1 n a 1 a .a1/n :::.ap /n z F I z D F I z D p q b .b /:::.b / p q b .b C n/:::.b C n/ nŠ 1 q nD0 1 q is an entire function of each parameter a1;:::;ap;b1;:::;bq.

F.2 Wright Functions

F.2.1 The Classical Wright Function

In this section we present some deﬁnitions and properties of the so-called Wright functions given in [Bat-1, Section 4.1], [Bat-3, Section 18.1], [KiSrTr06, Section 1.11], and [Mai10, Appendix F]. The simplest Wright function .˛;ˇI z/ is deﬁned for z;˛;ˇ 2 C by the series 2 3 ˇ ˇ X1 k 4 ˇ 5 1 z .˛;ˇI z/ D 01 ˇ z WD : (F.2.1) .˛kC ˇ/ kŠ .ˇ; ˛/ kD0

If ˛>1, this series is absolutely convergent for all z 2 C, while for ˛ D1 it is absolutely convergent for jzj <1and for jzjD1 and Re.ˇ/ > 1;see [KiSrTr06, Sec. 1.11]. Moreover, for ˛>1, .˛;ˇI z/ is an entire function of z. Using formulas for the order (B.2.3) and the type (B.2.4) of an entire function and Stirling’s asymptotic formula for the Gamma function (A.1.24) one can deduce 1 that for ˛>1 the Wright function .˛;ˇI z/ has order D and type ˛ C 1 1 1 D .˛ C 1/˛ .˛ C 1/ D ˛ .

The function .˛;ˇI z/ was introduced by Wright [Wri33],whoin[Wri34]and [Wri40b] investigated its asymptotic behaviour at inﬁnity provided that ˛>1. When ˛ D 1 and ˇ D C 1, the function .1; C 1I˙z2=4/ is expressed in terms of the Bessel functions J.z/ and I.z/, given in Appendix G by formulas (G.1) and (G.10): Â Ã Â Ã Â Ã Â Ã z2 2 z2 2 1; C 1I D J .z/; 1; C 1I D I .z/: (F.2.2) 4 z 4 z 362 F Higher Transcendental Functions

F.2.2 Mellin–Barnes Integral Representation and Asymptotics

The integral representation of (F.2.1) in terms of the Mellin–Barnes contour integral is given by Z 1 .s/ .˛;ˇI z/ D .z/s ds; (F.2.3) 2i L .ˇ ˛s/ where the path of integration L separates all the poles at s Dk.k2 N0/ to the left. If L D . i1; C i1/. 2 R/, then the representation (F.2.3) is valid if either of the following conditions holds:

.1 ˛/ 0<˛<1; jarg.z/j < ; z ¤ 0 (F.2.4) 2 or

˛ D 1; Re.ˇ/ > 1 C 2; arg.z/ D 0; z ¤ 0: (F.2.5)

The relation (F.2.3) means that the Mellin transform of (F.2.1)isgivenbythe formula .s/ MŒ.˛; ˇI t/.s/ D .Re.s/ > 0/: (F.2.6) .ˇ ˛s/

The Laplace transform of (F.2.1) is expressed in terms of the Mittag-Lefﬂer function: Â Ã 1 1 LŒ.˛; ˇI t/.s/ D E .˛ > 1I ˇ 2 CI Re.s/ > 0/; (F.2.7) s ˛;ˇ s see [Bat-3, Section 18.2]. If z 2 C and jarg.z/jÄ .0< </, then the asymptotic behaviour of .˛;ˇI z/ at inﬁnity is given by

.˛;ˇI z/ ÄÂ Ã " Â Ã !# 1 1 1=.1C˛/ D a .˛z/.1ˇ/=.1C˛/exp 1 C .˛z/1=.1C˛/ 1 C O ; 0 ˛ z (F.2.8)

1=2 where a0 D Œ2.˛ C 1/ .z !1/. The following differentiation formula for .˛;ˇI z/ is a direct consequence of the deﬁnition (F.2.1): Â Ã d n .˛;ˇI z/ D .˛;˛ C nˇI z/.n2 N/: (F.2.9) dz F.2 Wright Functions 363

F.2.3 The Bessel–Wright Function: Generalized Wright Functions and Fox–Wright Functions

When ˛ D , ˇ D C1,andz is replaced by z, the function .˛;ˇI z/ is denoted by J .z/:

X1 1 .z/k J .z/ WD .; C 1Iz/ D ; (F.2.10) .kC C 1/ kŠ kD0 and this function is known as the Bessel–Wright function, or the Wright generalized Bessel function,1 see [PrBrMa-V2]and[Kir94, p. 352]. When D 1, the Bessel function of the ﬁrst kind (G.1) is connected with (F.2.10)by Á Â Ã z z2 J .z/ D J 1 : (F.2.11) 2 4

p If D q is a rational number then [Kir94, p. 352] the Bessel–Wright function satisﬁes the following differential equation of order .p C q/ 2 3 Â Ã .z/q Y 1 d 4 p C q z d 5 J .z/ D 0; (F.2.12) qqpp q dz j j D1 where ( j q ;1Ä j Ä q 1; dj D j CC1q 1 q ;qÄ j Ä q C p:

The original differential equation for J .z/, equivalent to (F.2.12), was formulated by Wright [Wri33]: Â Ã Â Ã p q d Cp d .1/qz=J .z/ D z11= z J .z/: dz dz

There exists a further generalization of the Bessel–Wright function (the so-called generalized Bessel–Wright function) given in [Pat66, Pat67]: Á 1 2k z C2 X .1/k z J .z/ D 2 ;>0: (F.2.13) ; 2 .C k C C 1/ . C k C 1/ kD0

1Also misnamed the Bessel–Maitland function after E.M. Wright’s second name Maitland. 364 F Higher Transcendental Functions

This function generates the Lommel function, and is a special case of the four- parametric Mittag-Lefﬂer function. The more general function pq .z/ is deﬁned for z 2 C, complex al ;bj 2 C,and real ˛l , ˇj 2 R .l D 1; ;pI j D 1; ;q/by the series 2 3 ˇ Q .a ;˛ / 1 p l l 1;p ˇ X .a C ˛ k/ k 4 ˇ 5 Q lD1 l l z pq .z/ D pq ˇ z WD q (F.2.14) j D1 .bj C ˇj k/ kŠ .bl ;ˇl /1;q kD0

.z;al ;bj 2 CI ˛l ;ˇj 2 RI l D 1; ;pI j D 1; ;q/:

This general (Wright or, more appropriately, Fox–Wright) function was investigated by Fox [Fox28] and Wright [Wri35b,Wri40b,Wri40c], who presented its asymptotic expansion for large values of the argument z under the condition

Xq Xp ˇj ˛l > 1: (F.2.15) j D1 lD1

If these conditions are satisﬁed, the series in (F.2.14) is convergent for any z 2 C. This result follows from the assertion [KiSrTr06, Theor. 1.]:

Theorem F.1. Let al ;bj 2 C and ˛l , ˇj 2 R .l D 1; pI j D 1; ;q/and let

Xq Xp D ˇj ˛l ; (F.2.16) j D1 lD1 Yp Yq ˛l ˇj ı D j˛l j jˇj j ; (F.2.17) lD1 j D1 and

q p X X p q D b a C : (F.2.18) j l 2 j D1 lD1

(a) If >1, then the series in (F.2.14) is absolutely convergent for all z 2 C. (b) If D1, then the series in (F.2.14) is absolutely convergent for jzj <ıand for jzjDı and Re./ > 1=2.

When ˛l ;ˇj 2 R .l D 1; ;pI j D 1; ;q/, the generalized Wright function pq.z/ has the following integral representation as a Mellin–Barnes contour integral: " # Q ˇ Z p .al ;˛l /1;p ˇ 1 .s/ .a ˛ s/ ˇ Q lD1 l l s pq ˇ z D q .z/ ds; (F.2.19) 2i C .bl ;ˇl /1;q j D1 .bj ˇj s/ F.3 Meijer G-Functions 365 where the path of integration C separates all the poles at s Dk.k2 N0/ to the left and all the poles s D .al C nl /=˛l .l D 1; ;pI nl 2 N/ to the right. If C D . i1;C i1/. 2 R), then representation (F.2.19) is valid if either of the following conditions holds:

.1 / <1; jarg.z/j < ; z ¤ 0 (F.2.20) 2 or 1 D 1; . C 1/ C < Re./; arg.z/ D 0; z ¤ 0: (F.2.21) 2 Conditions for the representation (F.2.19) were also given for the case when C D L1 .L1/ is a loop situated in a horizontal strip starting at the point 1 C i'1 .1Ci'1/ and terminating at the point 1 C i'2 .1Ci'2/ with 1 <'1 < '2 < 1. If we put ˛l D 1; 1 Ä l Ä p,andˇl D 1; 1 Ä l Ä q, in representation (F.2.14) then we get the following relation of the generalized Wright function pq with the generalized hypergeometric function pFq : " # Q ˇ p .al ;1/1;p ˇ .a / ˇ QlD1 l pq ˇ z D q pFq .a1;:::;ap I b1;:::;bqI z/: (F.2.22) .bl ;1/1;q lD1 .bl /

F.3 Meijer G-Functions

F.3.1 Deﬁnition via Integrals: Existence

In [Mei36] more general classes of transcendental functions were introduced via generalization of the Gauss hypergeometric functions presented in the form of series (commonly known now as Meijer G-functions): Ä ˇ ˇ m;n ˇ a1;:::;ap Gp;q z ˇ : (F.3.1) b1;:::;bq

This deﬁnition is due to the relation between the generalized hypergeometric function and the Meijer G-functions (see, e.g., [Sla66, p. 42]): Ä ˇ ˇ m;n ˇ 1 a1;:::;1 ap .a1/:::.ap/ Gp;q z ˇ D pFq .aI bI z/: (F.3.2) 0; 1 b1;:::;1 bq .b1/:::.bq/ 366 F Higher Transcendental Functions

Later this deﬁnition was replaced by the Mellin–Barnes representation of the G-function Ä ˇ Z ˇ 1 m;n ˇ a1;:::;ap Gm;n s Gp;q z ˇ D p;q .s/z ds; (F.3.3) b1;:::;bq 2i L where L is a suitably chosen path, z 6D 0, zs WD expŒs.ln jzjCiargz/ with a single valued branch of arg z, and the integrand is deﬁned as Q Q m .b s/ n .1 a C s/ Gm;n Q kD1 k j DQ1 j p;q .s/ D q p : (F.3.4) kDmC1 .1 bk C s/ j DnC1 .aj s/

In (F.3.4) the empty product is assumed to be equal to 1, parameters m; n; p; q satisfy the relation 0 Ä n Ä q, 0 Ä n Ä p, and the complex numbers aj , bk are such that no pole of .bk s/;k D 1;:::;m, coincides with a pole of .1aj Cs/;j D 1;:::;n.

F.3.2 Basic Properties of the Meijer G-Functions

10. Symmetry. The G-functions are symmetric with respect to their parameters in the following sense: if the value of a parameter from the group .a1;:::;an/ (respectively, from the group .anC1;:::;ap/) is equal to the value of a parameter from the group .bmC1;:::;bq/ (respectively, to the value of a parameter from the group .b1;:::;bm/), then these parameters can be excluded from the deﬁnition of the G- function, i.e. the “order” of the G-function decreases. For instance, if a1 D bq, then Ä ˇ Ä ˇ ˇ ˇ m;n ˇ a1;:::;ap m;n1 ˇ a2;:::;ap Gp;q z ˇ D Gp1;q1 z ˇ : (F.3.5) b1;:::;bq b1;:::;bq1

20. Shift of parameters. The two properties below indicate how to “shift” parameters. They follow from the change of variable in the deﬁnition of the G-functions (F.3.3). Ä ˇ Ä ˇ ˇ ˇ m;n ˇ a1;:::;ap m;n ˇ a1 C ;:::;ap C z Gp;q z ˇ D Gp;q z ˇ I (F.3.6) b1;:::;bq b1 C ;:::;bq C Ä ˇ Ä ˇ ˇ ˇ m;n ˇ a1;:::;ap m;n 1 ˇ 1 b1;:::;1 bq Gp;q z ˇ D Gp;q ˇ : (F.3.7) b1;:::;bq z 1 a1;:::;1 ap

30. Multiplication formula. F.3 Meijer G-Functions 367

For any natural number r 2 N the following formula is valid: Ä ˇ ˇ m;n ˇ a1;:::;ap Gp;q z ˇ b1;:::;bq Ä ˇ zr ˇ c ;:::;c ;:::;c ;:::;c D .2/urvGmr;nr ˇ 1;1 1;r p;1 p;r ; pr;qr r.qp/ ˇ (F.3.8) r d1;1;:::;d1;r ;:::;dq;1;:::;dq;r where

a C l 1 b C l 1 c D j ;.lD 1;:::;r/I d D k ;.lD 1;:::;r/: j;l r k;l r

F.3.3 Special Cases

The basic elementary functions can be represented as G-function with special values of parameters. They mostly follow from the relation between the generalized hypergeometric function pFq and the Meijer G-function (F.3.2). Let us recall some of these representations.

0;1 exp z D G1;0 Œz j0I (F.3.9) Ä ˇ 2 ˇ p 1;0 z ˇ1 sin z D G ˇ ;0 I (F.3.10) 0;2 4 2 Ä ˇ 2 ˇ p 1;0 z ˇ 1 cos z D G ˇ0; I (F.3.11) 0;2 4 2 Ä ˇ ˇ 1;2 ˇ 1; 1 log .1 C z/ D G z ˇ .jzj <1/I (F.3.12) 2;2 1; 0 Ä ˇ ˇ 1 1;2 2 ˇ 1; 1 arcsin z D p G2;2 z ˇ 1 .jzj <1/I (F.3.13) 2 2 ;0 Ä ˇ 1 ˇ 1 1;2 2 ˇ 1; 2 arctan z D G2;2 z ˇ 1 .jzj <1/I (F.3.14) 2 2 ;0 Ä ˇ ˇ 1;0 ˇ C 1 z D G z ˇ .jzj <1/I (F.3.15) 1;1 Ä ˇ ˇ ˛ 1 1;1 ˇ 1 ˛ .1 ˙ z/ D G z ˇ .jzj <1;Re ˛>0/I (F.3.16) .˛/ 1;1 0 Ä ˇ ˇ ˛ 1;0 ˇ 1 ˛ .1 ˙ z/ D .1 ˛/G z ˇ .jzj <1;Re ˛<1/: (F.3.17) 1;1 0 368 F Higher Transcendental Functions

F.3.4 Relations to Fractional Calculus

Let us consider the Riemann–Liouville fractional integral (see (E.1.3)) Z x ˛ 1 ˛1 IaC .x/ WD .x / ./d; a0: .˛/ a

It is well-known (see, e.g. [SaKiMa93, Ch. 4]) that this integral can be extended into the complex domain. For this we ﬁx a point z 2 C and introduce the multi-valued function

.z /˛1:

By ﬁxing an arbitrary single-valued branch of this function in the complex plane cut along the line L starting at D a and ending at D1and containing the point D z we deﬁne the Riemann–Liouville fractional integral in the complex domain

Zz 1 I ˛ f.z/ D .z /˛1 f. /d ; (F.3.18) aC .˛/ a where the integration is performed along a part of the above-described contour (cut) L starting at D a and ending at D z. Analogously, the so-called “right- sided” Riemann–Liouville fractional integral (or Weyl type fractional integral) in the complex domain is deﬁned by

Z1 1 I ˛ f.z/ D .z /˛1 f. /d : (F.3.19) .˛/ z

The density f in both formulas (F.3.18)and(F.3.19)isassumedtobedeﬁnedin a neighbourhood of the contour L. If this function is analytic in the whole plane, then by the Cauchy Integral Theorem one can replace the contour L by the straight interval Œa; z andbytherayŒz; 1/, respectively. Let us present the formulas which show that the Riemann–Liouville fractional integration of the Meijer G-function yields the Meijer G-function with other values of parameters (see, e.g. [Kir94, p. 318], [PrBrMa-V3]).

Zz Ä ˇ ˇ ˛ m;n 1 ˛1 m;n ˇ a1;:::;ap IaC Gp;q .z/ D .z / Gp;q ˇ d .˛/ b1;:::;bq a Z1 Ä ˇ ˛1 ˇ ˛ .1 / m;n ˇ a1;:::;ap D z Gp;q z ˇ d .˛/ b1;:::;bq 0 F.3 Meijer G-Functions 369 Ä ˇ ˇ ˛ m;nC1 ˇ 0; a1;:::;ap D z GpC1;qC1 ˇ : (F.3.20) b1;:::;bq; ˛

This formula is valid when Re bk > 1; k D 1;:::;m,andp Ä q.Ifp D q, then it is valid for jzj <1Á .Forp C q<2.mC n/ we require additionally that pCq jarg zj < m C n 2 .

Z1 Ä ˇ ˇ ˛ m;n 1 ˛1 m;n ˇ a1;:::;ap I Gp;q .z/ D .z / Gp;q ˇ d .˛/ b1;:::;bq z Z1 Ä ˇ ˛1 ˇ ˛ . 1/ m;n ˇ a1;:::;ap D z Gp;q z ˇ d .˛/ b1;:::;bq 1 Ä ˇ ˇ ˛ mC1;n ˇ a1;:::;ap;0 D z GpC1;qC1 ˇ : (F.3.21) ˛; b1;:::;bq

This formula is valid when 01.Forp C q<2.mC n/ we require additionally pCq that jarg zj < m C n 2 .

F.3.5 Integral Transforms of G-Functions

The structure of the Meijer G-function is very similar to that of the Mellin integral transform. Hence, the Mellin transform of the Meijer G-function is up to a power factor equal to the ratio of the products of -functions (see, e.g., [BatErd54b, p. 301], [Kir94,p.318]).

Z1 Ä ˇ Á ˇ M m;n s1 m;n ˇ a1;:::;ap Gp;q .t/ .s/ D t Gp;q t ˇ dt b1;:::;bq 0 Qm Qn .bk C s/ .1 aj s/ sGm;n s kD1 j D1 D p;q .s/ D Qq Qp ; (F.3.22) .1 bk s/ .aj C s/ kDmC1 j DnC1 1 where, e.g., p Cq<2.mCn/; jarg j < m C n .p C q/ , min Re bk < 2 1ÄkÄm Re <1 max Re aj . Other conditions under which formula (F.3.22) is valid 1Äj Än can be found in [Luk1, pp. 157–159] and [BatErd54b, pp. 300–301]. We also mention several formulas for the Mellin integral transform presented in [BatErd54a, pp. 295–296]. 370 F Higher Transcendental Functions

The Laplace transform of the Meijer G-function can be determined by the relation:

Z1 Ä ˇ Á ˇ L m;n tx m;n ˇ a1;:::;ap Gp;q .x/ .t/ D e Gp;q x ˇ dx b1;:::;bq 0 Ä ˇ ˇ 1 m;nC1 ˇ 0; a1;:::;ap D t GpC1;q ˇ ; (F.3.23) t b1;:::;bq 1 where, e.g., p C q<2.mC n/; jarg j < m C n 2 .p C q/ , jarg tj <=2, Re bk 1; k D 1;:::;m. More general formulas for the Laplace transform of the Meijer G-function with different weights can also be found in [Luk1, pp. 166–169] and [BatErd54b, p. 302].

F.4 Fox H -Functions

F.4.1 Deﬁnition via Integrals: Existence

A straightforward generalization of the Meijer G-functions are the so-called Fox H-functions, introduced and studied by Fox in [Fox61]. According to a standard notation the Fox H-functions are deﬁned by Ä ˇ Z ˇ 1 m;n m;n ˇ .a1;˛1/;:::;.ap ;˛p/ Hm;n s Hp;q .z/ D Hp;q z ˇ D p;q .s/ z ds; .b1;ˇ1/;:::;.bq;ˇq / 2i L (F.4.1) where L is a suitable path in the complex plane C to be found later on, zs D expfs.log jzjCi argz/g,and

A.s/B.s/ Hm;n.s/ D ; (F.4.2) p;q C.s/D.s/ Ym Yn A.s/ D .bj ˇj s/; B.s/ D .1 aj C ˛j s/; (F.4.3) j D1 j D1 Yq Yp C.s/ D .1 bj C ˇj s/; D.s/ D .aj ˛j s/; (F.4.4) j DmC1 j DnC1

C with 0 Ä n Ä p, 1 Ä m Ä q, faj ;bj g2C, f˛j ;ˇj g2R .Asusual,anempty product, when it occurs, is taken to be equal to 1. Hence

n D 0 , B.s/ D 1; mD q , C.s/ D 1; nD p , D.s/ D 1: F.4 Fox H -Functions 371

Due to the occurrence of the factor zs in the integrand of (D.1.1), the H-function is, in general, multi-valued, but it can be made one-valued on the Riemann surface of log z by choosing a proper branch. We also note that when the ˛sandˇs are equal m;n to 1, we obtain the Meijer G-functions Gp;q .z/, thus the Meijer G-functions can be considered as special cases of the Fox H-functions: Ä ˇ Ä ˇ ˇ ˇ m;n ˇ .a1;1/;:::;.ap ;1/ m;n ˇ a1;:::;ap Hp;q z ˇ D Gp;q z ˇ : .b1;1/;:::;.bq ;1/ b1;:::;bq

The above integral representation of H-functions in terms of products and ratios of Gamma functions is known to be of Mellin–Barnes integral type. A compact notation is usually adopted for (F.4.1): Ä ˇ ˇ m;n m;n ˇ.aj ;˛j /j D1;:::;p Hp;q .z/ D Hp;q z ˇ : (F.4.5) .bj ;ˇj /j D1;:::;q

The singular points of the kernel H are the poles of the Gamma functions appearing in the expressions of A.s/ and B.s/, that we assume do not coincide. Denoting by P.A/ and P.B/ the sets of these poles, we write P.A/ \ P.B/ D;. The conditions for the existence of the H-functions can be determined by inspecting the convergence of the integral (F.4.1), which can depend on the selection of the contour L and on certain relations between the parameters fai ;˛i g (i D 1;:::;p) and fbj ;ˇj g (j D 1;:::;q). For the analysis of the general case we refer to the specialized treatises on H-functions, e.g., [MatSax73, MaSaHa10, SrGuGo82] and, in particular to the paper by Braaksma [Bra62], where an exhaustive discussion on the asymptotic expansions and analytical continuation of these functions can be found, see also [KilSai99]. In the following we limit ourselves to recalling the essential properties of the H-functions, preferring to analyze later in detail those functions related to fractional diffusion. As will be shown later, this phenomenon depends on one real independent variable and three parameters; in this case we shall have z D x 2 R and m Ä 2, n Ä 2, p Ä 3, q Ä 3. The contour L in (F.4.1) can be chosen as follows:

(i) L D Li1 chosen to go from i1 to Ci1 leaving to the right all the poles of P.A/, namely the poles sj;k D .bj Ck/=ˇj ; j D 1;2;:::;m; k D 0; 1; : : : of the functions appearing in A.s/, and to left all the poles of P.B/, namely the poles sj;l D .aj 1l/=ˇj ; j D 1;2;:::;n; l D 0; 1; : : : of the functions appearing in B.s/. (ii) L D LC1 is a loop beginning and ending at C1 and encircling once in the negative direction all the poles of P.A/, but none of the poles of P.B/. (iii) L D L1 is a loop beginning and ending at 1 and encircling once in the positive direction all the poles of P.B/, but none of the poles of P.A/. Braaksma has shown that, independently of the choice of L, the Mellin–Barnes integral makes sense and deﬁnes an analytic function of z in the following two cases 372 F Higher Transcendental Functions

Xq Xp >0;0

Via the following useful and important formula for the H-function Ä ˇ Ä ˇ ˇ ˇ m;n ˇ.aj ;˛j /1;p n;m 1 ˇ .1 bj ;ˇj /1;q Hp;q z ˇ D Hq;p ˇ (F.4.8) .bj ;ˇj /1;q z .1 aj ;˛j /1;p we can transform the H-function with <0and argument z to one with >0and argument 1=z. This property is useful when comparing the results of the theory of H-functions based on (F.4.1)usingzs with the theory that uses zs, which is often found in the literature. More detailed information on the existence of the H-function is presented, e.g., in [KilSai04](seealso[PrBrMa-V3]). In order to formulate this result we introduce a set of auxiliary parameters. Let m; n; p; q; ˛j ;aj ;ˇk ;bk (j D 1;:::;p; k D 1;:::;q). Deﬁne 8 ˆ Pn Pp Pm Pq ˆ a D ˛ ˛ C ˇ ˇ I ˆ j j k k ˆ j D1 j DnC1 kD1 kDmC1 ˆ Pm Pp ˆ ˆ a1 D ˇk ˛j I ˆ kD1 j DnC1 ˆ Pn Pq ˆ ˆ a D ˛j ˇkI ˆ 2 ˆ j D1 kDmC1 <ˆ Pq Pp D ˇk ˛j I ˆ kD1 j D1 (F.4.9) ˆ Qp Qq ˆ ˛j ˇk ˆ ı D ˛j ˇk I ˆ j D1 kD1 ˆ Pq Pp ˆ 1 ˆ D bk aj C .p q/I ˆ 2 ˆ kD1 j D1 ˆ Pm Pq Pn Pp ˆ D b b C a a I ˆ k k j j :ˆ kD1 kDmC1 j D1 j DnC1 1 c D m C n 2 .p C q/:

Theorem F.2 ([KilSai04, pp. 4–5]). Let the parameters a;;ı; be deﬁned as m;n in (F. 4 . 9 ). Then the Fox H-function Hp;q .z/ (as deﬁned in (F. 4 . 1 )–(F. 4 . 4 )) exists in the following cases:

L L m;n If D 1;>0; then Hp;q .z/ exists for all z W z 6D 0: (F.4.10) F.4 Fox H -Functions 373

L L m;n If D 1;D 0; then Hp;q .z/ exists for all z W 0ı: (F.4.14) L L m;n If D C1;D 0; Re <1; then Hp;q .z/ exists for z WjzjDı: (F.4.15) a If L D L ;a >0; then H m;n.z/ exists for all z Wjarg zj < ; z 6D 0: i1 p;q 2 (F.4.16) L L m;n If D i1;a D 0; C Re <1; then Hp;q .z/ exists for all z W arg z D 0; z 6D 0: (F.4.17)

Other important properties of the Fox H-functions, that can easily be derived from their deﬁnition, are included in the list below. (i) The H-function is symmetric in the set of pairs .a1;˛1/;:::;.an;˛n/, .anC1;˛nC1/;:::;.ap;˛p / and .b1;ˇ1/;:::;.bm;ˇm/, .bmC1;ˇmC1/;:::;.bq;ˇq /. (ii) If one of the .aj ;˛j /; j D 1;:::;n, is equal to one of the .bk;ˇk/; k D m C 1;:::;q; [or one of the pairs .aj ;˛j /; j D n C 1;:::;p, is equal to one of the .bk;ˇk/; k D 1;:::;m], then the H-function reduces to one of the lower order, that is, p; q and n [or m] decrease by unity. Provided n 1 and q>m,wehave Ä ˇ Ä ˇ ˇ ˇ m;n ˇ .aj ;˛j /1;p m;n1 ˇ .aj ;˛j /2;p Hp;q z ˇ D Hp1;q1 z ˇ ; .bk;ˇk /1;q1 .a1;˛1/ .bk;ˇk/1;q1 (F.4.18) Ä ˇ Ä ˇ ˇ ˇ m;n ˇ.aj ;˛j /1;p1 .b1;ˇ1/ m1;n ˇ.aj ;˛j /1;p1 Hp;q z ˇ D Hp1;q1 z ˇ : .b1;ˇ1/.bk;ˇk/2;q .bk;ˇk/2;q (F.4.19)

(iii) Ä ˇ Ä ˇ ˇ ˇ m;n ˇ.aj ;˛j /1;p m;n ˇ.aj C ˛j ;˛j /1;p z Hp;q z ˇ D Hp;q z ˇ : (F.4.20) .bk;ˇk/1;q .bk C ˇk ;ˇk/1;q

(iv) Ä ˇ Ä ˇ ˇ ˇ 1 m;n ˇ.aj ;˛j /1;p m;n c ˇ.aj ;c˛j /1;p Hp;q z ˇ D Hp;q z ˇ ;c>0: (F.4.21) c .bk;ˇk/1;q .bk;cˇk/1;q 374 F Higher Transcendental Functions

The convergent and asymptotic expansions (for z ! 0 or z !1) are mostly obtained by applying the residue theorem in the poles (assumed to be simple) of the Gamma functions appearing in A.s/ or B.s/ that are found inside the specially chosen path. In some cases (in particular if n D 0 , B.s/ D 1)wefindan exponential asymptotic behaviour. In the presence of a multiple pole s0 of order N the treatment becomes more cumbersome because we need to expand in a power series at the pole the product s N 1 of the involved functions, including z , and take the first N terms up to .s s0/ N 1 inclusive. Then the coefficient of .s s0/ is the required residue. Let us consider the case N D 2 (double pole) of interest for the fractional diffusion. Then the expansions for the Gamma functions are of the form 2 .s/D .s0/ 1 C .s0/.s s0/ C O .s s0/ ;s! s0 ;s0 ¤ 0; 1; 2;::: .1/k .s/D 1 C .k C 1/.s C k/ C O .s C k/2 ;s!k; .kC 1/.s C k/ where k D 0; 1; 2; : : : and .z/ denotes the logarithmic derivative of the function,

d 0.z/ .z/ D log .z/ D ; dz .z/ whereas the expansion of zs yields the logarithmic term

s s0 2 z D z 1 C log z .s s0/ C O..s s0/ / ;s! s0 :

F.4.2 Series Representations and Asymptotics: Recurrence Relations

Series representations of the Fox H-functions can be found by applying the Residue Theory to calculate the corresponding Mellin–Barnes integral. These calculations critically depend on the choice of the contour of integration L in (F.4.1)andthe distribution of poles of the integrand there. For simplicity, let us suppose that the poles of the Gamma functions .bk Cˇks/ and the poles of the Gamma functions .1 aj ˛j s/ do not coincide, i.e.

˛j .bk C r/ 6D ˇk.ai t 1/; j D 1;:::;n; k D 1;:::;m; r;t D 0;1;2;:::: (F.4.22) F.4 Fox H -Functions 375

This assumption allow us to choose one of the above described contours of integration L D L1,orL D LC1,orL D Li1. The series representations of Fox H-functions are based on the following theorem: Theorem F.3 ([KilSai04,p.5]). Let the conditions (F.4.22) be satisﬁed. Then the following assertions hold true. (i) In the cases (F.4.10) and (F.4.11)theH-function (F. 4 . 1 ) is analytic in z and can be calculated via the formula

Xm X1 h i m;n Hm;n s Hp;q .z/ D RessDbkr p;q .s/z ; (F.4.23) kD1 rD0

where the sum is calculated with respect to all poles bkr of the Gamma functions .bk C ˇks/. (ii) In the cases (F.4.13) and (F.4.14)theH-function (F. 4 . 1 ) is analytic in z and can be calculated via the formula

Xm X1 h i m;n Hm;n s Hp;q .z/ D RessDajt p;q .s/z ; (F.4.24) kD1 rD0

where the sum is calculated with respect to all poles ajt of the Gamma functions .1 aj ˛j s/. (iii) In the case (F.4.16)theH-function (F. 4 . 1 ) is analytic in z in the sector jarg zj < a 2 .

Theorem F.4 ([KilSai04,p.6]). Suppose the poles of the Gamma functions .bkC ˇks/ and the poles of the Gamma functions .1 aj ˛j s/ do not coincide.

(i) If all the poles of the Gamma functions .bk C ˇks/ are simple, and either m;n >0;z 6D 0,or D 0; 0 < jzj <ı, then the Fox H-function Hp;q .z/ has the power series expansion

Xm X1 m;n .bk Cr/=ˇk Hp;q .z/ D hkrz ; (F.4.25) kD1 rD0

where h i Hm;n hkr D lim .s bkr/ p;q .z/ s!bkr Qm Á Qn Á ˇi ˛i bi Œbk C r 1 ai C Œbk C r r ˇk ˇk .1/ iD1;i6Dk iD1 D Qp Á Qq Á: rŠˇk ˛i ˇi ai Œbk C r 1 bi C Œbk C r ˇk ˇk iDnC1 iDmC1 (F.4.26) 376 F Higher Transcendental Functions

(ii) If all the poles of the Gamma functions .1 aj ˛j s/ are simple, and either m;n <0;z 6D 0,or D 0; jzj >ı, then the Fox H-function Hp;q .z/ has the power series expansion

Xm X1 m;n .aj 1t/=˛j Hp;q .z/ D hjtz ; (F.4.27) kD1 rD0

where h i Hm;n hjt D lim .s ajt/ p;q .z/ s!ajt Qm Á Qn Á ˇi ˛i bi C Œ1 aj C t 1 ai C Œ1 aj C t t ˛j ˛j .1/ iD1 iD1;i6Dj D Qp Á Qq Á: tŠ˛j ˛i ˇi ai C Œ1 aj C t 1 bi Œ1 aj C t ˛j ˛j iDnC1 iDmC1 (F.4.28)

Theorem F.5 ([KilSai04,p.9]). Suppose the poles of the Gamma functions .bkC ˇks/ and the poles of the Gamma functions .1 aj ˛j s/ do not coincide. Let either <0;z 6D 0 or D 0; jzj >ı. m;n Then the Fox H-function Hp;q .z/ has the following power-logarithmic series expansion

Njt1 X0 X00 X m;n .aj 1t/=˛j .aj 1t/=˛j l Hp;q .z/ D hjtz C Hjtlz Œlog z ; (F.4.29) j;t j;t lD0 P 0 where summation in is performed over all j; t; j D 1;:::;n;t D 0; 1;P 2; : : :, 00 for which the poles of .1 aj ˛j s/ are simple, and summation in is performed over all values of parameters j; t for which the poles of .1 aj ˛j s/ have order Njt, the constants hjt are given by the formulas (F.4.28), and the constants Hjtl can be explicitly calculated (see [KilSai04,p.8]). From Theorems F.3 to F.5 follow corresponding asymptotic power- and power- m;n logarithmic type expansions at inﬁnity of the Fox H-functions Hp;q .z/ (see, e.g., [KilSai04]). Exponential asymptotic expansions in the case >0;a D 0 and in the case n D 0 are presented, for example, in [KilSai04, Sect. 1.6, 1.7] (see also [Bra62, MaSaHa10, SrGuGo82]). More detailed information on the asymptotics of the Fox H-functions at inﬁnity can be found in [KilSai04, Ch. 1]. The asymptotic behaviour of the Fox H-functions at zero is also discussed there. The following two three-term recurrence formulas are linear combinations of the H-function with the same values of parameters m; n; p; q in which some aj and bk are replaced by aj ˙ 1 and by bk ˙ 1, respectively. Such formulas are called contiguous relations in [SrGuGo82]. F.4 Fox H -Functions 377

Let m 1 and 1 Ä n Ä p 1, then the following recurrence relation holds: Ä ˇ ˇ m;n ˇ.aj ;˛j /j D1;:::;;p .b1˛p apˇ1 C ˇ1/Hp;q z ˇ .bj ;ˇj /j D1;:::;q Ä ˇ ˇ m;n ˇ .aj ;˛j /j D1;:::;p D ˛p Hp;q z ˇ .b1 C 1; ˇ1/; .bj ;ˇj /j D2;:::;q Ä ˇ ˇ m;n ˇ.aj ;˛j /j D1;:::;p1;.ap1;˛p/ ˇ1Hp;q z ˇ : (F.4.30) .bj ;ˇj /j D1;:::;q

Let n 1 and 1 Ä m Ä q 1, then the following recurrence relation holds: Ä ˇ ˇ m;n ˇ.aj ;˛j /j D1;:::;;p .bq˛1 a1ˇq C ˇq /Hp;q z ˇ .bj ;ˇj /j D1;:::;q Ä ˇ ˇ m;n ˇ.a1 1; ˛1/; .aj ;˛j /j D2;:::;;p D ˇpHp;q z ˇ .bj ;ˇj /j D1;:::;q Ä ˇ ˇ m;n ˇ .aj ;˛j /j D1;:::;p ˛1Hp;q z ˇ : (F.4.31) .bj ;ˇj /j D1;:::;q1;.bq C 1; ˇq/

A complete list of contiguous relations for H-functions can be found in [Bus72].

F.4.3 Special Cases

Most elementary functions can be represented as special cases of the Fox H- function. Let us present a list of such representations. Ä ˇ ˇ Á 1;0 ˇ 1 b=ˇ 1 H z ˇ D z exp z ˇ I (F.4.32) 0;1 .b; ˇ/ ˇ Ä ˇ ˇ 1;1 ˇ.1 a; 1/ a H z ˇ D .a/.1 C z/ D .a/1F0.aIz/I (F.4.33) 1;1 .0; 1/ Ä ˇ ˇ 1;0 ˇ.˛ C ˇ C 1; 1/ ˛ ˇ H z ˇ D z .1 z/ I (F.4.34) 1;1 .˛; 1/ Ä ˇ 2 ˇ 1;0 z ˇ 1 H0;2 ˇ 1 D p sin zI (F.4.35) 4 2 ;1 ; .0; 1/ Ä ˇ 2 ˇ 1;0 z ˇ 1 H0;2 ˇ 1 D p cos zI (F.4.36) 4 .0; 1/; 2 ;1 Ä ˇ 2 ˇ 1;0 z ˇ i H0;2 ˇ 1 D p sinh zI (F.4.37) 4 2 ;1 ; .0; 1/ 378 F Higher Transcendental Functions Ä ˇ 2 ˇ 1;0 z ˇ i H ˇ D p cosh zI (F.4.38) 0;2 4 .0; 1/; 1 ;1 Ä ˇ 2 ˇ 1;0 ˇ.1; 1/; .1; 1/ ˙H z ˇ D log .1 ˙ z/I (F.4.39) 2;2 .1; 1/; .0; 1/ " ˇ # ˇ ˇ 1 ;1 ; 1 ;1 H 1;2 z2 ˇ 2 2 D 2 arcsin zI (F.4.40) 2;2 ˇ 1 .0; 1/; 2 ;1 " ˇ # ˇ ˇ 1 ;1 ; .1; 1/ H 1;2 z2 ˇ 2 D 2 arctan z: (F.4.41) 2;2 ˇ 1 2 ;1 ; .0; 1/

A number of representation formulas relating some special functions to the Fox H-function is presented in [KilSai04, Sect. 2. 9]. Two of these formulas appear below. Ä ˇ ˇ 1;1 ˇ.0; 1/ H z ˇ D E˛;ˇ.z/; (F.4.42) 1;2 .0; 1/; .1 ˇ; ˛/ where E˛;ˇ is the Mittag-Lefﬂer function

X1 zk E .z/ D : ˛;ˇ .˛kC ˇ/ kD0 Ä ˇ ˇ 1 1;1 ˇ.1 ; 1/ H z ˇ D E .z/; Re >0; (F.4.43) ./ 1;2 .0; 1/; .1 ˇ; ˛/ ˛;ˇ

where E˛;ˇ is the three-parametric (Prabhakar) Mittag-Lefﬂer function

X1 ./ zk E .z/ D k : ˛;ˇ .˛kC ˇ/ kD0 Ä ˇ ˇ 1;0 ˇ H z ˇ D J .z/; (F.4.44) 0;2 .0; 1/; .; /

where J is the Bessel–Maitland (or the Bessel–Wright) function

X1 .z/k J .z/ D : .C k C 1/kŠ kD0 Ä ˇ 2 ˇ 1;1 z ˇ. C 2 ;1/ H1;3 ˇ D J;.z/; (F.4.45) 4 . C 2 ;1/;.2 ; 1/; .. C 2 / ; / F.4 Fox H -Functions 379

where J; is the generalized Bessel–Maitland (or the generalized Bessel–Wright) function C2C2k X1 .1/k z J .z/ D 2 : ; .C k C C 1/ .k C C 1/ kD0 2 3 Ä ˇ ˇ ˇ .al ;˛l /1;p ˇ 1;p ˇ .1 a1;˛1/;:::;.1 ap;˛p / 4 ˇ 5 Hp;qC1 z ˇ D pq ˇ z ; .0; 1/ ; .1 b1;ˇ1/;:::;.1 bq;ˇq/ .bl ;ˇl /1;q (F.4.46) where pq is the generalized Wright function 2 3 ˇ Q .a ;˛ / 1 p l l 1;p ˇ X .a C ˛ k/ zk 4 ˇ 5 Q lD1 l l pq ˇ z D q : j D1 .bj C ˇj k/ kŠ .bl ;ˇl /1;q kD0

F.4.4 Relations to Fractional Calculus

Following [KilSai04, Sect. 2.7] we present here two theorems describing the relationship of the Fox H-function to fractional calculus. Theorem F.6 ([KilSai04, pp. 52–53]). Let ˛ 2 C (Re ˛>0), ! 2 C, and >0. Let us assume that either a >0or a D 0 and Re <1. Then the following statements hold: (i) If Ä Re b min k C Re !>1; (F.4.47) 1ÄkÄm ˇk

for a >0or a D 0 and 0, while Ä Re b Re C 1=2 min k ; C Re !>1; (F.4.48) 1ÄkÄm ˇk

˛ for a D 0 and <0, then the Riemann–Liouville fractional integral I0C of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã ˇ ˛ ! m;n ˇ .a1;˛1/;:::;.ap ;˛p/ I0Ct Hp;q t ˇ .x/ .0; 1/ ; .b1;ˇ1/;:::;.bq;ˇq / Ä ˇ ˇ !C˛ m;nC1 ˇ .!; /;.a1;˛1/;:::;.ap;˛p/ D x HpC1;qC1 x ˇ : (F.4.49) .b1;ˇ1/;:::;.bq;ˇq/; .! ˛; / 380 F Higher Transcendental Functions

(ii) If Ä Re a 1 min j C Re ! C Re ˛<0; (F.4.50) 1Äj Än ˛k

for a >0or a D 0 and Ä 0, while Ä Re aj 1 Re C 1=2 min ; C Re ! C Re ˛<0; (F.4.51) 1ÄkÄm ˛j

˛ for a D 0 and >0, then the Riemann–Liouville fractional integral I of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã ˇ ˛ ! m;n ˇ .a1;˛1/;:::;.ap;˛p/ It Hp;q t ˇ .x/ .0; 1/ ; .b1;ˇ1/;:::;.bq ;ˇq/ Ä ˇ ˇ !C˛ mC1;n ˇ .a1;˛1/;:::;.ap;˛p/; .!; / D x HpC1;qC1 x ˇ : (F.4.52) .! ˛; /; .b1;ˇ1/;:::;.bq;ˇq /

Theorem F.7 ([KilSai04,p.55]). Let ˛ 2 C (Re ˛>0), ! 2 C, and >0.Letus assume that either a >0or a D 0 and Re <1. Then the following statements hold: (i) If either the condition in (F.4.44) is satisﬁed for a >0or a D 0 and 0, or the condition in (F.4.45) is satisﬁed for a D 0 and <0, then the ˛ Riemann–Liouville fractional derivative D0C of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã ˇ ˛ ! m;n ˇ .a1;˛1/;:::;.ap;˛p / D0Ct Hp;q t ˇ .x/ .0; 1/ ; .b1;ˇ1/;:::;.bq;ˇq/ Ä ˇ ˇ !˛ m;nC1 ˇ .!; /;.a1;˛1/;:::;.ap ;˛p/ D x HpC1;qC1 x ˇ : (F.4.53) .b1;ˇ1/;:::;.bq;ˇq/; .! C ˛; /

(ii) If Ä Re a 1 min j C Re ! C 1 fRe ˛g <0; (F.4.54) 1Äj Än ˛k

for a >0or a D 0 and Ä 0, while Ä Re aj 1 Re C 1=2 min ; C Re ! C 1 fRe ˛g <0; (F.4.55) 1ÄkÄm ˛j F.4 Fox H -Functions 381

for a D 0 and >0,wherefRe ˛g denotes the fractional part of the ˛ number Re ˛, then the Riemann–Liouville fractional derivative D of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã ˇ ˛ ! m;n ˇ .a1;˛1/;:::;.ap ;˛p/ Dt Hp;q t ˇ .x/ .0; 1/ ; .b1;ˇ1/;:::;.bq;ˇq / Ä ˇ ˇ !˛ mC1;n ˇ .a1;˛1/;:::;.ap ;˛p/; .!; / D x HpC1;qC1 x ˇ : (F.4.56) .! C ˛; /; .b1;ˇ1/;:::;.bq;ˇq /

F.4.5 Integral Transforms of H -Functions

Here we consider the Fox H-function where, in the deﬁnition (F.4.1)–(F.4.4), the poles of the Gamma functions .bk C ˇks/ and the poles of the Gamma functions .1 aj ˛j s/ do not coincide. The notation introduced in (F.4.9) will be useful for us in this subsection too. The ﬁrst result is related to the Mellin transform of the Fox H-function. This follows from Theorem F.2. and the Mellin inversion theorem (see, e.g., [Tit86, Sect. 1.5]). Theorem F.8 ([KilSai04,p.43]). Let a 0, and s 2 C be such that Ä Ä Re b 1 Re a min k < Re s< min j (F.4.57) 1ÄkÄm ˇk 1Äj Än ˛j when a >0, and, additionally

Re s C Re <1; when a D 0. Then the Mellin transform of the Fox H-function exists and the following relation holds: Â Ä ˇ Ã Ä ˇ ˇ.a ;˛ / .a ;˛ / ˇ M m;n ˇ j j 1;p Hm;n j j 1;p ˇ Hp;q x ˇ .s/ D p;q ˇ s ; (F.4.58) .bj ;ˇj /1;q .bj ;ˇj /1;q

Hm;n where p;q is the kernel in the Mellin–Barnes integral representation of the H-function. A number of more general formulas for the Mellin transforms of the Fox H- function are presented in [KilSai04,p.44],[MaSaHa10, pp. 39–40]. The next theorem gives the formula for the Laplace transform of the Fox H-function. 382 F Higher Transcendental Functions

Theorem F.9 ([KilSai04, p. 45]). Let either a >0,ora D 0; Re<1 be such that Ä Re b min k > 1; (F.4.59) 1ÄkÄm ˇk when a >0,ora D 0, 0, and " # Re b Re C 1 min k ; 2 > 1; (F.4.60) 1ÄkÄm ˇk when a D 0, <0. Then the Laplace transform of the Fox H-function exists and the following relation holds for all t 2 C; Re t>0: Â Ä ˇ Ã Ä ˇ ˇ.a ;˛ / 1 1 ˇ.0; 1/; .a ;˛ / L m;n ˇ j j 1;p m;nC1 ˇ j j 1;p Hp;q x ˇ .t/ D HpC1;q ˇ : (F.4.61) .bj ;ˇj /1;q t t .bj ;ˇj /1;q

A number of more general formulas for the Laplace transform of the Fox H- function are presented in [KilSai04, pp. 46–48].

F.5 Historical and Bibliographical Notes

The (classical) hypergeometric function 2F1 is commonly deﬁned by the following Gauss series representation (see, e.g., [NIST, p. 384])

X1 X1 .a/k .b/k k .c/ .aC k/ .b C k/ k 2F1.a; bI cI z/ D z D z : .c/kkŠ .a/.b/ .cC k/kŠ kD0 kD0

The term hypergeometric series was proposed by J. Wallis in his book Arithmetica Infinitorum (1655). Hypergeometric series were studied by L. Euler, and a systematic analysis of their properties was presented in C.-F. Gauss’s 1812 paper (see the reprint in the collection of Gauss’s works [Gauss, pp. 123–162]). Studies in the nineteenth century included those of E. Kummer [Kum36], and the fundamental characterization by Bernhard Riemann of the hypergeometric function by means of the differential equation it satisfies. Riemann showed that the second-order differential equation for 2F1, examined in the complex plane, could be characterized (on the Riemann sphere) by its three regular singularities. Historically, confluent hypergeometric functions were introduced as solutions of a degenerate form of the hypergeometric differential equation. Kummer’s confluent hypergeometric function M.aI bI z/ (known also as the ˚-function, see [Bat-1]) was F.5 Historical and Bibliographical Notes 383 introduced by Kummer in 1837 ([Kum37]) as a solution to (Kummer’s) differential equation

d2w dw z C .b z/ C aw D 0: dz2 dz

The function M.a;bI z/ can be represented in the form of a series too

X1 .a/k k M.a;bI z/ D z D 1F1.aI bI z/: .b/kkŠ kD0

Another (linearly independent) solution to Kummer’s differential equation U.aI bI z/ (known also as the -function, see [Bat-1]) was found by Tricomi in 1947 ([Tri47]). The function .˛;ˇI z/, called the Wright function, was introduced by Wright in 1933 (see [Wri33]) in relation to the asymptotic theory of partitions. An extended discussion of its properties and applications is given in [GoLuMa99]. Special attention is given to the key role of the Wright function in the theory of fractional partial differential equations. The generalized hypergeometric functions introduced by Pochhammer [Poc70] and Goursat [Gou83a, Gou83b] are solutions of linear differential equations of order n with polynomial coefficients. These functions were considered later by Pincherle. Thus, Pincherle’s paper [Pin88] is based on what he called the “duality principle”, which relates linear differential equations with rational coefficients to linear difference equations with rational coefficients. Let us recall that the phrase “rational coefficients” means that the coefficients are in general rational functions (i.e. a ratio of two polynomials) of the independent variable and, in particular, polynomials (for more details see [MaiPag03]). These integrals were used by Meijer in 1946 to introduce the G-function into mathematical analysis [Mei46]. From 1956 to 1970 a lot of work was done on this function, which can be seen from the bibliography of the book by Mathai and Saxena [MatSax73]. The H-functions, introduced by Fox [Fox61] in 1961 as symmetrical Fourier kernels, can be regarded as the extreme generalization of the generalized hypergeometric functions pFq beyond the Meijer G functions (see, e.g. [Sax09]). The importance of this function is appreciated by scientists, engineers and statisticians due to its vast potential of applications in diverse fields. These functions include, among others, the functions considered by Boersma [Boe62], Mittag-Leffler [ML1, ML2, ML3, ML4], the generalized Bessel function due to Wright [Wri34], the generalization of the hypergeometric functions studied by Fox (1928), and Wright [Wri35b, Wri40c], the Krätzel function [Kra79], the generalized Mittag-Leffler function due to Dzherbashyan [Dzh60], the generalized Mittag-Leffler function due to Prabhakar [Pra71] and to Kilbas and Saigo [KilSai95a], the multi-index Mittag-Leffler function due to Kiryakova [Kir94]andLuchko[Luc99](seealso 384 F Higher Transcendental Functions

[KilSai96]), etc. Except for the functions of Boersma [Boe62], the aforementioned functions cannot be obtained as special cases of the G-function of Meijer [Mei46], hence a study of the H-function will cover a wider range than the G-function and gives general, deeper, and useful results directly applicable in various problems of a physical, biological, engineering and earth sciences nature, such as ﬂuid ﬂow, rheology, diffusion in porous media, kinematics in viscoelastic media, relaxation and diffusion processes in complex systems, propagation of seismic waves, anomalous diffusion and turbulence, etc. See, Caputo [Cap69], Glöckle and Nonnenmacher [GloNon93], Mainardi et al. [MaLuPa01], Saichev and Zaslavsky [SaiZas97], Hilfer [Hil00], Metzler and Klafter [MetKla00], Podlubny [Pod99], Schneider [Sch86]and Schneider and Wyss [SchWys89] and others. A major contribution of Fox involves a systematic investigation of the asymptotic expansion of the generalized hypergeometric function (now called Wright functions, or generalized Wright functions,orFox–Wright functions):

Qp .aj C ˛j l/ X1 l j D1 z F ..a ;˛ /;:::;.a ;˛ /I .b ;ˇ /;:::;.b ;ˇ /I z/ D ; p q 1 1 p p 1 1 q q Qq lŠ lD0 .bk C ˇkl/ kD1

Pq where z 2 C, aj ;bk 2 C, ˛j ;ˇk 2 R, j D 1;:::;pI k D 1;:::;qI ˇk kD1 Pp ˛j 1. His method is an improvement of the approach by Barnes (see, j D1 e.g., [Barn07b]) who found an asymptotic expansion of the ordinary generalized hypergeometric function pFq .z/. In 1961 Fox introduced [Fox61]theH-function in the theory of special functions, which generalized the MacRobert’s E-function (see [Mac-R38], [Sla66, p. 42]), the generalized Wright hypergeometric function, and the Meijer G-function. In the mentioned paper he investigated the far-most generalized Fourier (or Mellin) kernel associated with the H-function and established many properties and special cases of this kernel. Like the Meijer G-functions, the Fox H-functions turn out to be related to the Mellin–Barnes integrals and to the Mellin transforms, but in a more general way. After Fox, the H-functions were carefully investigated by Braaksma [Bra62], who provided their convergent and asymptotic expansions in the complex plane, based on their Mellin–Barnes integral representation. More recently, the H-functions, being related to the Mellin transforms (see [Mari83]), have been recognized to play a fundamental role in probability theory and statistics (see e.g. [MaSaHa10, Sch86, SaxNon04, UchZol99, Uch03]), in fractional calculus [KilSai99, KiSrTr06, Kir94], and its applications [AnhLeo01, AnhLeo03, AnLeSa03, Hil00, MatSax73], including phenomena of non-standard (anomalous) relaxation and diffusion [GorMai98, Koc90]. Several books specially devoted to H-functions and their applications have been published recently. Among them are F.6 Exercises 385 the books by Kilbas and Saigo [KilSai04] and by Mathai, Saxena and Haubold [MaSaHa10]. In [MaPaSa05] the fundamental solutions of the Cauchy problem for the space- time fractional diffusion equation are expressed in terms of proper Fox H-functions, based on their Mellin–Barnes integral representations.

F.6 Exercises

F.6.1. Prove that the following equalities hold for all z; jzj <=4([NIST, p. 386]): (i) F.aI 1=2 C aI 1=2Itan2 z/ D .cos z/2a cos .2az/; sin ..1 2a/z/ (ii) F.aI 1=2 C aI 3=2Itan2 z/ D .cos z/2a . .1 2a/sin z F.6.2. Prove that the following equalities hold for all z; jzj <=2([NIST, p. 386]): (i) F.aI aI 1=2I sin2 z/ D cos .2az/; cos ..2a 1/z/ (ii) F.aI 1 aI 1=2I sin2 z/ D ; cos z sin ..2a 1/z/ (iii) F.aI 1 aI 3=2I sin2 z/ D . .2a 1/ sin z F.6.3. Prove the following representations of the Kummer conﬂuent hypergeometric function 1F1 and the Gauss hypergeometric function 2F1 in terms of the Fox H-function ([KilSai04,p.63]): Ä ˇ ˇ 1;1 ˇ .1 a; 1/ .a/ .i/ H z ˇ D 1F1.aI cIz/I 1;2 .0; 1/; .1 c;1/ .c/ Ä ˇ ˇ 1;2 ˇ.1 a; 1/; .1 b; 1/ .a/.b/ .ii/H z ˇ D 2F1.aI bI cIz/: 2;2 .0; 1/; .1 c;1/ .c/

F.6.4. Prove the following relations for the incomplete gamma- and Beta-functions ([Kir94, p. 334]):

Zz t ˛1 .i/ .˛; z/ WD e t dt D 1F1.˛I ˛ C 1Iz/I 0 Zz p1 q1 .ii/Bz.p; q/ WD t .1 t/ dt D 2F1.p; 1 qI p C 1I z/: 0 386 F Higher Transcendental Functions

F.6.5. Prove the following representation of the error function in terms of conﬂuent hypergeometric function ([Kir94,p.334]):

Zz 2 t 2 2z 1 z 2 erf.z/ WD p e dt D p 1F1. I Iz /: 2 2 0

F.6.6. Prove the following representation of the classical polynomials in terms of special cases of the hypergeometric functions pFq : (i) Laguerre polynomials ([MaSaHa10, p. 29])

ez dn ˚ « .1 C ˛/ L.˛/.z/ WD ezznC˛ D n F .nI ˛ C 1I z/I n z˛nŠ dzn nŠ 1 1

(ii) Jacobi polynomials ([MaSaHa10,p.28])

.1/n dn ˚ « P .˛;ˇ/.z/ WD .1 z/˛.1 C z/ˇ .z2 1/n n 2nnŠ dzn Â Ã .˛ C 1/ 1 z D n F n; n C ˛ C ˇ C 1I ˇ C 1I I nŠ 2 1 2

(iii) Legendre polynomials ([Kir94, p. 333]) Â 1 dn ˚ « P .z/ WD .1 z/˛Cn.1 C z/ˇCn D .1/n F n; n n 2nnŠ dzn 2 1 Ã 1 z C1I 1I ; jzj <1I 2

(iv) Tchebyshev polynomials ([Kir94, p. 333])

p Â Ã nŠ . 1 ; 1 / 1 1 z T .z/ WD cos.arccos z/ D P 2 2 .z/ D F n; nI I I n 1 n 2 1 2 2 .nC 2 /

(v) Bessel polynomials ([Kir94, p. 333])

n Á X .n/ .a C n 1/ z k Y .z;a;b/ WD k k D F n; a n kŠ b 2 0 kD0 Á z Cn 1II I b F.6 Exercises 387

(vi) Hermite polynomials ([NIST, p. 443]) Â Ã X .1/k.2z/n2k n n 1 1 H .z/ WD nŠ Œn=2 D .2z/n F ; C II : n kŠ.n 2k/Š 2 0 2 2 2 z2 kD0

F.6.7. Prove the following representations of some elementary functions in terms of the Meijer G-function ([Kir94, pp. 331–332]): Ä ˇ ˇ z 1;1 ˛ ˇ .i/ D a ˛ G az ˇ ˛ I 1 C az˛ 1;1 Ä ˇ ˛ ˇ ˇ z˛ ˇ 1;0 ˛ ˇˇ .ii/ z e D ˛ G z ˇ I 0;1 ˛ Â Ã Ä ˇ 1 C z ˇ 1 1;2 2 ˇ 1; 2 .iii/ log D G1;1 z ˇ 1 ; jzj <1I 1 z 2 ;0 Ä Ä ˇ p 12a ˇ 3 1 2a 1 1;2 ˇ 1 a; a .iv/ .1 C 1 z/ D p G z ˇ 2 : 2 222a 2;2 0; 1 2a

F.6.8. Prove the following series representation for the Fox H-function ([Bra62, p. 279]): Ä ˇ ˇ b1 1;n ˇ1 ˇ .a1;˛1/;:::;.ap ;˛p/ ˇ1z Hp;q z ˇ .b1;ˇ1/;:::;.bq;ˇq / Qn ÁÁ b1C 1 aj C ˛j X1 ˇ1 .z/ j D1 D ÁÁ ÁÁ: Š Qq Qp b1C b1C 1 bk C ˇk aj ˛j ˇ1 ˇ1 kD2 j DnC1 References

[Abe23] Abel, N.H.: Oplösning af et Par Opgaver ved Hjelp af bestemie Integraler. Magazin for Naturvidenskaberne Aargang 1, Bind 2, 11–27 (1823), in Norwe- gian. French translation “Solution de quelques problèmes à l’aide d’intégrales déﬁnies”. In: Sylov, L., Lie, S. (eds.) Oeuvres Complètes de Niels Henrik Abel (Deuxième Edition), I, Christiania, 11–27 (1881). Reprinted by Éditions Jacques Gabay, Sceaux (1992) [Abe26] Abel, N.H.: Auﬂoesung einer mechanischen Aufgabe. Journal für die reine und angewandte Mathematik (Crelle), 1, 153–157 (1826, in German). French translation “Résolution d’un problème de mécanique” In: Sylov, L., Lie, S. (eds.) Oeuvres Complètes de Niels Henrik Abel (Deuxième Edition), I, Christiania (1881), pp. 97–101. Reprinted by Éditions Jacques Gabay, Sceaux (1992). English translation “Solution of a mechanical problem” with comments by J.D. Tamarkin In: Smith, D.E. (ed.) A Source Book in Mathematics, pp. 656–662. Dover, New York (1959) [Abe26a] Abel, N.H.: Untersuchungen über die Reihe:

m m .m 1/ m .m 1/ .m 2/ 1 C x C x2 C x3 C 1 1 2 1 2 3 u. s. w. J. Reine Angew. Math. 1, 311–339 (1826); translation into English in: Researches on the series

m m.m 1/ m.m 1/.m 2/ 1 C x C x2 C x3 C : 1 1:2 1:2:3 Tokio Math. Ges. IV, 52–86 (1891) [AblFok97] Ablowitz, M.J., Fokas, A.S.: Complex Variables. Cambridge University Press, Cambridge (1997) [AbrSte72] Abramovitz, M., Stegun, I.A.: Handbook of Mathematical Functions, 10th printing. National Bureau of Standards, Washington, DC (1972) [Adv-07] Sabatier, J., Agrawal, O.P., Tenreiro Machado, J.A. (eds.): Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineer- ing. Springer, Berlin (2007) [Aga53] Agarwal, R.P.: A propos d’une note de M. Pierre Humbert. C. R. Acad. Sci. Paris 236, 2031–2032 (1953)

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