A CBRTAIN CLASS OF GENERATING FUNCTIONS INVOLVING UNILATERAL AND BILATERAL SERIES DISSERTATION SUBMITTED IN PARTIAL FULFItWENT OF THE REQUIREMENTS FOR TME^WARO OF TH^ DEGREE OF MATHEMATICS i' By BABITA AGRAWAL Under the Supervision of PROF. M.A. PATHAN DEPARTMENT OF MATHEMATICS ALIGARH MUSLIM UNIVERSITY ALIGARH (INDIA) 2003 A\a^* Azaa \^i>^. S'^^ '^^ "'^^''n, Un^S^"-^- 2 OJUL ?009 Dedicated to Ply Husband who has been a constant source of inspiration to me throughout the duration of my work italta (£^a^iauMil DEPARTMENT OF MATHEMATICS ALIGARH MUSLIM UNIVERSITY ALIGARH - 202 002 (U. P.), INDIA Phones: (Off.): 0571-2701019;(Res.):0571-2701282 Fax: 0571-2704229 E-maiI:[email protected] Prof. M. A. Pathan CERTIFICATE This is to certify that the contents of this dissertations entitled "A Certain Class Of Generating Functions Involving Unilateral And Bilateral Series" is the original work of Mrs. Babita Agarwal carried out under my supervision. Her work is suitable for the award of the degree of Master of Philosophy in Mathematics. 7)^<^^. Prof. M. A. Pathan (Supervisor) CONTENTS Acknowledgement 1 CHAPTER 1 : PRELIMINARIES 1-14 1.1 : INTRODUCTION 1 1.2 : THE GENERATING FUNCTIONS 1.3 : GAMMA FUNCTION 1.4 : FACTORIAL FUNCTION 1.5 : GENERALIZED HERMITE POLYNOMIALS 10 CHAPTER 2 : GENERALIZED HERMITE POLYNOMIALS 15 32 OF TWO VARIABLES 2.1 : INTRODUCTION 15 2.2 : HYPERGEOMETRIC FORM 17 2.3 : ANOTHER GENERATING RELATION 20 2.4 : n"' DIFFERENTIAL FORMULA 21 2.5 : INTEGRAL REPRESENTATION 28 CHAPTER 3 : GENERALIZED INCOMPLETE GAMMA 33-57 FUNCTION AND BETA FUNCTIONS 33 3.1 : INTROUCTION 36 3.2 : REPRESENTATION FOR r(z/, x; z) IN TERMS OF A KDF FUNCTION 3.3 : INTEGRAL REPRESENTATION OF THE 44 EXTENDED BETA FUNCTION 3.4 : PROPERTIES OF EXTENDED 47 BETA FUNCTION 3.5 : CONNECTION WITH OTHER 48 SPECIAL FUNCTIONS 3.6 : MELLLIN TRANSFORM REPRESENTATION 55 OF THE EXTENDED BETA FUNCTION 3.7 : THE EXTENDED BETA DISTRIBUTION 5G CHAPTER 4 : A CERTAIN CLASS OF GENERATING 58 71 FUNCTIONS INVOLVING BILATERAL AND UNILATERAL SERIES 4.1 : INTRODUCTION 58 4.2 : GENERATING FUNCTIONS INVOLVING 60 BILATERAL SERIES 4.3 : APPLICATIONS OF THE THEOREM 61 4.4 : A GENERAL THEOREM FOR PARTLY 66 BILATERAL AND PARTLY UNILATERAL GENERATING FUNCTIONS 4.5 : APPLICATION OF THEOREM 2 AND 68 THEOREM 3 CHAPTER 5 : ON GENERALIZATION OF BESSEL 72-80 POLYNOMIALS OF SEVERAL VARIABLES 5.1 : INTRODUCTION 72 5.2 : THE BESSEL POLYNOMIALS 73 . {ai,—,a„;l3)( ... \ 5.3 : SOME INTEGRAL REPRESENTATIONS 74 5.4 : GENERATING FUNCTIONS OF THE 77 BESSEL POLYNOMIALS 5.5 : PARTLY BILATERAL AND PARTLY UNILATERAL 79 GENERATING RELATIONS BIBLIOGRAPHY 80-8/ ACKNOWLEDGMENT It is a matter of great pleasure for me to express my deepest sense of thank- fuhu^ss and indebtedness to my supervisor Prof. M. A. Pathan, Department of Mathematics, Aligarli Muslim University, Aligarh, who took great pains in provid­ ing valuable guidence and constant inspiration throughout the preparation of this dissertation. I am extremely grateful to Dr. S. M. A. Zaidi, Chairman, Department of Mathematics, Aligarh Muslim University, Aligarh, who has always inspired me and provided requisite facilities for carrying out this work in the department. I am extremely grateful to Dr. (Mrs.) Subuhi Khan, Senior Lecturer and Dr. Nabiullah Khan, Department of Mathematics, Aligarh Mushm University, Aligarli, for their constant encouragement, suggestions and guidence. My sincere thanks are also due to my several teacher and colleagues in and outside; the department who have been a source of inspiration throughout this work. Last l)ut not the least, I express my infinite indebtness to my parents whose love and support are the base of my every success. The contribution that I have received by sustained cooperation of my husband can not be ignored. Date: Babita Agarwal CHAPTER 1 PRELIMINARIES 1.1 INTRODUCTION Mathematical functions are one of the basic building blocks of Mathematics. As such, there is an imperative that efficient, accurate and robust software is available for the evaluation of these functions. The functions of one real variables that are used can be split into three distinct classes: 1. Elementary functions, such as exp, sin, etc, which are usually supplied by the compiler being used or a library add-on, such as in Java. The software for these functions is nowadays extremely good. 2. Well-known special functions, such as error, Bessel, gamma, etc. which are provided in a wide range of commercial software libraries, and freely available packages. 3. Uncommon special functions. These are used much less often and so have not appeared in any standard packages. Special functions commonly arise in such areas of apphcation as heat conduc­ tion communication systems, electro-optics, non linear wave propagation, electro­ magnetic theory, quantum mechanics approximation theory, probability theory and electro circuit theory, among others. Special functions is strongly related to the second order ordinary and partial differential equation. Special functions provide a unique tool for developing simplified yet realistic models of physical problems, thus allowing for analytic solutions and hence a deeper insight into the problem under study. A vast mathematical literature has been devoted to the theory of these functions as constructed in the works of Euler, Gauss, Legendre, Hermite, Riemann, Chebyshev, Hardy, Watson, Ramanujan and other classical authors, for example Erdelyi, Magnus, Oberhettinger and Tricomi [23], McBride [45], Rainville [62], Srivastava and Manocha [67], Szego [73], Titchmarch [74] and Watson [77], [76]. Brief historical synopses can be found in Aksenov [4], Chitiara [19], Klein [39], Lavrent'ev and Shabat [42], Miller [46], Whittaker and Watson [78], Prudnikov [60]. The special functions of mathematical physics appear most often in solving partial differential equations by the method of separation of the variables, or in finding eigenfunctions of differential operators in certain curvilinear system of co­ ordinates. The special functions are the classes of functions listed in (l)-(4) (other terms, such as higher transcendental functions, are sometimes used). 1. The gamma function and related functions. 2. Fresenel's integral, the error function, the logarithmic integral and other func­ tions that can be expressed as indefinite integrals of elementary functions. 3. Elliptic Functions. 4. Solutions of linear ordinary differential equations of the second order derived by separating certain partial differential equations, e.g., Laplace's equations in various curvilinear co-ordinates. 1.2 GENERATING FUNCTIONS The name 'generating relation' was introduced by Laplace in 1812. Since then, the theory of generating relations has been developed into various directions and found wide applications in various branches of science and technology. Many authors choose to define the special functions in terms of generating relations. Also it may be used to determine a pure relation or a differential recurrence relation, to evaluate certain integrals etc. Generating relations play an important role in the investigation of various useful properties of the special function which they generate. They are used as ^-transform in solving certain classes of diflFerence equations which arise in a wide variety of problems in operation research (including for example, queuing theory and related stochastic processes). Generating relations can also be used with good effect for the determination of the asymptotic behaviour of the generated sequence (special func­ tion) {/n}^o Sisn -^ oohy suitable adapting Darboux's method. And, furthermore the existence of generating function for a sequence {/„} of numbers or functions oo may be useful for finding J2 fn by such summability methods as these due to Abel 71=0 and Casero. Linear Generating Functions Consider a function F{x,t) which has a formal (it need not converge) power series expansion in t oo F{x,t) = J2fn{x)e (1.2.1) n=0 the coefficient of t" in equation (1.2.1) is, in general, a function of x. We say that the expansion (1.2.1) of F{x,t) has generated the set fn{x) and that F{x,t) is a generating function for the fn{x). The slightly extend form of generating function F(x,t) of the set fn(x) is oo F{x,t) = '£cnfn{x)e (1.2.2) n=0 where c„; n = 0,1, 2, • • • be a specified sequence independent of x and t. Bilinear Generating Functions If three variable function F{x, y, t) possesses a formal power series expansion in t such that oo F{x,y,t) = Y.VnUx)Uy)e, (1.2.3) n=0 where the sequence {y„} is independent of x, y and t, then F(x,y,t) is called a bilinear generating function for the set {fn{x)}. More generally, if F{x, y, t) can be expanded in powers of t in the form F{x,y,t) = Y.^nfain){x)Uin){y)t'', (1.2-4) where the a{n) and 0{n) are functions of n which are not necessarily equal, we shall still call F{x,y,t) a bilinear generating function for the set {fn{x)}. Bilateral Generating Functions Suppose that a three variable function H{x, y, t) has a formal power series ex­ pansion in t such that oo H{x,y,t) = X;/in/n(x) g„(y) t" (1.2.5) n=0 where the sequence {/i„} is independent of a;, y and t and the sets of functions {/,i(x)}^Q and {gn(y)}^o ^^^ different. Then H{x,y,t) is called a bilateral gener­ ating function for the set {fni^)} or {gn{y)}- oo H{x,y,t) = Y^yn fain) {x) g0[n) [v] t", (1-2.6) n=0 where the sequence {?/„} is independent of x, y and i, the sets of function {fnix)}^^o and {gn{x)}'^^o are different, and a(n) and /3(n) are functions of n which are not necessarily equal.