DOCSLIB.ORG

Ramanujan's Lost Notebook

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Ramanujan's Lost Notebook Ramanujan’s Lost Notebook George E. Andrews r Bruce C. Berndt Ramanujan’s Lost Notebook Part III George E. Andrews Bruce C. Berndt Department of Mathematics Department of Mathematics The Pennsylvania State University University of Illinois at University Park, PA Urbana–Champaign USA Urbana, IL USA ISBN 978-1-4614-3809-0 ISBN 978-1-4614-3810-6 (eBook) DOI 10.1007/978-1-4614-3810-6 Springer New York Heidelberg Dordrecht London Library of Congress Control Number: 2005923547 Mathematics Subject Classification (2010): 11P82, 11P81, 11P84, 11P83, 33Dxx © Springer Science+Business Media New York 2012 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter devel- oped. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) The CEO’s of Ranks and Cranks Freeman Dyson Frank Garvan Oliver Atkin H.P.F. Swinnerton-Dyer I felt the joy of an explorer who suddenly discov- ers the key to the language lying hidden in the hieroglyphs which are beautiful in themselves. –Rabindranath Tagore, The Religion of Man Preface This is the third of five volumes that the authors plan to write in their exami- nation of all the claims made by S. Ramanujan in The Lost Notebook and Other Unpublished Papers, which was published by Narosa in 1988. This publication contains the “Lost Notebook,” which was discovered by the first author in the spring of 1976 at the library of Trinity College, Cambridge. Also included therein are other partial manuscripts, fragments, and letters that Ramanujan wrote to G.H. Hardy from nursing homes during 1917–1919. Our third volume contains ten chapters and focuses on some of the most important and influen- tial material in The Lost Notebook and Other Unpublished Papers. At center stage is the partition function p(n). In particular, three chapters are devoted to ranks and cranks of partitions. Ramanujan’s handwritten manuscript on the partition and tau functions is also examined. vii Contents 1 Introduction ............................................... 1 2 Ranks and Cranks, Part I ................................. 9 2.1 Introduction............................................ 9 2.2 Proof of Entry 2.1.1 ..................................... 17 2.3 Background for Entries 2.1.2 and 2.1.4 ..................... 18 2.4 Proof of Entry 2.1.2 ..................................... 24 2.5 Proof of Entry 2.1.4 ..................................... 32 2.6 Proof of Entry 2.1.5 ..................................... 33 3 Ranks and Cranks, Part II ................................ 45 3.1 Introduction............................................ 45 3.2 PreliminaryResults...................................... 47 3.3 The 2-Dissection for F (q)................................. 48 3.4 The 3-Dissection for F (q)................................. 51 3.5 The 5-Dissection for F (q)................................. 56 3.6 The 7-Dissection for F (q)................................. 60 3.7 The 11-Dissection for F (q)................................ 63 3.8 Conclusion ............................................. 69 4 Ranks and Cranks, Part III ................................ 71 4.1 Introduction............................................ 71 4.2 KeyFormulasonPage59................................. 71 4.3 Proofs of Entries 4.2.1 and 4.2.3 ........................... 72 4.4 Further Entries on Pages 58 and 59 . ..................... 74 4.5 Congruences for the Coefficients λn on Pages 179 and 180 . 75 4.6 Page 181: Partitions and Factorizations of Crank Coefficients . 82 4.7 Series on Pages 63 and 64 Related to Cranks . .......... 84 4.8 Ranks and Cranks: Ramanujan’s Influence Continues . ...... 86 4.8.1 Congruences and Related Work ..................... 87 ix xContents 4.8.2 Asymptotics and Related Analysis . ............. 87 4.8.3 Combinatorics . ................................ 88 4.8.4 Inequalities....................................... 88 4.8.5 Generalizations . ................................ 88 5 Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions ......................................... 89 5.0 Congruences for τ(n) .................................... 92 5.1 The Congruence p(5n +4)≡ 0(mod5)..................... 94 5.2 Divisibility of τ(n)by5 .................................. 95 5.3 The Congruence p(25n + 24) ≡ 0(mod25).................. 97 5.4 Congruences Modulo 5k .................................. 98 5.5 CongruencesModulo7................................... 99 5.6 CongruencesModulo7,Continued.........................100 5.7 CongruencesModulo49..................................102 5.8 CongruencesModulo49,Continued........................104 5.9 The Congruence p(11n +6)≡ 0(mod11)...................106 5.10CongruencesModulo11,Continued........................107 5.11 Divisibility by 2 or 3 . ................................110 5.12 Divisibility of τ(n).......................................113 5.13CongruencesModulo13..................................119 5.14 Congruences for p(n)Modulo13 ..........................121 5.15CongruencestoFurtherPrimeModuli .....................123 5.16 Congruences for p(n)Modulo17,19,23,29,or31...........125 5.17 Divisibility of τ(n)by23.................................127 5.18 The Congruence p(121n − 5) ≡ 0 (mod 121) . .............129 5.19 Divisibility of τ(n) for Almost All Values of n ...............130 5.20 The Congruence p(5n +4)≡ 0(mod5),Revisited............132 5.21 The Congruence p(25n + 24) ≡ 0 (mod 25), Revisited .........134 5.22 Congruences for p(n)ModuloHigherPowersof5............135 5.23 Congruences for p(n) Modulo Higher Powers of 5, Continued . 136 5.24 The Congruence p(7n +5)≡ 0(mod7).....................138 5.25Commentary............................................139 5.1 The Congruence p(5n +4)≡ 0(mod5).....................140 5.2 Divisibility of τ(n)by5 ..................................141 5.4 Congruences Modulo 5k ..................................143 5.5 CongruencesModulo7...................................144 5.6 CongruencesModulo7,Continued.........................144 5.7 CongruencesModulo49..................................144 5.8 CongruencesModulo49,Continued........................145 5.9 The Congruence p(11n +6)≡ 0(mod11)...................145 5.10CongruencesModulo11,Continued........................146 5.11 Divisibility by 2 or 3 . ................................147 5.12 Divisibility of τ(n).......................................150 5.13CongruencesModulo13..................................151 Contents xi 5.14 Congruences for p(n)Modulo13 ..........................153 5.15CongruencestoFurtherPrimeModuli .....................153 5.16 Congruences for p(n)Modulo17,19,23,29,or31...........159 5.17 Divisibility of τ(n)by23.................................176 5.18 The Congruence p(121n − 5) ≡ 0 (mod 121) .................177 5.19 Divisibility of τ(n) for Almost All Values of n ...............177 5.20 The Congruence p(5n +4)≡ 0(mod5),Revisited............178 5.23 Congruences for p(n) Modulo Higher Powers of 5, Continued . 179 5.24 The Congruence p(7n +5)≡ 0(mod7).....................179 6 Theorems about the Partition Function on Pages 189 and 182 ........................................................181 6.1 Introduction............................................181 6.2 TheIdentitiesforModulus5..............................183 6.3 TheIdentitiesforModulus7..............................186 6.4 Two Beautiful, False, but Correctable Claims of Ramanujan . 193 6.5 Page182...............................................195 6.6 FurtherRemarks........................................199 7 Congruences for Generalized Tau Functions on Page 178 ..205 7.1 Introduction............................................205 7.2 Proofs .................................................207 8 Ramanujan’s Forty Identities for the Rogers–Ramanujan Functions ..................................................217 8.1 Introduction............................................217
Load more

Recommended publications
  • Hypergeometric Functions and Feynman Diagrams 3
    Hypergeometric Functions and Feynman Diagrams Mikhail Kalmykov, Vladimir Bytev, Bernd Kniehl, Sven-Olaf Moch, Bennie Ward, and Scott Yost Abstract The relationship between Feynman diagrams and hypergeometric functions is discussed. Special attention is devoted to existing techniquesfor the constructionof the Y-expansion. As an example, we present a detailed discussion of the construction of the Y-expansion of the Appell function 3 around rational values of parameters via an iterative solution of differential equations. As a by-product,we have found that the one-loop massless pentagon diagram in dimension 3 = 3 2n is not expressible in terms of multiple polylogarithms. Another interesting example− is the Puiseux-type solution involving a differential operator generated by a hypergeometric function of three variables. The holonomic properties of the # hypergeometric functions are briefly discussed. Mikhail Kalmykov JINR, Dubna, Russia, e-mail: [email protected] Vladimir Bytev BLTP JINR, 141980 Dubna, Moscow region, Russia e-mail: [email protected] Bernd A. Kniehl, II. Institut fuer Theoretische Physik, Universitaet Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany e-mail: [email protected] Sven-Olaf Moch II. Institut fuer Theoretische Physik, Universitaet Hamburg, Luruper Chaussee 149, 22761 arXiv:2012.14492v3 [hep-th] 22 Jan 2021 Hamburg, Germany e-mail: [email protected] B.F.L. Ward Department of Physics, Baylor University, One Bear Place, # 97316, Waco, TX 76798-7316, USA e- mail: BFL{\protect\relax$\@@underline{\hbox{\}}\mathsurround\z@$\relax}[email protected] S.A. Yost Department of Physics, The Citadel, 171 Moultrie St., Charleston, SC 29409, USA e-mail: [email protected] 1 2 Authors Suppressed Due to Excessive Length 1 Introduction Recent interest in the analytical properties of Feynman diagrams has been motivated by processes at the LHC.
    [Show full text]
  • Some New Results for Humbert's Double
    Palestine Journal of Mathematics Vol. 8(2)(2019) , 146–158 © Palestine Polytechnic University-PPU 2019 SOME NEW RESULTS FOR HUMBERT’S DOUBLE HYPERGEOMETRIC SERIES 2 AND φ2 Ghazala Yasmin and Abdulghani Muhyi Communicated by P.K. Banerji MSC 2010 Classifications: Primary 33B15, 33C20; Secondary 33C70. Keywords and phrases: Gamma Function, Generalized Hypergeometric Function, Kummer’s Theorem, Generalized (Wright) Hypergeometric Function, Humbert’s Functions, Lavoie-Trottier Integral Formula. Abstract. This article aims to establish new expressions of Humbert’s functions 2 and φ2 with the help of the generalization of Kummer’s summation theorem. Some special cases are also obtained. Further, we present some interesting integrals involving the Humbert’s functions, which are expressed in terms of generalized (Wright) hypergeometric function. Also, some special cases were considered as an application of the presented integral formulas. 1 Introduction The theory of hypergeometric functions has a vast popularity and great interest. The usefulness of this functions lies in the fact that the solutions of many applied problems involving partial dif- ferential equations are obtainable with the help of such hypergeometric functions. For instance, the energy absorbed by some non ferromagnetic conductor sphere contained in an internal mag- netic field can be calculated with the help of such functions [12, 15]. Multi-variable hyper- geometric functions are used in physical and quantum chemical applications as well [14, 19]. Especially, many problems in gas dynamics lead to those of degenerate second order partial differential equations, which are then solvable in terms of multiple hypergeometric functions. Further, in the investigation of the boundary value problems for these partial differential equa- tions, we need decompositions for multivariate hypergeometric functions in terms of simpler hypergeometric functions of the Gauss and Humbert types.
    [Show full text]
  • A New Proof of a Reduction Formula for the Appell Series F3 Due to Bailey∗
    FACTA UNIVERSITATIS (NIS)ˇ Ser. Math. Inform. Vol. 34, No 5 (2019), 849–854 https://doi.org/10.22190/FUMI1905849M A NEW PROOF OF A REDUCTION FORMULA FOR THE APPELL SERIES F3 DUE TO BAILEY∗ Gradimir V. Milovanovi´cand Arjun K. Rathie c 2019 by University of Niˇs, Serbia | Creative Commons Licence: CC BY-NC-ND Abstract. In this short note, we provide a new proof of an interesting and useful reduction formula for the Appell series F3 due to Bailey [On the sum of a terminating 3F2(1), Quart. J. Math. Oxford Ser. (2) 4 (1953), 237–240]. Keywords: Appell series, Humbert series, Whipple summation theorem, reduction formula, special functions 1. Introduction Hypergeometric series and many their generalizations, including multiple series, play an important role in the applied mathematics and mathematical physics. In this short note we are interested only for multiple hypergeometric series of Appell type, precisely for Appell series F3, which is defined by (cf. [2, 3, 6]) ∞ ∞ ′ ′ m n ′ ′ (a)m(a )n(b)m(b )n w z (1.1) F3(a,a ,b,b ; c; w,z)= m n m=0 n=0 (c) + · m! n! X X for w < 1, z < 1, c =0, 1, 2,... | | | | 6 − − As usual (λ)k is the well known Pochhammer symbol (or the shifted factorial or the raised factorial, since (1)n = n!) defined by Γ(λ + k) (λ)k = = λ(λ + 1) (λ + k 1). Γ(λ) ··· − A survey on multiple hypergeometric series of Appell type with a rich list of references has been recently published by Schlosser [10].
    [Show full text]
  • Basic Hypergeometric Series Second Edition
    ENCYCLOPEDIA OF MATHEMATICS AND ITS ApPLICATIONS Editorial Board R. S. Doran, P. Flajolet, M. Ismail, T.-y' Lam, E. Lutwak Volume 96 Basic Hypergeometric Series Second Edition This revised and expanded new edition will continue to meet the need for an author­ itative, up-to-date, self contained, and comprehensive account of the rapidly growing field of basic hypergeometric series, or q-series. It contains almost all of the important summation and transformation formulas of basic hypergeometric series one needs to know for work in fields such as combinatorics, number theory, modular forms, quan­ tum groups and algebras, probability and statistics, coherent-state theory, orthogonal polynomials, or approximation theory. Simplicity, clarity, deductive proofs, thought­ fully designed exercises, and useful appendices are among its strengths. The first five chapters cover basic hypergeometric series and integrals, whilst the next five are de­ voted to applications in various areas including Askey-Wilson integrals and orthogonal polynomials, partitions in number theory, multiple series, and generating functions. Chapters 9 to 11 are new for the second edition, the final chapter containing a sim­ plified version of the main elements of the theta and elliptic hypergeometric series as a natural extension of the single-base q-series. Elsewhere some new material and exercises have been added to reflect recent developments, and the bibliography has been revised to maintain its comprehensive nature. ENCYCLOPEDIA OF MATHEMATICS AND ITS ApPLICATIONS All the titles listed below can be obtained from good booksellers or from Cambridge University Press. For a complete series listing visit http://publishing.cambridge.org/stm/mathematics/eom/.
    [Show full text]
  • Note on the Hurwitz–Lerch Zeta Function of Two Variables
    S S symmetry Article Note on the Hurwitz–Lerch Zeta Function of Two Variables Junesang Choi 1, Recep ¸Sahin 2, O˘guzYa˘gcı 2 and Dojin Kim 3,* 1 Department of Mathematics, Dongguk University, Gyeongju 38066, Korea; [email protected] 2 Department of Mathematics, Kırıkkale University, Kırıkkale 71450, Turkey; [email protected] (R.S.); [email protected] (O.Y.) 3 Department of Mathematics, Dongguk University, Seoul 04620, Korea * Correspondence: [email protected] Received: 3 August 2020; Accepted: 27 August 2020; Published: 28 August 2020 Abstract: A number of generalized Hurwitz–Lerch zeta functions have been presented and investigated. In this study, by choosing a known extended Hurwitz–Lerch zeta function of two variables, which has been very recently presented, in a systematic way, we propose to establish certain formulas and representations for this extended Hurwitz–Lerch zeta function such as integral representations, generating functions, derivative formulas and recurrence relations. We also point out that the results presented here can be reduced to yield corresponding results for several less generalized Hurwitz–Lerch zeta functions than the extended Hurwitz–Lerch zeta function considered here. For further investigation, among possibly various more generalized Hurwitz–Lerch zeta functions than the one considered here, two more generalized settings are provided. Keywords: beta function; gamma function; Pochhammer symbol; Hurwitz–Lerch zeta function; Hurwitz–Lerch zeta function of two variables; hypergeometric functions; confluent hypergeometric functions; Appell hypergeometric functions; Humbert hypergeometric functions of two variables; integral representations; generating functions; derivative formulas; recurrence relation 1. Introduction and Preliminaries The generalized (or Hurwitz) zeta function z(s, n) is defined by (see, e.g., [1], pp.
    [Show full text]
  • Some Results for Q-Functions of Many Variables
    REND. SEM. MAT. UNIV. PADOVA, Vol. 112 (2004) Some Results for q-Functions of Many Variables. THOMAS ERNST (*) ABSTRACT - We will give some new formulas and expansions for q-Appell-, q-Lau- ricella- and q-Kampé de Fériet functions. This is an area which has been explored by Jackson, R. P. Agarwal, W. A Al-Salam, Andrews, Kalnins, Mil- ler, Manocha, Jain, Srivastava, and Van der Jeugt. The multiple q-hypergeo- metric functions are defined by the q-shifted factorial and the tilde operator. By a method invented by the author [20]-[23], which also involves the Ward- Alsalam q-addition and the Jackson-Hahn q-addition, we are able to find q- analogues of corresponding formulas for the multiple hypergeometric case. In the process we give a new definition of q-hypergeometric series, illustrated by some examples, which elucidate the integration property of q-calculus, and helps to try to make a first systematic attempt to find summation- and reduc- tion theorems for multiple basic series. The Jackson G q-function will be fre- quently used. Two definitions of a generalized q-Kampé de Fériet function, in the spirit of Karlsson and Srivastava [69], which are symmetric in the varia- bles, and which allow q to be a vector are given. Various connections of tran- sformation formulas for multiple q-hypergeometric series with Lie algebras and finite groups known from the literature are cited. 1. Historical introduction. After Gauss’ introduction of the hypergeometric series in 1812, vari- ous versions of the multiple hypergeometric series (MHS) were develo- ped by people whose names are now associated to the functions they in- troduced.
    [Show full text]
  • The Different Tongues of Q-Calculus
    Proceedings of the Estonian Academy of Sciences, 2008, 57, 2, 81–99 doi: 10.3176/proc.2008.2.03 Available online at www.eap.ee/proceedings The different tongues of q-calculus Thomas Ernst Department of Mathematics, Uppsala University, P.O. Box 480, SE-751 06 Uppsala, Sweden; [email protected] Received 14 December 2007, in revised form 7 February 2008 Abstract. In this review paper we summarize the various dialects of q-calculus: quantum calculus, time scales, and partitions. The close connection between Gq(x) functions on the one hand, and elliptic functions and theta functions on the other hand will be shown. The advantages of the Heine notation will be illustrated by the (q-)Euler reflection formula, q-Appell functions, Carlitz– AlSalam polynomials, and the so-called q-addition. We conclude with some short biographies about famous scientists in q-calculus. Key words: elliptic functions, theta functions, q-Appell functions, q-addition, Carlitz–AlSalam polynomial. 1. INTRODUCTION q-Calculus is a generalization of many subjects, like hypergeometric series, complex analysis, and particle physics. In short, q-calculus is quite a popular subject today. It has developed various dialects like quantum calculus, time scales, partitions, and continued fractions. We will give several aspects of q-calculus to enlighten the strong interdisciplinary as well as mathematical character of this subject. The close connection between q-calculus on the one hand, and elliptic functions and theta functions on the other hand will be shown. Some new formulas will be given, such as the two restatements of Kummer’s 2F1(¡1) formula (in q-form) (36) and (37).
    [Show full text]
  • Multiple Hypergeometric Series [.5Em] Appell Series
    Appell series Contiguous relations PDE's Integral representations Transformations Reductions Extensions A curious integral More sums Multiple hypergeometric series Appell series and beyond Michael J. Schlosser Faculty of Mathematics Multiple hypergeometric series { Appell series and beyond Appell series Contiguous relations PDE's Integral representations Transformations Reductions Extensions A curious integral More sums Outline 1 Appell hypergeometric series 2 Contiguous relations 3 Partial differential equations 4 Integral representations 5 Transformations 6 Reduction formulae 7 Extensions 8 A curious integral 9 More sums Multiple hypergeometric series { Appell series and beyond Recall the Pochhammer symbol notation for the shifted factorial: ( a(a + 1) ::: (a + n − 1) if n = 1; 2;::: , (a)n := 1 if n = 0. The (generalized) hypergeometric series is defined by a1; a2;:::; ar X (a1)n (a2)n ::: (ar )n n r Fs ; x = x : b1; b2;:::; bs n!(b1)n ::: (bs )n n≥0 Goal: We would like to generalize the Gauß hypergeometric function a; b X (a)n (b)n n 2F1 ; x = x c n!(c)n n≥0 to a double series depending on two variables. Appell series Contiguous relations PDE's Integral representations Transformations Reductions Extensions A curious integral More sums Appell hypergeometric series Multiple hypergeometric series { Appell series and beyond Goal: We would like to generalize the Gauß hypergeometric function a; b X (a)n (b)n n 2F1 ; x = x c n!(c)n n≥0 to a double series depending on two variables. Appell series Contiguous relations PDE's Integral representations Transformations Reductions Extensions A curious integral More sums Appell hypergeometric series Recall the Pochhammer symbol notation for the shifted factorial: ( a(a + 1) ::: (a + n − 1) if n = 1; 2;::: , (a)n := 1 if n = 0.
    [Show full text]
  • Quantum Modular Forms and Singular Combinatorial Series with Distinct
    QUANTUM MODULAR FORMS AND SINGULAR COMBINATORIAL SERIES WITH DISTINCT ROOTS OF UNITY AMANDA FOLSOM, MIN-JOO JANG, SAM KIMPORT, AND HOLLY SWISHER Abstract. Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of SL2pZq, up to nontrivial error terms; however, their domains (the upper half-plane H, and the rationals Q, respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta functions, vertex algebras, and more. In this paper we study the pn ` 1q-variable combinatorial rank generating function Rnpx1, x2,...,xn; qq for n-marked Durfee symbols. These are n ` 1 dimensional multisums for n ą 1, and specialize to the ordinary two-variable partition rank generating function when n “ 1. The mock modular properties of Rn when viewed as a function of τ P H, 2πiτ with q “ e , for various n and fixed parameters x1, x2, ¨ ¨ ¨ , xn, have been studied in a series of papers. Namely, by Bringmann and Ono when n “ 1 and x1 a root of unity; by Bringmann when n “ 2 and x1 “ x2 “ 1; by Bringmann, Garvan, and Mahlburg for n ě 2 and x1 “ x2 “¨¨¨“ xn “ 1; and by the first and third authors for n ě 2 and the xj suitable roots of unity (1 ď j ď n). The quantum modular properties of R1 readily follow from existing results.
    [Show full text]
  • Lauricella Hypergeometric Series Over Finite Fields
    LAURICELLA HYPERGEOMETRIC SERIES OVER FINITE FIELDS BING HE Abstract. In this paper we give a finite field analogue of the Lauricella hypergeomet- ric series and obtain some transformation and reduction formulae and several generating functions for the Lauricella hypergeometric series over finite fields. These generalize some known results of Li et al as well as several other well-known results. 1. Introduction ∗ Let q be a power of a prime. Then Fq and Fbq are denoted the finite field of q elements and ∗ the group of multiplicative characters of Fq respectively. Setting χ(0) = 0 for all characters, ∗ we extend the domain of all characters χ of Fq to Fq: Let χ and " denote the inverse of χ and the trivial character respectively. See [2] and [7, Chapter 8] for more information about characters. Following [1], we define the generalized hypergeometric function as 1 a0; a1; : : : ; an X (a0)k(a1)k ··· (an)k k n+1Fn x := x ; b1; : : : ; bn k!(b1)k ··· (bn)k k=0 where (z)k is the Pochhammer symbol given by (z)0 = 1; (z)k = z(z + 1) ··· (z + k − 1) for k ≥ 1: It was Greene who in [6] developed the theory of hypergeometric functions over finite fields and established a number of transformation and summation identities for hypergeomet- ric series over finite fields which are analogues to those in the classical case. Greene, in particular, introduced the notation G A; B BC(−1) X 2F1 x = "(x) B(y)BC(1 − y)A(1 − xy) C q y for A; B; C 2 Fbq and x 2 Fq; that is a finite field analogue of the integral representation of Gauss hypergeometric series [1]: 1 a; b Γ(c) b c−b −a dt 2F1 x = t (1 − t) (1 − tx) ; c Γ(b)Γ(c − b) ˆ0 t(1 − t) 2010 Mathematics Subject Classification.
    [Show full text]
  • Integral Transforms Involving the Product of Humbert and Bessel Functions and Its Application
    AIMS Mathematics, 5(2): 1260–1274. DOI:10.3934/math.2020086 Received: 10 October 2019 Accepted: 06 January 2020 http://www.aimspress.com/journal/Math Published: 19 January 2020 Research article Integral transforms involving the product of Humbert and Bessel functions and its application A. Belafhal1, N. Nossir1, L. Dalil-Essakali1 and T. Usman2;∗ 1 LPNAMME, Laser Physics Group, Department of Physics, Faculty of Sciences, Choua¨ıb Doukkali University, P. B 20, 24000 El Jadida, Morocco 2 Department of Mathematics, School of Basic and Applied Sciences, Lingaya’s Vidyapeeth, Faridabad-121002, Haryana, India * Correspondence: Email: [email protected]; Tel: +917417803673. Abstract: In this paper, we develop some integral transforms involving a product of Humbert and Bessel functions with a weight e−γx2 . These integral transforms will be evaluated in terms of hypergeometric functions. Various transformation formulae are also evaluated in terms of Appell functions to complete this study. Some special cases of the evaluated integrals yield some infinite series of generalized hypergeometric and Appell functions. As application, one of our main results is investigated to give an expression of the Generalized Humbert-Gaussian beams (GHGBs) propagating through a paraxial ABCD optical system. Keywords: integral transforms; Bessel functions; Humbert functions; Appell functions; Hypergeometric functions Mathematics Subject Classification: 33B15, 33C10, 33C15 1. Introduction Integral transforms involving special functions have gained considerable attention in the bibliography. In the last decade, many papers are investigated to study the integral transforms involving the product of Bessel and other special functions (see [4,8,9]). In view of this, it is worth investigating a general integral transform involving Humbert and Bessel functions with a weight e−γx2 .
    [Show full text]
  • Mathematics 2003
    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
    [Show full text]