About this Journal Submit a Manuscript Table of Contents
ISRN Algebra
Volume 2011 (2011), Article ID 248519, 10 pages
http://dx.doi.org/10.5402/2011/248519
Research Article

Certain Transformation Formulae for Polybasic Hypergeometric Series

Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad 211 004, India

Received 4 August 2011; Accepted 21 August 2011

Academic Editors: A. Kiliçman, H. Rosengren, and A. Salemi

Copyright © 2011 Pankaj Srivastava and Mohan Rudravarapu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Making use of Bailey's transformation and certain known summations of truncated series, an attempt has been made to establish transformation formulae involving polybasic hypergeometric series.

1. Introduction

The remarkable contribution in the field of hypergeometric and basic hypergeometric series mainly due to Bailey [1] has appeared in Proceeding of London Mathematical society in 1947. The key result of the paper later on recognized as Bailley's transformation is as follows: if𝛽𝑛=𝑛𝑟=0𝛼𝑟𝑢𝑛𝑟𝑣𝑛+𝑟,𝛾𝑛=𝑟=𝑛𝛿𝑟𝑢𝑟𝑛𝑣𝑛+𝑟,then𝑛=0𝛼𝑛𝛾𝑛=𝑛=0𝛽𝑛𝛿𝑛,(1.1) where 𝛼𝑟,𝛿𝑟,𝑢𝑟,𝑣𝑟 are functions of 𝑟 only, such that the series for 𝛾𝑛 exists. Bailey's paper [2] published in the London Mathematical society in 1949, that strengthened the importance of Bailey’s transformation. The main result of the paper [2] was recognized as Bailey’s lemma during the 20th century. Making use of celebrated transformation, Bailey [1, 2] developed a number of transformations for both ordinary and basic hypergeometric series, and later on he successfully used these transformations to obtain a number of identities of the Rogers-Ramanujan type. The extensive use of Bailey transformation appeared in the papers of Slater [3, 4] and these papers were published in 1951 and 1952, respectively. Slater established 130 identities of the Rogers-Ramanujan type in [3, 4]. The platform provided by Bailey and Slater motivated a number of mathematicians namely Agarwal [5, 6], Andrews [79], Andrews and Warner [10], Bressoud et al. [11, 12], Denis et al. [13], Joshi and Vyas [14], Schilling and Warnaar [15], Singh [16], Srivastava [17], Verma and Jain [18, 19] and due to the contribution of these mathematicians, literatures of ordinary and basic hypergeometric series were enriched. In the present paper, making use of certain known summations of truncated series, an attempt has been made to establish transformation formulae involving poly-basic hypergeometric series.

2. Definitions and Notations

For real or complex 𝑞(|𝑞|<1), put (𝜆;𝑞)=𝑛=0(1𝜆𝑞𝑛).(2.1) Let (𝜆;𝑞)𝜇 be defined by (𝜆;𝑞)𝜇=(𝜆;𝑞)(𝜆𝑞𝜇;𝑞).(2.2)

For arbitrary parameters 𝜆 and 𝜇, so that (𝜆;𝑞)𝑛=1,𝑛=0,(1𝜆)(1𝜆𝑞)1𝜆𝑞𝑛1,𝑛𝜀(1,2,3),(2.3) the generalized basic hypergeometric series is defined by: 𝑟𝜙𝑠𝑎1,𝑎2,,𝑎𝑟𝑏;𝑞;𝑧1,𝑏2,,𝑏𝑠=𝑛=0𝑎1,𝑎2,,𝑎𝑟;𝑞𝑛𝑧𝑛𝑞,𝑏1,𝑏2,,𝑏𝑠;𝑞𝑛,(2.4) where (𝑎1,𝑎2,,𝑎𝑟;𝑞)𝑛=(𝑎1;𝑞)𝑛(𝑎2;𝑞)𝑛(𝑎𝑟;𝑞)𝑛 and max(|𝑞|,|𝑧|<1) for convergence.

The truncated basic hypergeometric series is defined by 𝑟𝜙𝑠𝑎1,𝑎2,,𝑎𝑟𝑏;𝑞;𝑧1,𝑏2,,𝑏𝑠𝑁=𝑁𝑛=0𝑎1,𝑎2,,𝑎𝑟;𝑞𝑛𝑧𝑛𝑞,𝑏1,𝑏2,,𝑏𝑠;𝑞𝑛.(2.5)

The polybasic hypergeometric series is defined by (cf. Gasper and Rahman [20, (3.9.1) page 85]): Φ𝑎1,𝑎2,,𝑎𝑟𝑐1,1,,𝑐1,𝑟1;;𝑐𝑚,1,,𝑐𝑚,𝑟𝑚;𝑞,𝑞1,,𝑞𝑚𝑏;𝑧1,𝑏2,,𝑏𝑟1𝑑1,1,,𝑑1,𝑟1;;𝑑𝑚,1,,𝑑𝑚,𝑟𝑚=𝑛=0𝑎1,𝑎2,,𝑎𝑟;𝑞𝑛𝑧𝑛𝑞,𝑏1,𝑏2,,𝑏𝑟1;𝑞𝑛𝑚𝑗=1𝑐𝑗,1,,𝑐𝑗,𝑟𝑗;𝑞𝑗𝑛𝑑𝑗,1,,𝑑𝑗,𝑟𝑗;𝑞𝑗𝑛,(2.6) where max(|𝑧|,|𝑞|,|𝑞1|,,|𝑞𝑚|)<1 for convergence.

The other notations appearing in this paper have their usual meaning. We will use the following summation formulae in our analysis: 2𝜙1𝑎,𝑦;𝑎𝑦𝑞;𝑞,𝑞𝑛=(𝑎𝑞,𝑦𝑞;𝑞)𝑛(𝑞,𝑎𝑦𝑞;𝑞)𝑛,(2.7) see [5, App.II(8)] 4𝜙3𝛼,𝑞𝛼,𝑞𝛼,𝑒;𝛼,𝛼,𝛼𝑞𝑒;1𝑞,𝑒𝑛=(𝛼𝑞,𝑒𝑞;𝑞)𝑛(𝑞,𝛼𝑞/𝑒;𝑞)𝑛𝑒𝑛,(2.8) see [5, App.II(8)] 6𝜙5𝛼,𝑞𝛼,𝑞𝛼,𝛽,𝛾,𝛿;𝛼,𝛼,𝛼𝑞𝛽,𝛼𝑞𝛾,𝛼𝑞𝛿;𝑞,𝑞𝑛=(𝛼𝑞,𝛽𝑞,𝛾𝑞,𝛿𝑞;𝑞)𝑛(𝑞,𝛼𝑞/𝛽,𝛼𝑞/𝛾,𝛼𝑞/𝛿;𝑞)𝑛,(2.9) see [5, App.II(25)] provided 𝛼=𝛽𝛾𝛿, 𝑛𝑟=0(1𝑎𝑝𝑟𝑞𝑟)(𝑎;𝑝)𝑟(𝑐;𝑞)𝑟𝑐𝑟(1𝑎)(𝑞;𝑞)𝑟(𝑎𝑝/𝑐;𝑝)𝑟=(𝑎𝑝;𝑝)𝑛(𝑐𝑞;𝑞)𝑛(𝑞;𝑞)𝑛(𝑎𝑝/𝑐;𝑝)𝑛𝑐𝑛,(2.10) see [20, App.II(II.34)] 𝑛𝑟=0(1𝑎𝑝𝑟𝑞𝑟)(1𝑏𝑝𝑟𝑞𝑟)(𝑎,𝑏;𝑝)𝑟(𝑐,𝑎/𝑏𝑐;𝑞)𝑟𝑞𝑟(1𝑎)(1𝑏)(𝑞,𝑎𝑞/𝑏;𝑞)𝑟(𝑎𝑝/𝑐,𝑏𝑐𝑝;𝑝)𝑟=(𝑎𝑝,𝑏𝑝;𝑝)𝑛(𝑐𝑞,𝑎𝑞/𝑏𝑐;𝑞)𝑛(𝑞,𝑎𝑞/𝑏;𝑞)𝑛(𝑎𝑝/𝑐,𝑏𝑐𝑝;𝑝)𝑛,(2.11)see [20, App.II(II.35)] 𝑛𝑟=0(1𝑎𝑑𝑝𝑟𝑞𝑟)(1𝑏𝑝𝑟/𝑑𝑞𝑟)(𝑎,𝑏;𝑝)𝑟𝑐,𝑎𝑑2/𝑏𝑐;𝑞𝑟𝑞𝑟(1𝑎𝑑)(1𝑏/𝑑)(𝑑𝑞,𝑎𝑑𝑞/𝑏;𝑞)𝑟(𝑎𝑑𝑝/𝑐,𝑏𝑐𝑝/𝑑;𝑝)𝑟=(1𝑎)(1𝑏)(1𝑐)1𝑎𝑑2/𝑏𝑐×𝑑(1𝑎𝑑)(1𝑏/𝑑)(1𝑐/𝑑)(1𝑎𝑑/𝑏𝑐)(𝑎𝑝,𝑏𝑝;𝑝)𝑛𝑐𝑞,𝑎𝑑2𝑞/𝑏𝑐;𝑞𝑛(𝑑𝑞,𝑎𝑑𝑞/𝑏;𝑞)𝑛(𝑎𝑑𝑝/𝑐,𝑏𝑐𝑝/𝑑;𝑝)𝑛(𝑐/𝑎𝑑,𝑑/𝑏𝑐;𝑝)1(1/𝑑,𝑏/𝑎𝑑;𝑞)11/𝑐,𝑏𝑐/𝑎𝑑2;𝑞1(1/𝑎,1/𝑏;𝑝)1,(2.12)which is 𝑚=0, case of [20, App. II (II. 36)].

3. Main Results

In this section we have established the following main results. Φ=[]𝛼𝑞,𝛽𝑞𝑎,𝑦;𝛼𝛽𝑞𝑝,𝑎𝑦𝑝;𝑞,𝑝;𝑝𝑎𝑝,𝑦𝑝;𝑝[]𝑝,𝑎𝑦𝑝;𝑝[]𝛼𝑞,𝛽𝑞;𝑞[]𝑞,𝛼𝛽𝑞;𝑞𝑞(1𝛼)(1𝛽)Φ(1𝑞)(1𝛼𝛽𝑞)𝑎𝑝,𝑦𝑝𝛼𝑞,𝛽𝑞;𝑎𝑦𝑝𝑞2,𝛼𝛽𝑞2;,Φ𝑝,𝑞;𝑞(3.1)𝛼𝑞,𝑒𝑞𝑎,𝑦;𝛼𝑞𝑒𝑝𝑝,𝑎𝑦𝑝;𝑞,𝑝;𝑒=1𝛼𝑞2(1𝑒)𝑒(1𝑞)(1𝛼𝑞/𝑒)×Φ𝑎𝑝,𝑦𝑝𝛼𝑞,𝑞2𝛼,𝑞2𝛼,𝑒𝑞;𝑎𝑦𝑝𝑞2,𝑞𝛼,𝑞𝛼,𝛼𝑞2𝑒;1𝑝,𝑞;𝑒,Φ(3.2)𝛼𝑞,𝛽𝑞,𝛾𝑞,𝛿𝑞𝑎,𝑦;𝛼𝑞𝛽,𝛼𝑞𝛾,𝛼𝑞𝛿=[]𝑝,𝑎𝑦𝑝;𝑞,𝑝;𝑝𝑎𝑝,𝑦𝑝;𝑝[]𝛼𝑞,𝛽𝑞,𝛾𝑞,𝛿𝑞;𝑞[]𝑝,𝑎𝑦𝑝𝑝[]𝑞,𝛼𝑞/𝛽,𝛼𝑞/𝛾,𝛼𝑞/𝛿;𝑞1𝑞2𝛼(1𝛽)(1𝛾)(1𝛿)𝑞(1𝑞)(1𝛼𝑞/𝛽)(1𝛼𝑞/𝛾)(1𝛼𝑞/𝛿)×Φ𝑎𝑝,𝑦𝑝𝛼𝑞,𝑞2𝛼,𝑞2𝛼,𝛽𝑞,𝛾𝑞,𝛿𝑞;𝑎𝑦𝑝𝑞2,𝑞𝛼,𝑞𝛼,𝛼𝑞2𝛽,𝛼𝑞2𝛾,𝛼𝑞2𝛿;,Φ𝑝,𝑞;𝑞(3.3)𝑥,𝑦𝑎𝑝𝑐𝑝;𝑥𝑦𝑃𝑎𝑝𝑐𝑃𝑞;𝑃,𝑝,𝑞;𝑐=(1𝑎𝑝𝑞)(1𝑐)(1𝑞)(1𝑎𝑝/𝑐)𝑐×Φ𝑥𝑃,𝑦𝑃𝑎𝑝𝑐𝑞𝑎𝑝2𝑞2;𝑥𝑦𝑃𝑎𝑝2𝑐𝑞21𝑎𝑝𝑞;𝑃,𝑝,𝑞,𝑝𝑞;𝑐,Φ(3.4)𝑥,𝑦𝑎𝑝,𝑏𝑝𝑐𝑞,𝑎𝑞;𝑏𝑐𝑥𝑦𝑃𝑎𝑝𝑐,𝑏𝑐𝑝𝑞,𝑎𝑞𝑏;=[]𝑃,𝑝,𝑞;𝑃𝑥𝑃,𝑦𝑃;𝑃[]𝑎𝑝,𝑏𝑝;𝑝[]𝑐𝑞,𝑎𝑞/𝑏𝑐;𝑞[]𝑃,𝑥𝑦𝑃;𝑃[]𝑞,𝑎𝑞/𝑏;𝑞[]𝑎𝑝/𝑐,𝑏𝑐𝑝;𝑝(1𝑎𝑝𝑞)(1𝑏𝑝/𝑞)(1𝑐)(1𝑎/𝑏𝑐)𝑞(1𝑞)(1𝑎𝑞/𝑏)(1𝑎𝑝/𝑐)(1𝑏𝑐𝑝)×Φ𝑥𝑃,𝑦𝑃𝑎𝑝2𝑞2𝑏𝑝2𝑞2𝑎𝑝,𝑏𝑝𝑐𝑞,𝑎𝑞;𝑏𝑐𝑥𝑦𝑃𝑎𝑝𝑞𝑏𝑝𝑞𝑎𝑝2𝑐,𝑏𝑐𝑝2𝑞2,𝑎𝑞2𝑏;𝑝𝑃,𝑝𝑞,𝑞,Φ,𝑝,𝑞;𝑞(3.5)𝑥,𝑦𝑎𝑝,𝑏𝑝𝑐𝑞,𝑎𝑑2𝑞;𝑏𝑐𝑥𝑦𝑃𝑎𝑑𝑝𝑐,𝑏𝑐𝑝𝑑𝑑𝑞,𝑎𝑑𝑞𝑏;=[]𝑃,𝑝,𝑞;𝑃𝑥𝑃,𝑦𝑃;𝑃[]𝑎𝑝,𝑏𝑝;𝑝𝑐𝑞,𝑎𝑑2𝑞/𝑏𝑐;𝑞[]𝑃,𝑥𝑦𝑃;𝑃[]𝑑𝑞,𝑎𝑑𝑞/𝑏;𝑞[]𝑎𝑑𝑝/𝑐,𝑏𝑐𝑝/𝑑;𝑝𝑑𝑞(1𝑎𝑑𝑝𝑞)(1𝑏𝑝/𝑑𝑞)(1𝑐/𝑑)(1𝑎𝑑/𝑏𝑐)(1𝑑𝑞)(1𝑎𝑑𝑞/𝑏)(1𝑎𝑑𝑝/𝑐)(1𝑏𝑐𝑝/𝑑)×Φ𝑥𝑃,𝑦𝑃𝑎𝑑𝑝2𝑞2𝑏𝑝2𝑑𝑞2𝑎𝑝,𝑏𝑝𝑐𝑞,𝑎𝑑2𝑞;𝑏𝑐𝑥𝑦𝑃𝑎𝑑𝑝𝑞𝑏𝑝𝑑𝑞𝑎𝑑𝑝2𝑐,𝑏𝑐𝑝2𝑑𝑑𝑞2,𝑎𝑑𝑞2𝑏;𝑝𝑃,𝑝𝑞,𝑞.,𝑝,𝑞;𝑞(3.6)

4. Proof of Main Results

Taking 𝑢𝑟=𝑣𝑟=1 in (1.1), Bailey's transformation takes the following form: If𝛽𝑛=𝑛𝑟=0𝛼𝑟,𝛾(4.1)𝑛=𝑟=0𝛿𝑟,(4.2)then𝑛=0𝛼𝑛𝛾𝑛=𝑛=0𝛽𝑛𝛿𝑛.(4.3)

Proof of Result (3.1). Taking 𝛼𝑟=(𝛼,𝛽;𝑞)𝑟𝑞𝑟/(𝑞,𝛼𝛽𝑞;𝑞)𝑟 and 𝛿𝑟=(𝑎,𝑦;𝑝)𝑟𝑝𝑟/(𝑝,𝑎𝑦𝑝;𝑝)𝑟 in (4.1) and (4.2), respectively, and making use of (2.7), we get 𝛽𝑛=(𝛼𝑞,𝛽𝑞;𝑞)𝑛(𝑞,𝛼𝛽𝑞;𝑞)𝑛,𝛾𝑛=(𝑎𝑝,𝑦𝑝;𝑝)(𝑝,𝑎𝑦𝑝;𝑝)(1𝑎𝑦)(1𝑝𝑛)(𝑎,𝑦;𝑝)𝑛(1𝑎)(1𝑦)(𝑝,𝑎𝑦;𝑝)𝑛.(4.4) Putting these values in (4.3), we get the following transformation: Φ+𝛼𝑞,𝛽𝑞𝑎,𝑦;𝛼𝛽𝑞𝑝,𝑎𝑦𝑝;𝑞,𝑝;𝑝(1𝑎𝑦)Φ=((1𝑎)(1𝑦)𝛼,𝛽𝑎,𝑦;𝛼𝛽𝑞𝑝,𝑎𝑦;𝑞,𝑝;𝑞𝑎𝑝,𝑦𝑝;𝑝)(𝑝,𝑎𝑦𝑝;𝑝)(𝛼𝑞,𝛽𝑞;𝑞)(𝑞,𝛼𝛽𝑞;𝑞)+(1𝑎𝑦)Φ,(1𝑎)(1𝑦)𝛼,𝛽𝑎,𝑦;𝛼𝛽𝑞𝑝,𝑎𝑦;𝑞,𝑝;𝑝𝑞(4.5) which on simplification gives the result (3.1).

Proof of Result (3.2). Taking 𝛼𝑟=(𝛼,𝑞𝛼,𝑞𝛼,𝑒;𝑞)𝑟/(𝑞,𝛼,𝛼,𝛼𝑞/𝑒;𝑞)𝑟𝑒𝑟 and 𝛿𝑟=(𝑎,𝑦;𝑝)𝑟𝑝𝑟/(𝑝,𝑎𝑦𝑝;𝑝)𝑟  in (4.1) and (4.2), respectively, and making use of (2.8) and (2.7), we get 𝛽𝑛=(𝛼𝑞,𝑒𝑞;𝑞)𝑛(𝑞,𝛼𝑞/𝑒;𝑞)𝑛𝑒𝑛,𝛾𝑛=(𝑎𝑝,𝑦𝑝;𝑝)(𝑝,𝑎𝑦𝑝;𝑝)(1𝑎𝑦)(1𝑝𝑛)(𝑎,𝑦;𝑝)𝑛(1𝑎)(1𝑦)(𝑝,𝑎𝑦;𝑝)𝑛.(4.6) Substituting these values in (4.3), we get the following transformation for |𝑒|>1: Φ𝛼𝑞,𝑒𝑞𝑎,𝑦;𝛼𝑞𝑒𝑝𝑝,𝑎𝑦𝑝;𝑞,𝑝;𝑒=(1𝑎𝑦)(1𝑎)(1𝑒)×Φ𝛼,𝑞𝛼,𝑞𝛼,𝑒𝑎,𝑦;𝛼,𝛼,𝛼𝑞𝑒𝑝𝑝,𝑎𝑦;𝑞,𝑝;𝑒(1𝑎𝑦)Φ(1𝑎)(1𝑦)𝛼,𝑞𝛼,𝑞𝛼,𝑒𝑎,𝑦;𝛼,𝛼,𝛼𝑞𝑒1𝑝,𝑎𝑦;𝑞,𝑝;𝑒,(4.7) which on simplification gives result (3.2).

Proof of Result (3.3). Taking 𝛼𝑟=(𝛼,𝑞𝛼,𝑞𝛼,𝛽,𝛾,𝛿;𝑞)𝑟𝑞𝑟/(𝑞,𝛼,𝛼,𝛼𝑞/𝛽,𝛼𝑞/𝛾,𝛼𝑞/𝛿;𝑞)𝑟, where 𝛼=𝛽𝛾𝛿 and 𝛿𝑟=(𝑎,𝑦;𝑝)𝑟𝑝𝑟/(𝑝,𝑎𝑦𝑝;𝑝)𝑟 in (4.1) and (4.2), respectively, and making use of (2.9) and (2.7), we get 𝛽𝑛=(𝛼𝑞,𝛽𝑞,𝛾𝑞,𝛿𝑞;𝑞)𝑛(𝑞,𝛼𝑞/𝛽,𝛼𝑞/𝛾,𝛼𝑞/𝛿;𝑞)𝑛,𝛾𝑛=(𝑎𝑝,𝑦𝑝;𝑝)(𝑝,𝑎𝑦𝑝;𝑝)(1𝑎𝑦)(1𝑝𝑛)(𝑎,𝑦;𝑝)𝑛(1𝑎)(1𝑦)(𝑝,𝑎𝑦;𝑝)𝑛.(4.8) Substituting these values in (4.3), we get the following transformation for 𝛼=𝛽𝛾𝛿: Φ𝛼𝑞,𝛽𝑞,𝛾𝑞,𝛿𝑞𝑎,𝑦;𝛼𝑞𝛽,𝛼𝑞𝛾,𝛼𝑞𝛿+𝑝,𝑎𝑦𝑝;𝑞,𝑝;𝑝(1𝑎𝑦)(1𝑎)(1𝑦)×Φ𝛼,𝑞𝛼,𝑞𝛼,𝛽,𝛾,𝛿𝑎,𝑦;𝛼,𝛼,𝛼𝑞𝛽,𝛼𝑞𝛾,𝛼𝑞𝛿=𝑝,𝑎𝑦;𝑞,𝑝;𝑞(𝑎𝑝,𝑦𝑝;𝑝)(𝑝,𝑎𝑦𝑝;𝑝)×(𝛼𝑞,𝛽𝑞,𝛾𝑞,𝛿𝑞;𝑞)(𝑞,𝛼𝑞/𝛽,𝛼𝑞/𝛾,𝛼𝑞/𝛿;𝑞)+(1𝑎𝑦)(1𝑎)(1𝑦)×Φ𝛼,𝑞𝛼,𝑞𝛼,𝛽,𝛾,𝛿𝑎,𝑦;𝛼,𝛼,𝛼𝑞𝛽,𝛼𝑞𝛾,𝛼𝑞𝛿,𝑝,𝑎𝑦;𝑞,𝑝;𝑝𝑞(4.9) which on simplification gives result (3.3).

Proof of Result (3.4). Taking 𝛼𝑟=(𝑎𝑝𝑞;𝑝𝑞)𝑟(𝑎;𝑝)𝑟(𝑐;𝑞)𝑟𝑐𝑟/((𝑎;𝑝𝑞)𝑟(𝑞;𝑞)𝑟(𝑎𝑝/𝑐;𝑝)𝑟) and 𝛿𝑟=(𝑥,𝑦;𝑃)𝑟𝑃𝑟/(𝑃,𝑥𝑦𝑃;𝑃)𝑟 in (4.1) and (4.2), respectively and making use of (2.10) and (2.7), we get 𝛽𝑛=(𝑎𝑝;𝑝)𝑛(𝑐𝑞;𝑞)𝑛𝑐𝑛(𝑞;𝑞)𝑛(𝑎𝑝/𝑐;𝑝)𝑛,𝛾𝑛=(𝑥𝑃,𝑦𝑃;𝑃)(𝑃,𝑥𝑦𝑃;𝑃)(1𝑥𝑦)(1𝑃𝑛)(𝑥,𝑦;𝑃)𝑛(1𝑥)(1𝑦)(𝑃,𝑥𝑦;𝑃)𝑛.(4.10) Putting these values in (4.3), we get the following transformation for |𝑐|>1: Φ𝑥,𝑦𝑎𝑝𝑐𝑞;𝑥𝑦𝑃𝑎𝑝𝑐𝑃𝑞;𝑃,𝑝,𝑞;𝑐=(1𝑥𝑦)(1𝑥)(1𝑦)×Φ𝑥,𝑦𝑎𝑝𝑞𝑎𝑐;𝑥𝑦𝑎𝑎𝑝𝑐𝑃𝑞;𝑃,𝑝𝑞,𝑝,𝑞;𝑐(1𝑥𝑦)(1𝑥)(1𝑦)×Φ𝑥,𝑦𝑎𝑝𝑞𝑎𝑐;𝑥𝑦𝑎𝑎𝑝𝑐1𝑞;𝑃,𝑝𝑞,𝑝,𝑞;𝑐,(4.11) which on simplification gives result (3.4).

Proof of Result (3.5). Taking 𝛼𝑟=(𝑎𝑝𝑞;𝑝𝑞)𝑟(𝑏𝑝/𝑞;𝑝/𝑞)𝑟(𝑎,𝑏;𝑝)𝑟(𝑐,𝑎/𝑏𝑐;𝑞)𝑟𝑞𝑟/((𝑎;𝑝𝑞)𝑟(𝑏;𝑝/𝑞)𝑟 (𝑞,𝑎𝑞/𝑏;𝑞)𝑟(𝑎𝑝/𝑐,𝑏𝑐𝑝;𝑝)𝑟) and 𝛿𝑟=(𝑥,𝑦;𝑃)𝑟𝑃𝑟/(𝑃,𝑥𝑦𝑃;𝑃)𝑟 in (4.1) and (4.2), respectively, and making use of (2.11) and (2.7), we get 𝛽𝑛=(𝑎𝑝,𝑏𝑝;𝑝)𝑛(𝑐𝑞,𝑎𝑞/𝑏𝑐;𝑞)𝑛(𝑞,𝑎𝑞/𝑏;𝑞)𝑛(𝑎𝑝/𝑐,𝑏𝑐𝑝;𝑝)𝑛,𝛾𝑛=(𝑥𝑃,𝑦𝑃;𝑃)(𝑃,𝑥𝑦𝑃;𝑃)(1𝑥𝑦)(1𝑃𝑛)(𝑥,𝑦;𝑃)𝑛(1𝑥)(1𝑦)(𝑃,𝑥𝑦;𝑃)𝑛.(4.12) Putting these values in (4.3), we get the following transformation: Φ𝑥,𝑦𝑎𝑝,𝑏𝑝𝑐𝑞,𝑎𝑞;𝑏𝑐𝑥𝑦𝑃𝑎𝑝𝑐,𝑏𝑐𝑝𝑞,𝑎𝑞𝑏;+𝑃,𝑝,𝑞;𝑃(1𝑥𝑦)(1𝑥)(1𝑦)×Φ𝑥,𝑦𝑎𝑝𝑞𝑏𝑝𝑞𝑎𝑎,𝑏𝑐,;𝑏𝑐𝑥𝑦𝑎𝑏𝑎𝑝𝑐,𝑏𝑐𝑝𝑞,𝑎𝑞𝑏;𝑝𝑃,𝑝𝑞,𝑞=,𝑝,𝑞;𝑞(𝑥𝑃,𝑦𝑃;𝑃)(𝑃,𝑥𝑦𝑃;𝑃)(𝑎𝑝,𝑏𝑝;𝑝)(𝑞,𝑎𝑞/𝑏;𝑞)(𝑐𝑞,𝑎𝑞/𝑏𝑐;𝑞)(𝑎𝑝/𝑐,𝑏𝑐𝑝;𝑝)+(1𝑥𝑦)(1𝑥)(1𝑦)×Φ𝑥,𝑦𝑎𝑝𝑞𝑏𝑝𝑞𝑎𝑎,𝑏𝑐,;𝑏𝑐𝑥𝑦𝑎𝑏𝑎𝑝𝑐,𝑏𝑐𝑝𝑞,𝑎𝑞𝑏;𝑝𝑃,𝑝𝑞,𝑞,,𝑝,𝑞;𝑃𝑞(4.13) which on simplification gives result (3.5).

Proof of Result (3.6). Taking 𝛼𝑟=(𝑎𝑑𝑝𝑞;𝑝𝑞)𝑟(𝑏𝑝/𝑑𝑞;𝑝/𝑞)𝑟(𝑎,𝑏;𝑝)𝑟(𝑐,𝑎𝑑2/𝑏𝑐;𝑞)𝑟𝑞𝑟/((𝑎𝑑;𝑝𝑞)𝑟(𝑏/𝑑;𝑝/𝑞)𝑟(𝑑𝑞,𝑎𝑑𝑞/𝑏;𝑞)𝑟(𝑎𝑑𝑝/𝑐,𝑏𝑐𝑝/𝑑;𝑝)𝑟) and 𝛿𝑟=(𝑥,𝑦;𝑃)𝑟𝑃𝑟/(𝑃,𝑥𝑦𝑃;𝑃)𝑟 in (4.1) and (4.2), respectively, and making use of (2.12) and (2.7), we get 𝛽𝑛=(1𝑎)(1𝑏)(1𝑐)1𝑎𝑑2/𝑏𝑐𝑑×(1𝑎𝑑)(1𝑏/𝑑)(1𝑐/𝑑)(1𝑎𝑑/𝑏𝑐)(𝑎𝑝,𝑏𝑝;𝑝)𝑛𝑐𝑞,𝑎𝑑2𝑞/𝑏𝑐;𝑞𝑛(𝑑𝑞,𝑎𝑑𝑞/𝑏;𝑞)𝑛(𝑎𝑑𝑝/𝑐,𝑏𝑐𝑝/𝑑;𝑝)𝑛(𝑏𝑎𝑑)(𝑐𝑎𝑑)(𝑑𝑏𝑐)(1𝑑)𝑑(1𝑎)(1𝑏)(1𝑐)𝑏𝑐𝑎𝑑2𝛾𝑛=(𝑥𝑃,𝑦𝑃;𝑃)(𝑃,𝑥𝑦𝑃;𝑃)(1𝑥𝑦)(1𝑃𝑛)(𝑥,𝑦;𝑃)𝑛(1𝑥)(1𝑦)(𝑃,𝑥𝑦;𝑃)𝑛.,(4.14) Putting these values in (4.3), we get the following transformation: (1𝑎)(1𝑏)(1𝑐)1𝑎𝑑2/𝑏𝑐𝑑(1𝑎𝑑)(1𝑏/𝑑)(1𝑐/𝑑)(1𝑎𝑑/𝑏𝑐)×Φ𝑥,𝑦𝑎𝑝,𝑏𝑝𝑐𝑞,𝑎𝑑2𝑞;𝑏𝑐𝑥𝑦𝑃𝑎𝑑𝑝𝑐,𝑏𝑐𝑝𝑑𝑑𝑞,𝑎𝑑𝑞𝑏;+𝑃,𝑝,𝑞;𝑃(1𝑥𝑦)(1𝑥)(1𝑦)×Φ𝑥,𝑦𝑎𝑑𝑝𝑞𝑏𝑝𝑑𝑞𝑎,𝑏𝑐,𝑎𝑑2;𝑏𝑏𝑐𝑥𝑦𝑎𝑑𝑑𝑎𝑑𝑝𝑐,𝑏𝑐𝑝𝑑𝑑𝑞,𝑎𝑑𝑞𝑏;𝑝𝑃,𝑝𝑞,𝑞=,𝑝,𝑞;𝑞(1𝑎)(1𝑏)(1𝑐)1𝑎𝑑2/𝑏𝑐×𝑑(1𝑎𝑑)(1𝑏/𝑑)(1𝑐/𝑑)(1𝑎𝑑/𝑏𝑐)(𝑥𝑃,𝑦𝑃;𝑃)(𝑎𝑝,𝑏𝑝;𝑝)𝑐𝑞,𝑎𝑑2𝑞/𝑏𝑐;𝑞(𝑃,𝑥𝑦𝑃;𝑃)(𝑑𝑞,𝑎𝑑𝑞/𝑏;𝑞)(𝑎𝑑𝑝/𝑐,𝑏𝑐𝑝/𝑑;𝑝)+(1𝑥𝑦)(1𝑥)(1𝑦)×Φ𝑥,𝑦𝑎𝑑𝑝𝑞𝑏𝑝𝑑𝑞𝑎,𝑏𝑐,𝑎𝑑2;𝑏𝑏𝑐𝑥𝑦𝑎𝑑𝑑𝑎𝑑𝑝𝑐,𝑏𝑐𝑝𝑑𝑑𝑞,𝑎𝑑𝑞𝑏;𝑝𝑃,𝑝𝑞,𝑞,,𝑝,𝑞;𝑃𝑞(4.15) which on simplification gives result (3.6).

References

  1. W. N. Bailey, “Some identities in combinatory analysis,” Proceedings of the London Mathematical Society, vol. 49, no. 2, pp. 421–425, 1947. View at Publisher · View at Google Scholar
  2. W. N. Bailey, “Identities of the Rogers-Ramanujan type,” Proceedings of the London Mathematical Society, vol. 50, pp. 1–10, 1948. View at Publisher · View at Google Scholar
  3. L. J. Slater, “A new proof of Rogers's transformations of infinite series,” Proceedings of the London Mathematical Society, vol. 53, pp. 460–475, 1951. View at Publisher · View at Google Scholar
  4. L. J. Slater, “Further identies of the Rogers-Ramanujan type,” Proceedings of the London Mathematical Society, vol. 54, pp. 147–167, 1952. View at Publisher · View at Google Scholar
  5. R. P. Agarwal, “Generalized hypergeometric series and its applications to the theory of combinatorial analysis and partition theory,” unpublished monograph.
  6. R. P. Agarwal, Resonance Of Ramanujan's Mathematics, Vol .I, New Age International, New Delhi, India, 1996.
  7. G. E. Andrews, “A general theory of identities of the Rogers-Ramanujan type,” Bulletin of the American Mathematical Society, vol. 80, pp. 1033–1052, 1974. View at Publisher · View at Google Scholar
  8. G. E. Andrews, “An analytic generalization of the Rogers-Ramanujan identities for odd moduli,” Proceedings of the National Academy of Sciences of the United States of America, vol. 71, pp. 4082–4085, 1974. View at Publisher · View at Google Scholar
  9. G. E. Andrews, “Bailey's transform, lemma, chains and tree,” in Special Functions 2000: Current Perspective and Future Directions, Tempe, AZ, vol. 30 of NATO Science Series. Series II, Mathematics, Physics and Chemistry, pp. 1–22, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2001. View at Publisher · View at Google Scholar
  10. G. E. Andrews and S. O. Warnaar, “The Bailey transform and false theta functions,” The Ramanujan Journal, vol. 14, no. 1, pp. 173–188, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  11. D. M. Bressoud, “Some identities for terminatingq-series,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 89, no. 2, pp. 211–223, 1981. View at Publisher · View at Google Scholar
  12. D. M. Bressoud, M. E. H. Ismail, and D. Stanton, “Change of base in Bailey pairs,” The Ramanujan Journal, vol. 4, no. 4, pp. 435–453, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  13. R. Y. Denis, S. N. Singh, and S. P. Singh, “On certain transformation formulae for abnormal q-series,” South East Asian Journal of Mathematics and Mathematical Sciences, vol. 1, no. 3, pp. 7–19, 2003.
  14. C. M. Joshi and Y. Vyas, “Extensions of Bailey's transform and applications,” International Journal of Mathematics and Mathematical Sciences, no. 12, pp. 1909–1923, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  15. A. Schilling and S. O. Warnaar, “A higher-level Bailey lemma,” International Journal of Modern Physics B, vol. 11, no. 1-2, pp. 189–195, 1997. View at Publisher · View at Google Scholar
  16. U. B. Singh, “A note on a transformation of Bailey,” The Quarterly Journal of Mathematics, vol. 45, no. 177, pp. 111–116, 1994. View at Publisher · View at Google Scholar
  17. P. Srivastava, “A note on Bailey's transform,” South East Asian Journal of Mathematics and Mathematical Sciences, vol. 2, no. 2, pp. 9–14, 2004.
  18. A. Verma and V. K. Jain, “Transformations between basic hypergeometric series on different bases and identities of Rogers-Ramanujan type,” Journal of Mathematical Analysis and Applications, vol. 76, no. 1, pp. 230–269, 1980. View at Publisher · View at Google Scholar
  19. A. Verma and V. K. Jain, “Transformations of nonterminating basic hypergeometric series, their contour integrals and applications to Rogers-Ramanujan identities,” Journal of Mathematical Analysis and Applications, vol. 87, no. 1, pp. 9–44, 1982. View at Publisher · View at Google Scholar
  20. G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, New York, NY, USA, 1991.