International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 248519 | https://doi.org/10.5402/2011/248519

Pankaj Srivastava, Mohan Rudravarapu, "Certain Transformation Formulae for Polybasic Hypergeometric Series", International Scholarly Research Notices, vol. 2011, Article ID 248519, 10 pages, 2011. https://doi.org/10.5402/2011/248519

Certain Transformation Formulae for Polybasic Hypergeometric Series

Academic Editor: A. Salemi
Received04 Aug 2011
Accepted21 Aug 2011
Published20 Oct 2011

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=𝑛𝑟=0𝛼𝑟𝑢𝑛−𝑟𝑣𝑛+𝑟,𝛾𝑛=âˆžî“ğ‘Ÿ=𝑛𝛿𝑟𝑢𝑟−𝑛𝑣𝑛+𝑟,thenâˆžî“ğ‘›=0𝛼𝑛𝛾𝑛=âˆžî“ğ‘›=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 [7–9], 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 ğ‘ž(|ğ‘ž|<1), put (𝜆;ğ‘ž)∞=âˆžî‘ğ‘›=0(1âˆ’ğœ†ğ‘žğ‘›).(2.1) Let (𝜆;ğ‘ž)𝜇 be defined by (𝜆;ğ‘ž)𝜇=(𝜆;ğ‘ž)∞(ğœ†ğ‘žğœ‡;ğ‘ž)∞.(2.2)

For arbitrary parameters 𝜆 and 𝜇, so that (𝜆;ğ‘ž)𝑛=1,𝑛=0,(1−𝜆)(1âˆ’ğœ†ğ‘ž)⋯1âˆ’ğœ†ğ‘žğ‘›âˆ’1,𝑛𝜀(1,2,3…),(2.3) the generalized basic hypergeometric series is defined by: ğ‘Ÿğœ™ğ‘ âŽ¡âŽ¢âŽ¢âŽ£ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘Ÿğ‘;ğ‘ž;𝑧1,𝑏2,…,ğ‘ğ‘ âŽ¤âŽ¥âŽ¥âŽ¦=âˆžî“ğ‘›=0î€·ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘Ÿî€¸;ğ‘žğ‘›ğ‘§ğ‘›î€·ğ‘ž,𝑏1,𝑏2,…,𝑏𝑠;ğ‘žğ‘›,(2.4) where (ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘Ÿ;ğ‘ž)𝑛=(ğ‘Ž1;ğ‘ž)𝑛(ğ‘Ž2;ğ‘ž)𝑛⋯(ğ‘Žğ‘Ÿ;ğ‘ž)𝑛 and max(|ğ‘ž|,|𝑧|<1) for convergence.

The truncated basic hypergeometric series is defined by ğ‘Ÿğœ™ğ‘ âŽ¡âŽ¢âŽ¢âŽ£ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘Ÿğ‘;ğ‘ž;𝑧1,𝑏2,…,ğ‘ğ‘ âŽ¤âŽ¥âŽ¥âŽ¦ğ‘=𝑁𝑛=0î€·ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘Ÿî€¸;ğ‘žğ‘›ğ‘§ğ‘›î€·ğ‘ž,𝑏1,𝑏2,…,𝑏𝑠;ğ‘žğ‘›.(2.5)

The polybasic hypergeometric series is defined by (cf. Gasper and Rahman [20, (3.9.1) page 85]): Î¦âŽ¡âŽ¢âŽ¢âŽ£ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘Ÿâˆ¶ğ‘1,1,…,𝑐1,𝑟1;…;𝑐𝑚,1,…,𝑐𝑚,𝑟𝑚;ğ‘ž,ğ‘ž1,…,ğ‘žğ‘šğ‘;𝑧1,𝑏2,…,𝑏𝑟−1∶𝑑1,1,…,𝑑1,𝑟1;…;𝑑𝑚,1,…,𝑑𝑚,ğ‘Ÿğ‘šâŽ¤âŽ¥âŽ¥âŽ¦=âˆžî“ğ‘›=0î€·ğ‘Ž1,ğ‘Ž2,…,ğ‘Žğ‘Ÿî€¸;ğ‘žğ‘›ğ‘§ğ‘›î€·ğ‘ž,𝑏1,𝑏2,…,𝑏𝑟−1;ğ‘žğ‘›ğ‘šî‘ğ‘—=1𝑐𝑗,1,…,𝑐𝑗,𝑟𝑗;ğ‘žğ‘—î‚ğ‘›î‚€ğ‘‘ğ‘—,1,…,𝑑𝑗,𝑟𝑗;ğ‘žğ‘—î‚ğ‘›,(2.6) where max(|𝑧|,|ğ‘ž|,|ğ‘ž1|,…,|ğ‘žğ‘š|)<1 for convergence.

The other notations appearing in this paper have their usual meaning. We will use the following summation formulae in our analysis: 2𝜙1âŽ¡âŽ¢âŽ¢âŽ£âŽ¤âŽ¥âŽ¥âŽ¦ğ‘Ž,𝑦;ğ‘Žğ‘¦ğ‘ž;ğ‘ž,ğ‘žğ‘›=(ğ‘Žğ‘ž,ğ‘¦ğ‘ž;ğ‘ž)𝑛(ğ‘ž,ğ‘Žğ‘¦ğ‘ž;ğ‘ž)𝑛,(2.7) see [5, App.II(8)] 4𝜙3âŽ¡âŽ¢âŽ¢âŽ£âˆšğ›¼,ğ‘žâˆšğ›¼,âˆ’ğ‘žâˆšğ›¼,𝑒;√𝛼,−𝛼,ğ›¼ğ‘žğ‘’;1ğ‘ž,ğ‘’âŽ¤âŽ¥âŽ¥âŽ¦ğ‘›=(ğ›¼ğ‘ž,ğ‘’ğ‘ž;ğ‘ž)𝑛(ğ‘ž,ğ›¼ğ‘ž/𝑒;ğ‘ž)𝑛𝑒𝑛,(2.8) see [5, App.II(8)] 6𝜙5âŽ¡âŽ¢âŽ¢âŽ¢âŽ£âˆšğ›¼,ğ‘žâˆšğ›¼,âˆ’ğ‘žâˆšğ›¼,𝛽,𝛾,𝛿;√𝛼,−𝛼,ğ›¼ğ‘žğ›½,ğ›¼ğ‘žğ›¾,ğ›¼ğ‘žğ›¿;âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦ğ‘ž,ğ‘žğ‘›=(ğ›¼ğ‘ž,ğ›½ğ‘ž,ğ›¾ğ‘ž,ğ›¿ğ‘ž;ğ‘ž)𝑛(ğ‘ž,ğ›¼ğ‘ž/𝛽,ğ›¼ğ‘ž/𝛾,ğ›¼ğ‘ž/𝛿;ğ‘ž)𝑛,(2.9) see [5, App.II(25)] provided 𝛼=𝛽𝛾𝛿, 𝑛𝑟=0(1âˆ’ğ‘Žğ‘ğ‘Ÿğ‘žğ‘Ÿ)(ğ‘Ž;𝑝)𝑟(𝑐;ğ‘ž)𝑟𝑐−𝑟(1âˆ’ğ‘Ž)(ğ‘ž;ğ‘ž)𝑟(ğ‘Žğ‘/𝑐;𝑝)𝑟=(ğ‘Žğ‘;𝑝)𝑛(ğ‘ğ‘ž;ğ‘ž)𝑛(ğ‘ž;ğ‘ž)𝑛(ğ‘Žğ‘/𝑐;𝑝)𝑛𝑐𝑛,(2.10) see [20, App.II(II.34)] 𝑛𝑟=0(1âˆ’ğ‘Žğ‘ğ‘Ÿğ‘žğ‘Ÿ)(1âˆ’ğ‘ğ‘ğ‘Ÿğ‘žâˆ’ğ‘Ÿ)(ğ‘Ž,𝑏;𝑝)𝑟(𝑐,ğ‘Ž/𝑏𝑐;ğ‘ž)ğ‘Ÿğ‘žğ‘Ÿ(1âˆ’ğ‘Ž)(1−𝑏)(ğ‘ž,ğ‘Žğ‘ž/𝑏;ğ‘ž)𝑟(ğ‘Žğ‘/𝑐,𝑏𝑐𝑝;𝑝)𝑟=(ğ‘Žğ‘,𝑏𝑝;𝑝)𝑛(ğ‘ğ‘ž,ğ‘Žğ‘ž/𝑏𝑐;ğ‘ž)𝑛(ğ‘ž,ğ‘Žğ‘ž/𝑏;ğ‘ž)𝑛(ğ‘Žğ‘/𝑐,𝑏𝑐𝑝;𝑝)𝑛,(2.11)see [20, App.II(II.35)] 𝑛𝑟=0(1âˆ’ğ‘Žğ‘‘ğ‘ğ‘Ÿğ‘žğ‘Ÿ)(1−𝑏𝑝𝑟/ğ‘‘ğ‘žğ‘Ÿ)(ğ‘Ž,𝑏;𝑝)𝑟𝑐,ğ‘Žğ‘‘2/𝑏𝑐;ğ‘žğ‘Ÿğ‘žğ‘Ÿ(1âˆ’ğ‘Žğ‘‘)(1−𝑏/𝑑)(ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘ž/𝑏;ğ‘ž)𝑟(ğ‘Žğ‘‘ğ‘/𝑐,𝑏𝑐𝑝/𝑑;𝑝)𝑟=(1âˆ’ğ‘Ž)(1−𝑏)(1−𝑐)1âˆ’ğ‘Žğ‘‘2/𝑏𝑐×𝑑(1âˆ’ğ‘Žğ‘‘)(1−𝑏/𝑑)(1−𝑐/𝑑)(1âˆ’ğ‘Žğ‘‘/𝑏𝑐)(ğ‘Žğ‘,𝑏𝑝;𝑝)ğ‘›î€·ğ‘ğ‘ž,ğ‘Žğ‘‘2î€¸ğ‘ž/𝑏𝑐;ğ‘žğ‘›(ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘ž/𝑏;ğ‘ž)𝑛(ğ‘Žğ‘‘ğ‘/𝑐,𝑏𝑐𝑝/𝑑;𝑝)𝑛−(𝑐/ğ‘Žğ‘‘,𝑑/𝑏𝑐;𝑝)1(1/𝑑,𝑏/ğ‘Žğ‘‘;ğ‘ž)11/𝑐,𝑏𝑐/ğ‘Žğ‘‘2;ğ‘ž1(1/ğ‘Ž,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⎡⎢⎢⎣⎤⎥⎥⎦=[]ğ›¼ğ‘ž,ğ›½ğ‘žâˆ¶ğ‘Ž,𝑦;ğ›¼ğ›½ğ‘žâˆ¶ğ‘,ğ‘Žğ‘¦ğ‘;ğ‘ž,𝑝;ğ‘ğ‘Žğ‘,𝑦𝑝;ğ‘âˆž[]𝑝,ğ‘Žğ‘¦ğ‘;ğ‘âˆž[]ğ›¼ğ‘ž,ğ›½ğ‘ž;ğ‘žâˆž[]ğ‘ž,ğ›¼ğ›½ğ‘ž;ğ‘žâˆžâˆ’ğ‘ž(1−𝛼)(1−𝛽)Φ⎡⎢⎢⎣(1âˆ’ğ‘ž)(1âˆ’ğ›¼ğ›½ğ‘ž)ğ‘Žğ‘,ğ‘¦ğ‘âˆ¶ğ›¼ğ‘ž,ğ›½ğ‘ž;ğ‘Žğ‘¦ğ‘âˆ¶ğ‘ž2,ğ›¼ğ›½ğ‘ž2;⎤⎥⎥⎦,Î¦âŽ¡âŽ¢âŽ¢âŽ£ğ‘,ğ‘ž;ğ‘ž(3.1)ğ›¼ğ‘ž,ğ‘’ğ‘žâˆ¶ğ‘Ž,𝑦;ğ›¼ğ‘žğ‘’ğ‘âˆ¶ğ‘,ğ‘Žğ‘¦ğ‘;ğ‘ž,𝑝;ğ‘’âŽ¤âŽ¥âŽ¥âŽ¦î€·=−1âˆ’ğ›¼ğ‘ž2(1−𝑒)ğ‘’âŽ¡âŽ¢âŽ¢âŽ£(1âˆ’ğ‘ž)(1âˆ’ğ›¼ğ‘ž/𝑒)Ã—Î¦ğ‘Žğ‘,ğ‘¦ğ‘âˆ¶ğ›¼ğ‘ž,ğ‘ž2√𝛼,âˆ’ğ‘ž2√𝛼,ğ‘’ğ‘ž;ğ‘Žğ‘¦ğ‘âˆ¶ğ‘ž2√,ğ‘žâˆšğ›¼,âˆ’ğ‘žğ›¼,ğ›¼ğ‘ž2𝑒;1𝑝,ğ‘ž;ğ‘’âŽ¤âŽ¥âŽ¥âŽ¦,Φ⎡⎢⎢⎢⎣(3.2)ğ›¼ğ‘ž,ğ›½ğ‘ž,ğ›¾ğ‘ž,ğ›¿ğ‘žâˆ¶ğ‘Ž,𝑦;ğ›¼ğ‘žğ›½,ğ›¼ğ‘žğ›¾,ğ›¼ğ‘žğ›¿âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦=[]∶𝑝,ğ‘Žğ‘¦ğ‘;ğ‘ž,𝑝;ğ‘ğ‘Žğ‘,𝑦𝑝;ğ‘âˆž[]ğ›¼ğ‘ž,ğ›½ğ‘ž,ğ›¾ğ‘ž,ğ›¿ğ‘ž;ğ‘žâˆž[]𝑝,ğ‘Žğ‘¦ğ‘âˆ¶ğ‘âˆž[]ğ‘ž,ğ›¼ğ‘ž/𝛽,ğ›¼ğ‘ž/𝛾,ğ›¼ğ‘ž/𝛿;ğ‘žâˆžâˆ’î€·1âˆ’ğ‘ž2𝛼(1−𝛽)(1−𝛾)(1−𝛿)ğ‘žâŽ¡âŽ¢âŽ¢âŽ¢âŽ£(1âˆ’ğ‘ž)(1âˆ’ğ›¼ğ‘ž/𝛽)(1âˆ’ğ›¼ğ‘ž/𝛾)(1âˆ’ğ›¼ğ‘ž/𝛿)Ã—Î¦ğ‘Žğ‘,ğ‘¦ğ‘âˆ¶ğ›¼ğ‘ž,ğ‘ž2√𝛼,âˆ’ğ‘ž2√𝛼,ğ›½ğ‘ž,ğ›¾ğ‘ž,ğ›¿ğ‘ž;ğ‘Žğ‘¦ğ‘âˆ¶ğ‘ž2√,ğ‘žâˆšğ›¼,âˆ’ğ‘žğ›¼,ğ›¼ğ‘ž2𝛽,ğ›¼ğ‘ž2𝛾,ğ›¼ğ‘ž2𝛿;⎤⎥⎥⎥⎦,Î¦âŽ¡âŽ¢âŽ¢âŽ£ğ‘,ğ‘ž;ğ‘ž(3.3)𝑥,ğ‘¦âˆ¶ğ‘Žğ‘âˆ¶ğ‘ğ‘;ğ‘¥ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘ğ‘ğ‘ƒâˆ¶ğ‘ž;𝑃,𝑝,ğ‘ž;ğ‘âŽ¤âŽ¥âŽ¥âŽ¦=(1âˆ’ğ‘Žğ‘ğ‘ž)(1−𝑐)⎡⎢⎢⎣(1âˆ’ğ‘ž)(1âˆ’ğ‘Žğ‘/𝑐)𝑐×Φ𝑥𝑃,ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘âˆ¶ğ‘ğ‘žâˆ¶ğ‘Žğ‘2ğ‘ž2;ğ‘¥ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘2ğ‘âˆ¶ğ‘ž21âˆ¶ğ‘Žğ‘ğ‘ž;𝑃,𝑝,ğ‘ž,ğ‘ğ‘ž;ğ‘âŽ¤âŽ¥âŽ¥âŽ¦,Φ⎡⎢⎢⎣(3.4)𝑥,ğ‘¦âˆ¶ğ‘Žğ‘,ğ‘ğ‘âˆ¶ğ‘ğ‘ž,ğ‘Žğ‘ž;ğ‘ğ‘ğ‘¥ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘ğ‘,ğ‘ğ‘ğ‘âˆ¶ğ‘ž,ğ‘Žğ‘žğ‘;⎤⎥⎥⎦=[]𝑃,𝑝,ğ‘ž;𝑃𝑥𝑃,𝑦𝑃;ğ‘ƒâˆž[]ğ‘Žğ‘,𝑏𝑝;ğ‘âˆž[]ğ‘ğ‘ž,ğ‘Žğ‘ž/𝑏𝑐;ğ‘žâˆž[]𝑃,𝑥𝑦𝑃;ğ‘ƒâˆž[]ğ‘ž,ğ‘Žğ‘ž/𝑏;ğ‘žâˆž[]ğ‘Žğ‘/𝑐,𝑏𝑐𝑝;ğ‘âˆžâˆ’(1âˆ’ğ‘Žğ‘ğ‘ž)(1−𝑏𝑝/ğ‘ž)(1−𝑐)(1âˆ’ğ‘Ž/𝑏𝑐)ğ‘žâŽ¡âŽ¢âŽ¢âŽ¢âŽ£(1âˆ’ğ‘ž)(1âˆ’ğ‘Žğ‘ž/𝑏)(1âˆ’ğ‘Žğ‘/𝑐)(1−𝑏𝑐𝑝)×Φ𝑥𝑃,ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘2ğ‘ž2∶𝑏𝑝2ğ‘ž2âˆ¶ğ‘Žğ‘,ğ‘ğ‘âˆ¶ğ‘ğ‘ž,ğ‘Žğ‘ž;ğ‘ğ‘ğ‘¥ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘ğ‘žâˆ¶ğ‘ğ‘ğ‘žâˆ¶ğ‘Žğ‘2𝑐,𝑏𝑐𝑝2âˆ¶ğ‘ž2,ğ‘Žğ‘ž2𝑏;𝑝𝑃,ğ‘ğ‘ž,ğ‘žâŽ¤âŽ¥âŽ¥âŽ¥âŽ¦,Φ⎡⎢⎢⎢⎣,𝑝,ğ‘ž;ğ‘ž(3.5)𝑥,ğ‘¦âˆ¶ğ‘Žğ‘,ğ‘ğ‘âˆ¶ğ‘ğ‘ž,ğ‘Žğ‘‘2ğ‘ž;ğ‘ğ‘ğ‘¥ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘‘ğ‘ğ‘,ğ‘ğ‘ğ‘ğ‘‘âˆ¶ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘žğ‘;⎤⎥⎥⎥⎦=[]𝑃,𝑝,ğ‘ž;𝑃𝑥𝑃,𝑦𝑃;ğ‘ƒâˆž[]ğ‘Žğ‘,𝑏𝑝;ğ‘âˆžî€ºğ‘ğ‘ž,ğ‘Žğ‘‘2î€»ğ‘ž/𝑏𝑐;ğ‘žâˆž[]𝑃,𝑥𝑦𝑃;ğ‘ƒâˆž[]ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘ž/𝑏;ğ‘žâˆž[]ğ‘Žğ‘‘ğ‘/𝑐,𝑏𝑐𝑝/𝑑;ğ‘âˆžâˆ’ğ‘‘ğ‘ž(1âˆ’ğ‘Žğ‘‘ğ‘ğ‘ž)(1−𝑏𝑝/ğ‘‘ğ‘ž)(1−𝑐/𝑑)(1âˆ’ğ‘Žğ‘‘/𝑏𝑐)(⎡⎢⎢⎢⎣1âˆ’ğ‘‘ğ‘ž)(1âˆ’ğ‘Žğ‘‘ğ‘ž/𝑏)(1âˆ’ğ‘Žğ‘‘ğ‘/𝑐)(1−𝑏𝑐𝑝/𝑑)×Φ𝑥𝑃,ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘‘ğ‘2ğ‘ž2∶𝑏𝑝2ğ‘‘ğ‘ž2âˆ¶ğ‘Žğ‘,ğ‘ğ‘âˆ¶ğ‘ğ‘ž,ğ‘Žğ‘‘2ğ‘ž;ğ‘ğ‘ğ‘¥ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘‘ğ‘ğ‘žâˆ¶ğ‘ğ‘âˆ¶ğ‘‘ğ‘žğ‘Žğ‘‘ğ‘2𝑐,𝑏𝑐𝑝2ğ‘‘âˆ¶ğ‘‘ğ‘ž2,ğ‘Žğ‘‘ğ‘ž2𝑏;𝑝𝑃,ğ‘ğ‘ž,ğ‘žâŽ¤âŽ¥âŽ¥âŽ¥âŽ¦.,𝑝,ğ‘ž;ğ‘ž(3.6)

4. Proof of Main Results

Taking 𝑢𝑟=𝑣𝑟=1 in (1.1), Bailey's transformation takes the following form: If𝛽𝑛=𝑛𝑟=0𝛼𝑟,𝛾(4.1)𝑛=âˆžî“ğ‘Ÿ=0𝛿𝑟,(4.2)thenâˆžî“ğ‘›=0𝛼𝑛𝛾𝑛=âˆžî“ğ‘›=0𝛽𝑛𝛿𝑛.(4.3)

Proof of Result (3.1). Taking 𝛼𝑟=(𝛼,𝛽;ğ‘ž)ğ‘Ÿğ‘žğ‘Ÿ/(ğ‘ž,ğ›¼ğ›½ğ‘ž;ğ‘ž)𝑟 and 𝛿𝑟=(ğ‘Ž,𝑦;𝑝)𝑟𝑝𝑟/(𝑝,ğ‘Žğ‘¦ğ‘;𝑝)𝑟 in (4.1) and (4.2), respectively, and making use of (2.7), we get 𝛽𝑛=(ğ›¼ğ‘ž,ğ›½ğ‘ž;ğ‘ž)𝑛(ğ‘ž,ğ›¼ğ›½ğ‘ž;ğ‘ž)𝑛,𝛾𝑛=(ğ‘Žğ‘,𝑦𝑝;𝑝)∞(𝑝,ğ‘Žğ‘¦ğ‘;𝑝)∞−(1âˆ’ğ‘Žğ‘¦)(1−𝑝𝑛)(ğ‘Ž,𝑦;𝑝)𝑛(1âˆ’ğ‘Ž)(1−𝑦)(𝑝,ğ‘Žğ‘¦;𝑝)𝑛.(4.4) Putting these values in (4.3), we get the following transformation: Φ⎡⎢⎢⎣⎤⎥⎥⎦+ğ›¼ğ‘ž,ğ›½ğ‘žâˆ¶ğ‘Ž,𝑦;ğ›¼ğ›½ğ‘žâˆ¶ğ‘,ğ‘Žğ‘¦ğ‘;ğ‘ž,𝑝;𝑝(1âˆ’ğ‘Žğ‘¦)Φ⎡⎢⎢⎣⎤⎥⎥⎦=((1âˆ’ğ‘Ž)(1−𝑦)𝛼,ğ›½âˆ¶ğ‘Ž,𝑦;ğ›¼ğ›½ğ‘žâˆ¶ğ‘,ğ‘Žğ‘¦;ğ‘ž,𝑝;ğ‘žğ‘Žğ‘,𝑦𝑝;𝑝)∞(𝑝,ğ‘Žğ‘¦ğ‘;𝑝)∞(ğ›¼ğ‘ž,ğ›½ğ‘ž;ğ‘ž)∞(ğ‘ž,ğ›¼ğ›½ğ‘ž;ğ‘ž)∞+(1âˆ’ğ‘Žğ‘¦)Φ⎡⎢⎢⎣⎤⎥⎥⎦,(1âˆ’ğ‘Ž)(1−𝑦)𝛼,ğ›½âˆ¶ğ‘Ž,𝑦;ğ›¼ğ›½ğ‘žâˆ¶ğ‘,ğ‘Žğ‘¦;ğ‘ž,𝑝;ğ‘ğ‘ž(4.5) which on simplification gives the result (3.1).

Proof of Result (3.2). Taking 𝛼𝑟√=(𝛼,ğ‘žâˆšğ›¼,âˆ’ğ‘žğ›¼,𝑒;ğ‘ž)𝑟√/(ğ‘ž,√𝛼,−𝛼,ğ›¼ğ‘ž/𝑒;ğ‘ž)𝑟𝑒𝑟 and 𝛿𝑟=(ğ‘Ž,𝑦;𝑝)𝑟𝑝𝑟/(𝑝,ğ‘Žğ‘¦ğ‘;𝑝)𝑟  in (4.1) and (4.2), respectively, and making use of (2.8) and (2.7), we get 𝛽𝑛=(ğ›¼ğ‘ž,ğ‘’ğ‘ž;ğ‘ž)𝑛(ğ‘ž,ğ›¼ğ‘ž/𝑒;ğ‘ž)𝑛𝑒𝑛,𝛾𝑛=(ğ‘Žğ‘,𝑦𝑝;𝑝)∞(𝑝,ğ‘Žğ‘¦ğ‘;𝑝)∞−(1âˆ’ğ‘Žğ‘¦)(1−𝑝𝑛)(ğ‘Ž,𝑦;𝑝)𝑛(1âˆ’ğ‘Ž)(1−𝑦)(𝑝,ğ‘Žğ‘¦;𝑝)𝑛.(4.6) Substituting these values in (4.3), we get the following transformation for |𝑒|>1: Î¦âŽ¡âŽ¢âŽ¢âŽ£ğ›¼ğ‘ž,ğ‘’ğ‘žâˆ¶ğ‘Ž,𝑦;ğ›¼ğ‘žğ‘’ğ‘âˆ¶ğ‘,ğ‘Žğ‘¦ğ‘;ğ‘ž,𝑝;ğ‘’âŽ¤âŽ¥âŽ¥âŽ¦=(1âˆ’ğ‘Žğ‘¦)⎡⎢⎢⎣√(1âˆ’ğ‘Ž)(1−𝑒)×Φ𝛼,ğ‘žâˆšğ›¼,âˆ’ğ‘žâˆšğ›¼,ğ‘’âˆ¶ğ‘Ž,𝑦;√𝛼,−𝛼,ğ›¼ğ‘žğ‘’ğ‘âˆ¶ğ‘,ğ‘Žğ‘¦;ğ‘ž,𝑝;ğ‘’âŽ¤âŽ¥âŽ¥âŽ¦âˆ’(1âˆ’ğ‘Žğ‘¦)Φ⎡⎢⎢⎣√(1âˆ’ğ‘Ž)(1−𝑦)𝛼,ğ‘žâˆšğ›¼,âˆ’ğ‘žâˆšğ›¼,ğ‘’âˆ¶ğ‘Ž,𝑦;√𝛼,−𝛼,ğ›¼ğ‘žğ‘’1∶𝑝,ğ‘Žğ‘¦;ğ‘ž,𝑝;ğ‘’âŽ¤âŽ¥âŽ¥âŽ¦,(4.7) which on simplification gives result (3.2).

Proof of Result (3.3). Taking 𝛼𝑟√=(𝛼,ğ‘žâˆšğ›¼,âˆ’ğ‘žğ›¼,𝛽,𝛾,𝛿;ğ‘ž)ğ‘Ÿğ‘žğ‘Ÿâˆš/(ğ‘ž,√𝛼,−𝛼,ğ›¼ğ‘ž/𝛽,ğ›¼ğ‘ž/𝛾,ğ›¼ğ‘ž/𝛿;ğ‘ž)𝑟, where 𝛼=𝛽𝛾𝛿 and 𝛿𝑟=(ğ‘Ž,𝑦;𝑝)𝑟𝑝𝑟/(𝑝,ğ‘Žğ‘¦ğ‘;𝑝)𝑟 in (4.1) and (4.2), respectively, and making use of (2.9) and (2.7), we get 𝛽𝑛=(ğ›¼ğ‘ž,ğ›½ğ‘ž,ğ›¾ğ‘ž,ğ›¿ğ‘ž;ğ‘ž)𝑛(ğ‘ž,ğ›¼ğ‘ž/𝛽,ğ›¼ğ‘ž/𝛾,ğ›¼ğ‘ž/𝛿;ğ‘ž)𝑛,𝛾𝑛=(ğ‘Žğ‘,𝑦𝑝;𝑝)∞(𝑝,ğ‘Žğ‘¦ğ‘;𝑝)∞−(1âˆ’ğ‘Žğ‘¦)(1−𝑝𝑛)(ğ‘Ž,𝑦;𝑝)𝑛(1âˆ’ğ‘Ž)(1−𝑦)(𝑝,ğ‘Žğ‘¦;𝑝)𝑛.(4.8) Substituting these values in (4.3), we get the following transformation for 𝛼=𝛽𝛾𝛿: Î¦âŽ¡âŽ¢âŽ¢âŽ¢âŽ£ğ›¼ğ‘ž,ğ›½ğ‘ž,ğ›¾ğ‘ž,ğ›¿ğ‘žâˆ¶ğ‘Ž,𝑦;ğ›¼ğ‘žğ›½,ğ›¼ğ‘žğ›¾,ğ›¼ğ‘žğ›¿âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦+∶𝑝,ğ‘Žğ‘¦ğ‘;ğ‘ž,𝑝;𝑝(1âˆ’ğ‘Žğ‘¦)⎡⎢⎢⎢⎣√(1âˆ’ğ‘Ž)(1−𝑦)×Φ𝛼,ğ‘žâˆšğ›¼,âˆ’ğ‘žâˆšğ›¼,𝛽,𝛾,ğ›¿âˆ¶ğ‘Ž,𝑦;√𝛼,−𝛼,ğ›¼ğ‘žğ›½,ğ›¼ğ‘žğ›¾,ğ›¼ğ‘žğ›¿âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦=∶𝑝,ğ‘Žğ‘¦;ğ‘ž,𝑝;ğ‘ž(ğ‘Žğ‘,𝑦𝑝;𝑝)∞(𝑝,ğ‘Žğ‘¦ğ‘;𝑝)∞×(ğ›¼ğ‘ž,ğ›½ğ‘ž,ğ›¾ğ‘ž,ğ›¿ğ‘ž;ğ‘ž)∞(ğ‘ž,ğ›¼ğ‘ž/𝛽,ğ›¼ğ‘ž/𝛾,ğ›¼ğ‘ž/𝛿;ğ‘ž)∞+(1âˆ’ğ‘Žğ‘¦)⎡⎢⎢⎢⎣√(1âˆ’ğ‘Ž)(1−𝑦)×Φ𝛼,ğ‘žâˆšğ›¼,âˆ’ğ‘žâˆšğ›¼,𝛽,𝛾,ğ›¿âˆ¶ğ‘Ž,𝑦;√𝛼,−𝛼,ğ›¼ğ‘žğ›½,ğ›¼ğ‘žğ›¾,ğ›¼ğ‘žğ›¿âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦,∶𝑝,ğ‘Žğ‘¦;ğ‘ž,𝑝;ğ‘ğ‘ž(4.9) which on simplification gives result (3.3).

Proof of Result (3.4). Taking 𝛼𝑟=(ğ‘Žğ‘ğ‘ž;ğ‘ğ‘ž)𝑟(ğ‘Ž;𝑝)𝑟(𝑐;ğ‘ž)𝑟𝑐−𝑟/((ğ‘Ž;ğ‘ğ‘ž)𝑟(ğ‘ž;ğ‘ž)𝑟(ğ‘Žğ‘/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=(ğ‘Žğ‘;𝑝)𝑛(ğ‘ğ‘ž;ğ‘ž)𝑛𝑐−𝑛(ğ‘ž;ğ‘ž)𝑛(ğ‘Žğ‘/𝑐;𝑝)𝑛,𝛾𝑛=(𝑥𝑃,𝑦𝑃;𝑃)∞(𝑃,𝑥𝑦𝑃;𝑃)∞−(1−𝑥𝑦)(1−𝑃𝑛)(𝑥,𝑦;𝑃)𝑛(1−𝑥)(1−𝑦)(𝑃,𝑥𝑦;𝑃)𝑛.(4.10) Putting these values in (4.3), we get the following transformation for |𝑐|>1: Î¦âŽ¡âŽ¢âŽ¢âŽ£ğ‘¥,ğ‘¦âˆ¶ğ‘Žğ‘âˆ¶ğ‘ğ‘ž;ğ‘¥ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘ğ‘ğ‘ƒâˆ¶ğ‘ž;𝑃,𝑝,ğ‘ž;ğ‘âŽ¤âŽ¥âŽ¥âŽ¦=(1−𝑥𝑦)⎡⎢⎢⎣(1−𝑥)(1−𝑦)×Φ𝑥,ğ‘¦âˆ¶ğ‘Žğ‘ğ‘žâˆ¶ğ‘Žâˆ¶ğ‘;ğ‘¥ğ‘¦âˆ¶ğ‘Žâˆ¶ğ‘Žğ‘ğ‘ğ‘ƒâˆ¶ğ‘ž;𝑃,ğ‘ğ‘ž,𝑝,ğ‘ž;ğ‘âŽ¤âŽ¥âŽ¥âŽ¦âˆ’(1−𝑥𝑦)⎡⎢⎢⎣(1−𝑥)(1−𝑦)×Φ𝑥,ğ‘¦âˆ¶ğ‘Žğ‘ğ‘žâˆ¶ğ‘Žâˆ¶ğ‘;ğ‘¥ğ‘¦âˆ¶ğ‘Žâˆ¶ğ‘Žğ‘ğ‘1âˆ¶ğ‘ž;𝑃,ğ‘ğ‘ž,𝑝,ğ‘ž;ğ‘âŽ¤âŽ¥âŽ¥âŽ¦,(4.11) which on simplification gives result (3.4).

Proof of Result (3.5). Taking 𝛼𝑟=(ğ‘Žğ‘ğ‘ž;ğ‘ğ‘ž)𝑟(𝑏𝑝/ğ‘ž;𝑝/ğ‘ž)𝑟(ğ‘Ž,𝑏;𝑝)𝑟(𝑐,ğ‘Ž/𝑏𝑐;ğ‘ž)ğ‘Ÿğ‘žğ‘Ÿ/((ğ‘Ž;ğ‘ğ‘ž)𝑟(𝑏;𝑝/ğ‘ž)𝑟 (ğ‘ž,ğ‘Žğ‘ž/𝑏;ğ‘ž)𝑟(ğ‘Žğ‘/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=(ğ‘Žğ‘,𝑏𝑝;𝑝)𝑛(ğ‘ğ‘ž,ğ‘Žğ‘ž/𝑏𝑐;ğ‘ž)𝑛(ğ‘ž,ğ‘Žğ‘ž/𝑏;ğ‘ž)𝑛(ğ‘Žğ‘/𝑐,𝑏𝑐𝑝;𝑝)𝑛,𝛾𝑛=(𝑥𝑃,𝑦𝑃;𝑃)∞(𝑃,𝑥𝑦𝑃;𝑃)∞−(1−𝑥𝑦)(1−𝑃𝑛)(𝑥,𝑦;𝑃)𝑛(1−𝑥)(1−𝑦)(𝑃,𝑥𝑦;𝑃)𝑛.(4.12) Putting these values in (4.3), we get the following transformation: Î¦âŽ¡âŽ¢âŽ¢âŽ£ğ‘¥,ğ‘¦âˆ¶ğ‘Žğ‘,ğ‘ğ‘âˆ¶ğ‘ğ‘ž,ğ‘Žğ‘ž;ğ‘ğ‘ğ‘¥ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘ğ‘,ğ‘ğ‘ğ‘âˆ¶ğ‘ž,ğ‘Žğ‘žğ‘;⎤⎥⎥⎦+𝑃,𝑝,ğ‘ž;𝑃(1−𝑥𝑦)⎡⎢⎢⎢⎣(1−𝑥)(1−𝑦)×Φ𝑥,ğ‘¦âˆ¶ğ‘Žğ‘ğ‘žâˆ¶ğ‘ğ‘ğ‘žğ‘Žâˆ¶ğ‘Ž,𝑏∶𝑐,;ğ‘ğ‘ğ‘¥ğ‘¦âˆ¶ğ‘Žâˆ¶ğ‘âˆ¶ğ‘Žğ‘ğ‘,ğ‘ğ‘ğ‘âˆ¶ğ‘ž,ğ‘Žğ‘žğ‘;𝑝𝑃,ğ‘ğ‘ž,ğ‘žâŽ¤âŽ¥âŽ¥âŽ¥âŽ¦=,𝑝,ğ‘ž;ğ‘ž(𝑥𝑃,𝑦𝑃;𝑃)∞(𝑃,𝑥𝑦𝑃;𝑃)∞(ğ‘Žğ‘,𝑏𝑝;𝑝)∞(ğ‘ž,ğ‘Žğ‘ž/𝑏;ğ‘ž)∞(ğ‘ğ‘ž,ğ‘Žğ‘ž/𝑏𝑐;ğ‘ž)∞(ğ‘Žğ‘/𝑐,𝑏𝑐𝑝;𝑝)∞+(1−𝑥𝑦)(⎡⎢⎢⎢⎣1−𝑥)(1−𝑦)×Φ𝑥,ğ‘¦âˆ¶ğ‘Žğ‘ğ‘žâˆ¶ğ‘ğ‘ğ‘žğ‘Žâˆ¶ğ‘Ž,𝑏∶𝑐,;ğ‘ğ‘ğ‘¥ğ‘¦âˆ¶ğ‘Žâˆ¶ğ‘âˆ¶ğ‘Žğ‘ğ‘,ğ‘ğ‘ğ‘âˆ¶ğ‘ž,ğ‘Žğ‘žğ‘;𝑝𝑃,ğ‘ğ‘ž,ğ‘žâŽ¤âŽ¥âŽ¥âŽ¥âŽ¦,,𝑝,ğ‘ž;ğ‘ƒğ‘ž(4.13) which on simplification gives result (3.5).

Proof of Result (3.6). Taking 𝛼𝑟=(ğ‘Žğ‘‘ğ‘ğ‘ž;ğ‘ğ‘ž)𝑟(𝑏𝑝/ğ‘‘ğ‘ž;𝑝/ğ‘ž)𝑟(ğ‘Ž,𝑏;𝑝)𝑟(𝑐,ğ‘Žğ‘‘2/𝑏𝑐;ğ‘ž)ğ‘Ÿğ‘žğ‘Ÿ/((ğ‘Žğ‘‘;ğ‘ğ‘ž)𝑟(𝑏/𝑑;𝑝/ğ‘ž)𝑟(ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘ž/𝑏;ğ‘ž)𝑟(ğ‘Žğ‘‘ğ‘/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=(1âˆ’ğ‘Ž)(1−𝑏)(1−𝑐)1âˆ’ğ‘Žğ‘‘2/𝑏𝑐𝑑×(1âˆ’ğ‘Žğ‘‘)(1−𝑏/𝑑)(1−𝑐/𝑑)(1âˆ’ğ‘Žğ‘‘/𝑏𝑐)(ğ‘Žğ‘,𝑏𝑝;𝑝)ğ‘›î€·ğ‘ğ‘ž,ğ‘Žğ‘‘2î€¸ğ‘ž/𝑏𝑐;ğ‘žğ‘›(ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘ž/𝑏;ğ‘ž)𝑛(ğ‘Žğ‘‘ğ‘/𝑐,𝑏𝑐𝑝/𝑑;𝑝)𝑛−(ğ‘âˆ’ğ‘Žğ‘‘)(ğ‘âˆ’ğ‘Žğ‘‘)(𝑑−𝑏𝑐)(1−𝑑)𝑑(1âˆ’ğ‘Ž)(1−𝑏)(1−𝑐)ğ‘ğ‘âˆ’ğ‘Žğ‘‘2𝛾𝑛=(𝑥𝑃,𝑦𝑃;𝑃)∞(𝑃,𝑥𝑦𝑃;𝑃)∞−(1−𝑥𝑦)(1−𝑃𝑛)(𝑥,𝑦;𝑃)𝑛(1−𝑥)(1−𝑦)(𝑃,𝑥𝑦;𝑃)𝑛.,(4.14) Putting these values in (4.3), we get the following transformation: (1âˆ’ğ‘Ž)(1−𝑏)(1−𝑐)1âˆ’ğ‘Žğ‘‘2/ğ‘ğ‘âŽ¡âŽ¢âŽ¢âŽ¢âŽ£ğ‘‘(1âˆ’ğ‘Žğ‘‘)(1−𝑏/𝑑)(1−𝑐/𝑑)(1âˆ’ğ‘Žğ‘‘/𝑏𝑐)×Φ𝑥,ğ‘¦âˆ¶ğ‘Žğ‘,ğ‘ğ‘âˆ¶ğ‘ğ‘ž,ğ‘Žğ‘‘2ğ‘ž;ğ‘ğ‘ğ‘¥ğ‘¦ğ‘ƒâˆ¶ğ‘Žğ‘‘ğ‘ğ‘,ğ‘ğ‘ğ‘ğ‘‘âˆ¶ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘žğ‘;⎤⎥⎥⎥⎦+𝑃,𝑝,ğ‘ž;𝑃(1−𝑥𝑦)⎡⎢⎢⎢⎣(1−𝑥)(1−𝑦)×Φ𝑥,ğ‘¦âˆ¶ğ‘Žğ‘‘ğ‘ğ‘žâˆ¶ğ‘ğ‘ğ‘‘ğ‘žâˆ¶ğ‘Ž,𝑏∶𝑐,ğ‘Žğ‘‘2;ğ‘ğ‘ğ‘ğ‘¥ğ‘¦âˆ¶ğ‘Žğ‘‘âˆ¶ğ‘‘âˆ¶ğ‘Žğ‘‘ğ‘ğ‘,ğ‘ğ‘ğ‘ğ‘‘âˆ¶ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘žğ‘;𝑝𝑃,ğ‘ğ‘ž,ğ‘žâŽ¤âŽ¥âŽ¥âŽ¥âŽ¦=,𝑝,ğ‘ž;ğ‘ž(1âˆ’ğ‘Ž)(1−𝑏)(1−𝑐)1âˆ’ğ‘Žğ‘‘2/𝑏𝑐×𝑑(1âˆ’ğ‘Žğ‘‘)(1−𝑏/𝑑)(1−𝑐/𝑑)(1âˆ’ğ‘Žğ‘‘/𝑏𝑐)(𝑥𝑃,𝑦𝑃;𝑃)∞(ğ‘Žğ‘,𝑏𝑝;𝑝)âˆžî€·ğ‘ğ‘ž,ğ‘Žğ‘‘2î€¸ğ‘ž/𝑏𝑐;ğ‘žâˆž(𝑃,𝑥𝑦𝑃;𝑃)∞(ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘ž/𝑏;ğ‘ž)∞(ğ‘Žğ‘‘ğ‘/𝑐,𝑏𝑐𝑝/𝑑;𝑝)∞+(1−𝑥𝑦)⎡⎢⎢⎢⎣(1−𝑥)(1−𝑦)×Φ𝑥,ğ‘¦âˆ¶ğ‘Žğ‘‘ğ‘ğ‘žâˆ¶ğ‘ğ‘ğ‘‘ğ‘žâˆ¶ğ‘Ž,𝑏∶𝑐,ğ‘Žğ‘‘2;ğ‘ğ‘ğ‘ğ‘¥ğ‘¦âˆ¶ğ‘Žğ‘‘âˆ¶ğ‘‘âˆ¶ğ‘Žğ‘‘ğ‘ğ‘,ğ‘ğ‘ğ‘ğ‘‘âˆ¶ğ‘‘ğ‘ž,ğ‘Žğ‘‘ğ‘žğ‘;𝑝𝑃,ğ‘ğ‘ž,ğ‘žâŽ¤âŽ¥âŽ¥âŽ¥âŽ¦,,𝑝,ğ‘ž;ğ‘ƒğ‘ž(4.15) which on simplification gives result (3.6).

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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.


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