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=π‘›ξ“π‘Ÿ=0π›Όπ‘Ÿπ‘’π‘›βˆ’π‘Ÿπ‘£π‘›+π‘Ÿ,𝛾𝑛=βˆžξ“π‘Ÿ=π‘›π›Ώπ‘Ÿπ‘’π‘Ÿβˆ’π‘›π‘£π‘›+π‘Ÿ,thenβˆžξ“π‘›=0𝛼𝑛𝛾𝑛=βˆžξ“π‘›=0𝛽𝑛𝛿𝑛,(1.1) where π›Όπ‘Ÿ,π›Ώπ‘Ÿ,π‘’π‘Ÿ,π‘£π‘Ÿ are functions of π‘Ÿ 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 π‘ž(|π‘ž|<1), put (πœ†;π‘ž)∞=βˆžξ‘π‘›=0(1βˆ’πœ†π‘žπ‘›).(2.1) Let (πœ†;π‘ž)πœ‡ be defined by (πœ†;π‘ž)πœ‡=(πœ†;π‘ž)∞(πœ†π‘žπœ‡;π‘ž)∞.(2.2)

For arbitrary parameters πœ† and πœ‡, so that (πœ†;π‘ž)𝑛=ξ‚»ξ€·1,𝑛=0,(1βˆ’πœ†)(1βˆ’πœ†π‘ž)β‹―1βˆ’πœ†π‘žπ‘›βˆ’1ξ€Έ,π‘›πœ€(1,2,3…),(2.3) the generalized basic hypergeometric series is defined by: π‘Ÿπœ™π‘ βŽ‘βŽ’βŽ’βŽ£π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿπ‘;π‘ž;𝑧1,𝑏2,…,π‘π‘ βŽ€βŽ₯βŽ₯⎦=βˆžξ“π‘›=0ξ€·π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿξ€Έ;π‘žπ‘›π‘§π‘›ξ€·π‘ž,𝑏1,𝑏2,…,𝑏𝑠;π‘žπ‘›,(2.4) where (π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿ;π‘ž)𝑛=(π‘Ž1;π‘ž)𝑛(π‘Ž2;π‘ž)𝑛⋯(π‘Žπ‘Ÿ;π‘ž)𝑛 and max(|π‘ž|,|𝑧|<1) for convergence.

The truncated basic hypergeometric series is defined by π‘Ÿπœ™π‘ βŽ‘βŽ’βŽ’βŽ£π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿπ‘;π‘ž;𝑧1,𝑏2,…,π‘π‘ βŽ€βŽ₯βŽ₯βŽ¦π‘=𝑁𝑛=0ξ€·π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿξ€Έ;π‘žπ‘›π‘§π‘›ξ€·π‘ž,𝑏1,𝑏2,…,𝑏𝑠;π‘žπ‘›.(2.5)

The polybasic hypergeometric series is defined by (cf. Gasper and Rahman [20, (3.9.1) page 85]): Ξ¦βŽ‘βŽ’βŽ’βŽ£π‘Ž1,π‘Ž2,…,π‘Žπ‘ŸβˆΆπ‘1,1,…,𝑐1,π‘Ÿ1;…;π‘π‘š,1,…,π‘π‘š,π‘Ÿπ‘š;π‘ž,π‘ž1,…,π‘žπ‘šπ‘;𝑧1,𝑏2,…,π‘π‘Ÿβˆ’1βˆΆπ‘‘1,1,…,𝑑1,π‘Ÿ1;…;π‘‘π‘š,1,…,π‘‘π‘š,π‘Ÿπ‘šβŽ€βŽ₯βŽ₯⎦=βˆžξ“π‘›=0ξ€·π‘Ž1,π‘Ž2,…,π‘Žπ‘Ÿξ€Έ;π‘žπ‘›π‘§π‘›ξ€·π‘ž,𝑏1,𝑏2,…,π‘π‘Ÿβˆ’1ξ€Έ;π‘žπ‘›π‘šξ‘π‘—=1𝑐𝑗,1,…,𝑐𝑗,π‘Ÿπ‘—;π‘žπ‘—ξ‚π‘›ξ‚€π‘‘π‘—,1,…,𝑑𝑗,π‘Ÿπ‘—;π‘žπ‘—ξ‚π‘›,(2.6) where max(|𝑧|,|π‘ž|,|π‘ž1|,…,|π‘žπ‘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: 2πœ™1⎑⎒⎒⎣⎀βŽ₯βŽ₯βŽ¦π‘Ž,𝑦;π‘Žπ‘¦π‘ž;π‘ž,π‘žπ‘›=(π‘Žπ‘ž,π‘¦π‘ž;π‘ž)𝑛(π‘ž,π‘Žπ‘¦π‘ž;π‘ž)𝑛,(2.7) see [5, App.II(8)] 4πœ™3βŽ‘βŽ’βŽ’βŽ£βˆšπ›Ό,π‘žβˆšπ›Ό,βˆ’π‘žβˆšπ›Ό,𝑒;βˆšπ›Ό,βˆ’π›Ό,π›Όπ‘žπ‘’;1π‘ž,π‘’βŽ€βŽ₯βŽ₯βŽ¦π‘›=(π›Όπ‘ž,π‘’π‘ž;π‘ž)𝑛(π‘ž,π›Όπ‘ž/𝑒;π‘ž)𝑛𝑒𝑛,(2.8) see [5, App.II(8)] 6πœ™5βŽ‘βŽ’βŽ’βŽ’βŽ£βˆšπ›Ό,π‘žβˆšπ›Ό,βˆ’π‘žβˆšπ›Ό,𝛽,𝛾,𝛿;βˆšπ›Ό,βˆ’π›Ό,π›Όπ‘žπ›½,π›Όπ‘žπ›Ύ,π›Όπ‘žπ›Ώ;⎀βŽ₯βŽ₯βŽ₯βŽ¦π‘ž,π‘žπ‘›=(π›Όπ‘ž,π›½π‘ž,π›Ύπ‘ž,π›Ώπ‘ž;π‘ž)𝑛(π‘ž,π›Όπ‘ž/𝛽,π›Όπ‘ž/𝛾,π›Όπ‘ž/𝛿;π‘ž)𝑛,(2.9) see [5, App.II(25)] provided 𝛼=𝛽𝛾𝛿, π‘›ξ“π‘Ÿ=0(1βˆ’π‘Žπ‘π‘Ÿπ‘žπ‘Ÿ)(π‘Ž;𝑝)π‘Ÿ(𝑐;π‘ž)π‘Ÿπ‘βˆ’π‘Ÿ(1βˆ’π‘Ž)(π‘ž;π‘ž)π‘Ÿ(π‘Žπ‘/𝑐;𝑝)π‘Ÿ=(π‘Žπ‘;𝑝)𝑛(π‘π‘ž;π‘ž)𝑛(π‘ž;π‘ž)𝑛(π‘Žπ‘/𝑐;𝑝)𝑛𝑐𝑛,(2.10) see [20, App.II(II.34)] π‘›ξ“π‘Ÿ=0(1βˆ’π‘Žπ‘π‘Ÿπ‘žπ‘Ÿ)(1βˆ’π‘π‘π‘Ÿπ‘žβˆ’π‘Ÿ)(π‘Ž,𝑏;𝑝)π‘Ÿ(𝑐,π‘Ž/𝑏𝑐;π‘ž)π‘Ÿπ‘žπ‘Ÿ(1βˆ’π‘Ž)(1βˆ’π‘)(π‘ž,π‘Žπ‘ž/𝑏;π‘ž)π‘Ÿ(π‘Žπ‘/𝑐,𝑏𝑐𝑝;𝑝)π‘Ÿ=(π‘Žπ‘,𝑏𝑝;𝑝)𝑛(π‘π‘ž,π‘Žπ‘ž/𝑏𝑐;π‘ž)𝑛(π‘ž,π‘Žπ‘ž/𝑏;π‘ž)𝑛(π‘Žπ‘/𝑐,𝑏𝑐𝑝;𝑝)𝑛,(2.11)see [20, App.II(II.35)] π‘›ξ“π‘Ÿ=0(1βˆ’π‘Žπ‘‘π‘π‘Ÿπ‘žπ‘Ÿ)(1βˆ’π‘π‘π‘Ÿ/π‘‘π‘žπ‘Ÿ)(π‘Ž,𝑏;𝑝)π‘Ÿξ€·π‘,π‘Žπ‘‘2ξ€Έ/𝑏𝑐;π‘žπ‘Ÿπ‘žπ‘Ÿ(1βˆ’π‘Žπ‘‘)(1βˆ’π‘/𝑑)(π‘‘π‘ž,π‘Žπ‘‘π‘ž/𝑏;π‘ž)π‘Ÿ(π‘Žπ‘‘π‘/𝑐,𝑏𝑐𝑝/𝑑;𝑝)π‘Ÿ=ξ€·(1βˆ’π‘Ž)(1βˆ’π‘)(1βˆ’π‘)1βˆ’π‘Žπ‘‘2ξ€Έ/𝑏𝑐×𝑑(1βˆ’π‘Žπ‘‘)(1βˆ’π‘/𝑑)(1βˆ’π‘/𝑑)(1βˆ’π‘Žπ‘‘/𝑏𝑐)(π‘Žπ‘,𝑏𝑝;𝑝)π‘›ξ€·π‘π‘ž,π‘Žπ‘‘2ξ€Έπ‘ž/𝑏𝑐;π‘žπ‘›(π‘‘π‘ž,π‘Žπ‘‘π‘ž/𝑏;π‘ž)𝑛(π‘Žπ‘‘π‘/𝑐,𝑏𝑐𝑝/𝑑;𝑝)π‘›βˆ’(𝑐/π‘Žπ‘‘,𝑑/𝑏𝑐;𝑝)1(1/𝑑,𝑏/π‘Žπ‘‘;π‘ž)1ξ€·1/𝑐,𝑏𝑐/π‘Žπ‘‘2ξ€Έ;π‘ž1(1/π‘Ž,1/𝑏;𝑝)1ξƒ°,(2.12)which is π‘š=0, case of [20, App. II (II. 36)].

3. Main Results

In this section we have established the following main results. Φ⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=[]π›Όπ‘ž,π›½π‘žβˆΆπ‘Ž,𝑦;π›Όπ›½π‘žβˆΆπ‘,π‘Žπ‘¦π‘;π‘ž,𝑝;π‘π‘Žπ‘,𝑦𝑝;π‘βˆž[]𝑝,π‘Žπ‘¦π‘;π‘βˆž[]π›Όπ‘ž,π›½π‘ž;π‘žβˆž[]π‘ž,π›Όπ›½π‘ž;π‘žβˆžβˆ’π‘ž(1βˆ’π›Ό)(1βˆ’π›½)Φ⎑⎒⎒⎣(1βˆ’π‘ž)(1βˆ’π›Όπ›½π‘ž)π‘Žπ‘,π‘¦π‘βˆΆπ›Όπ‘ž,π›½π‘ž;π‘Žπ‘¦π‘βˆΆπ‘ž2,π›Όπ›½π‘ž2;⎀βŽ₯βŽ₯⎦,Ξ¦βŽ‘βŽ’βŽ’βŽ£π‘,π‘ž;π‘ž(3.1)π›Όπ‘ž,π‘’π‘žβˆΆπ‘Ž,𝑦;π›Όπ‘žπ‘’π‘βˆΆπ‘,π‘Žπ‘¦π‘;π‘ž,𝑝;π‘’βŽ€βŽ₯βŽ₯βŽ¦ξ€·=βˆ’1βˆ’π›Όπ‘ž2ξ€Έ(1βˆ’π‘’)π‘’βŽ‘βŽ’βŽ’βŽ£(1βˆ’π‘ž)(1βˆ’π›Όπ‘ž/𝑒)Γ—Ξ¦π‘Žπ‘,π‘¦π‘βˆΆπ›Όπ‘ž,π‘ž2βˆšπ›Ό,βˆ’π‘ž2βˆšπ›Ό,π‘’π‘ž;π‘Žπ‘¦π‘βˆΆπ‘ž2√,π‘žβˆšπ›Ό,βˆ’π‘žπ›Ό,π›Όπ‘ž2𝑒;1𝑝,π‘ž;π‘’βŽ€βŽ₯βŽ₯⎦,Φ⎑⎒⎒⎒⎣(3.2)π›Όπ‘ž,π›½π‘ž,π›Ύπ‘ž,π›Ώπ‘žβˆΆπ‘Ž,𝑦;π›Όπ‘žπ›½,π›Όπ‘žπ›Ύ,π›Όπ‘žπ›ΏβŽ€βŽ₯βŽ₯βŽ₯⎦=[]βˆΆπ‘,π‘Žπ‘¦π‘;π‘ž,𝑝;π‘π‘Žπ‘,𝑦𝑝;π‘βˆž[]π›Όπ‘ž,π›½π‘ž,π›Ύπ‘ž,π›Ώπ‘ž;π‘žβˆž[]𝑝,π‘Žπ‘¦π‘βˆΆπ‘βˆž[]π‘ž,π›Όπ‘ž/𝛽,π›Όπ‘ž/𝛾,π›Όπ‘ž/𝛿;π‘žβˆžβˆ’ξ€·1βˆ’π‘ž2𝛼(1βˆ’π›½)(1βˆ’π›Ύ)(1βˆ’π›Ώ)π‘žβŽ‘βŽ’βŽ’βŽ’βŽ£(1βˆ’π‘ž)(1βˆ’π›Όπ‘ž/𝛽)(1βˆ’π›Όπ‘ž/𝛾)(1βˆ’π›Όπ‘ž/𝛿)Γ—Ξ¦π‘Žπ‘,π‘¦π‘βˆΆπ›Όπ‘ž,π‘ž2βˆšπ›Ό,βˆ’π‘ž2βˆšπ›Ό,π›½π‘ž,π›Ύπ‘ž,π›Ώπ‘ž;π‘Žπ‘¦π‘βˆΆπ‘ž2√,π‘žβˆšπ›Ό,βˆ’π‘žπ›Ό,π›Όπ‘ž2𝛽,π›Όπ‘ž2𝛾,π›Όπ‘ž2𝛿;⎀βŽ₯βŽ₯βŽ₯⎦,Ξ¦βŽ‘βŽ’βŽ’βŽ£π‘,π‘ž;π‘ž(3.3)π‘₯,π‘¦βˆΆπ‘Žπ‘βˆΆπ‘π‘;π‘₯π‘¦π‘ƒβˆΆπ‘Žπ‘π‘π‘ƒβˆΆπ‘ž;𝑃,𝑝,π‘ž;π‘βŽ€βŽ₯βŽ₯⎦=(1βˆ’π‘Žπ‘π‘ž)(1βˆ’π‘)⎑⎒⎒⎣(1βˆ’π‘ž)(1βˆ’π‘Žπ‘/𝑐)𝑐×Φπ‘₯𝑃,π‘¦π‘ƒβˆΆπ‘Žπ‘βˆΆπ‘π‘žβˆΆπ‘Žπ‘2π‘ž2;π‘₯π‘¦π‘ƒβˆΆπ‘Žπ‘2π‘βˆΆπ‘ž21βˆΆπ‘Žπ‘π‘ž;𝑃,𝑝,π‘ž,π‘π‘ž;π‘βŽ€βŽ₯βŽ₯⎦,Φ⎑⎒⎒⎣(3.4)π‘₯,π‘¦βˆΆπ‘Žπ‘,π‘π‘βˆΆπ‘π‘ž,π‘Žπ‘ž;𝑏𝑐π‘₯π‘¦π‘ƒβˆΆπ‘Žπ‘π‘,π‘π‘π‘βˆΆπ‘ž,π‘Žπ‘žπ‘;⎀βŽ₯βŽ₯⎦=[]𝑃,𝑝,π‘ž;𝑃π‘₯𝑃,𝑦𝑃;π‘ƒβˆž[]π‘Žπ‘,𝑏𝑝;π‘βˆž[]π‘π‘ž,π‘Žπ‘ž/𝑏𝑐;π‘žβˆž[]𝑃,π‘₯𝑦𝑃;π‘ƒβˆž[]π‘ž,π‘Žπ‘ž/𝑏;π‘žβˆž[]π‘Žπ‘/𝑐,𝑏𝑐𝑝;π‘βˆžβˆ’(1βˆ’π‘Žπ‘π‘ž)(1βˆ’π‘π‘/π‘ž)(1βˆ’π‘)(1βˆ’π‘Ž/𝑏𝑐)π‘žβŽ‘βŽ’βŽ’βŽ’βŽ£(1βˆ’π‘ž)(1βˆ’π‘Žπ‘ž/𝑏)(1βˆ’π‘Žπ‘/𝑐)(1βˆ’π‘π‘π‘)Γ—Ξ¦π‘₯𝑃,π‘¦π‘ƒβˆΆπ‘Žπ‘2π‘ž2βˆΆπ‘π‘2π‘ž2βˆΆπ‘Žπ‘,π‘π‘βˆΆπ‘π‘ž,π‘Žπ‘ž;𝑏𝑐π‘₯π‘¦π‘ƒβˆΆπ‘Žπ‘π‘žβˆΆπ‘π‘π‘žβˆΆπ‘Žπ‘2𝑐,𝑏𝑐𝑝2βˆΆπ‘ž2,π‘Žπ‘ž2𝑏;𝑝𝑃,π‘π‘ž,π‘žβŽ€βŽ₯βŽ₯βŽ₯⎦,Φ⎑⎒⎒⎒⎣,𝑝,π‘ž;π‘ž(3.5)π‘₯,π‘¦βˆΆπ‘Žπ‘,π‘π‘βˆΆπ‘π‘ž,π‘Žπ‘‘2π‘ž;𝑏𝑐π‘₯π‘¦π‘ƒβˆΆπ‘Žπ‘‘π‘π‘,π‘π‘π‘π‘‘βˆΆπ‘‘π‘ž,π‘Žπ‘‘π‘žπ‘;⎀βŽ₯βŽ₯βŽ₯⎦=[]𝑃,𝑝,π‘ž;𝑃π‘₯𝑃,𝑦𝑃;π‘ƒβˆž[]π‘Žπ‘,𝑏𝑝;π‘βˆžξ€Ίπ‘π‘ž,π‘Žπ‘‘2ξ€»π‘ž/𝑏𝑐;π‘žβˆž[]𝑃,π‘₯𝑦𝑃;π‘ƒβˆž[]π‘‘π‘ž,π‘Žπ‘‘π‘ž/𝑏;π‘žβˆž[]π‘Žπ‘‘π‘/𝑐,𝑏𝑐𝑝/𝑑;π‘βˆžβˆ’π‘‘π‘ž(1βˆ’π‘Žπ‘‘π‘π‘ž)(1βˆ’π‘π‘/π‘‘π‘ž)(1βˆ’π‘/𝑑)(1βˆ’π‘Žπ‘‘/𝑏𝑐)(⎑⎒⎒⎒⎣1βˆ’π‘‘π‘ž)(1βˆ’π‘Žπ‘‘π‘ž/𝑏)(1βˆ’π‘Žπ‘‘π‘/𝑐)(1βˆ’π‘π‘π‘/𝑑)Γ—Ξ¦π‘₯𝑃,π‘¦π‘ƒβˆΆπ‘Žπ‘‘π‘2π‘ž2βˆΆπ‘π‘2π‘‘π‘ž2βˆΆπ‘Žπ‘,π‘π‘βˆΆπ‘π‘ž,π‘Žπ‘‘2π‘ž;𝑏𝑐π‘₯π‘¦π‘ƒβˆΆπ‘Žπ‘‘π‘π‘žβˆΆπ‘π‘βˆΆπ‘‘π‘žπ‘Žπ‘‘π‘2𝑐,𝑏𝑐𝑝2π‘‘βˆΆπ‘‘π‘ž2,π‘Žπ‘‘π‘ž2𝑏;𝑝𝑃,π‘π‘ž,π‘žβŽ€βŽ₯βŽ₯βŽ₯⎦.,𝑝,π‘ž;π‘ž(3.6)

4. Proof of Main Results

Taking π‘’π‘Ÿ=π‘£π‘Ÿ=1 in (1.1), Bailey's transformation takes the following form: If𝛽𝑛=π‘›ξ“π‘Ÿ=0π›Όπ‘Ÿ,𝛾(4.1)𝑛=βˆžξ“π‘Ÿ=0π›Ώπ‘Ÿ,(4.2)thenβˆžξ“π‘›=0𝛼𝑛𝛾𝑛=βˆžξ“π‘›=0𝛽𝑛𝛿𝑛.(4.3)

Proof of Result (3.1). Taking π›Όπ‘Ÿ=(𝛼,𝛽;π‘ž)π‘Ÿπ‘žπ‘Ÿ/(π‘ž,π›Όπ›½π‘ž;π‘ž)π‘Ÿ and π›Ώπ‘Ÿ=(π‘Ž,𝑦;𝑝)π‘Ÿπ‘π‘Ÿ/(𝑝,π‘Žπ‘¦π‘;𝑝)π‘Ÿ in (4.1) and (4.2), respectively, and making use of (2.7), we get 𝛽𝑛=(π›Όπ‘ž,π›½π‘ž;π‘ž)𝑛(π‘ž,π›Όπ›½π‘ž;π‘ž)𝑛,𝛾𝑛=(π‘Žπ‘,𝑦𝑝;𝑝)∞(𝑝,π‘Žπ‘¦π‘;𝑝)βˆžβˆ’(1βˆ’π‘Žπ‘¦)(1βˆ’π‘π‘›)(π‘Ž,𝑦;𝑝)𝑛(1βˆ’π‘Ž)(1βˆ’π‘¦)(𝑝,π‘Žπ‘¦;𝑝)𝑛.(4.4) Putting these values in (4.3), we get the following transformation: Φ⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦+π›Όπ‘ž,π›½π‘žβˆΆπ‘Ž,𝑦;π›Όπ›½π‘žβˆΆπ‘,π‘Žπ‘¦π‘;π‘ž,𝑝;𝑝(1βˆ’π‘Žπ‘¦)Φ⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦=((1βˆ’π‘Ž)(1βˆ’π‘¦)𝛼,π›½βˆΆπ‘Ž,𝑦;π›Όπ›½π‘žβˆΆπ‘,π‘Žπ‘¦;π‘ž,𝑝;π‘žπ‘Žπ‘,𝑦𝑝;𝑝)∞(𝑝,π‘Žπ‘¦π‘;𝑝)∞(π›Όπ‘ž,π›½π‘ž;π‘ž)∞(π‘ž,π›Όπ›½π‘ž;π‘ž)∞+(1βˆ’π‘Žπ‘¦)Φ⎑⎒⎒⎣⎀βŽ₯βŽ₯⎦,(1βˆ’π‘Ž)(1βˆ’π‘¦)𝛼,π›½βˆΆπ‘Ž,𝑦;π›Όπ›½π‘žβˆΆπ‘,π‘Žπ‘¦;π‘ž,𝑝;π‘π‘ž(4.5) which on simplification gives the result (3.1).

Proof of Result (3.2). Taking π›Όπ‘Ÿβˆš=(𝛼,π‘žβˆšπ›Ό,βˆ’π‘žπ›Ό,𝑒;π‘ž)π‘Ÿβˆš/(π‘ž,βˆšπ›Ό,βˆ’π›Ό,π›Όπ‘ž/𝑒;π‘ž)π‘Ÿπ‘’π‘Ÿ and π›Ώπ‘Ÿ=(π‘Ž,𝑦;𝑝)π‘Ÿπ‘π‘Ÿ/(𝑝,π‘Žπ‘¦π‘;𝑝)π‘Ÿβ€‰β€‰in (4.1) and (4.2), respectively, and making use of (2.8) and (2.7), we get 𝛽𝑛=(π›Όπ‘ž,π‘’π‘ž;π‘ž)𝑛(π‘ž,π›Όπ‘ž/𝑒;π‘ž)𝑛𝑒𝑛,𝛾𝑛=(π‘Žπ‘,𝑦𝑝;𝑝)∞(𝑝,π‘Žπ‘¦π‘;𝑝)βˆžβˆ’(1βˆ’π‘Žπ‘¦)(1βˆ’π‘π‘›)(π‘Ž,𝑦;𝑝)𝑛(1βˆ’π‘Ž)(1βˆ’π‘¦)(𝑝,π‘Žπ‘¦;𝑝)𝑛.(4.6) Substituting these values in (4.3), we get the following transformation for |𝑒|>1: Ξ¦βŽ‘βŽ’βŽ’βŽ£π›Όπ‘ž,π‘’π‘žβˆΆπ‘Ž,𝑦;π›Όπ‘žπ‘’π‘βˆΆπ‘,π‘Žπ‘¦π‘;π‘ž,𝑝;π‘’βŽ€βŽ₯βŽ₯⎦=(1βˆ’π‘Žπ‘¦)⎑⎒⎒⎣√(1βˆ’π‘Ž)(1βˆ’π‘’)×Φ𝛼,π‘žβˆšπ›Ό,βˆ’π‘žβˆšπ›Ό,π‘’βˆΆπ‘Ž,𝑦;βˆšπ›Ό,βˆ’π›Ό,π›Όπ‘žπ‘’π‘βˆΆπ‘,π‘Žπ‘¦;π‘ž,𝑝;π‘’βŽ€βŽ₯βŽ₯βŽ¦βˆ’(1βˆ’π‘Žπ‘¦)Φ⎑⎒⎒⎣√(1βˆ’π‘Ž)(1βˆ’π‘¦)𝛼,π‘žβˆšπ›Ό,βˆ’π‘žβˆšπ›Ό,π‘’βˆΆπ‘Ž,𝑦;βˆšπ›Ό,βˆ’π›Ό,π›Όπ‘žπ‘’1βˆΆπ‘,π‘Žπ‘¦;π‘ž,𝑝;π‘’βŽ€βŽ₯βŽ₯⎦,(4.7) which on simplification gives result (3.2).

Proof of Result (3.3). Taking π›Όπ‘Ÿβˆš=(𝛼,π‘žβˆšπ›Ό,βˆ’π‘žπ›Ό,𝛽,𝛾,𝛿;π‘ž)π‘Ÿπ‘žπ‘Ÿβˆš/(π‘ž,βˆšπ›Ό,βˆ’π›Ό,π›Όπ‘ž/𝛽,π›Όπ‘ž/𝛾,π›Όπ‘ž/𝛿;π‘ž)π‘Ÿ, where 𝛼=𝛽𝛾𝛿 and π›Ώπ‘Ÿ=(π‘Ž,𝑦;𝑝)π‘Ÿπ‘π‘Ÿ/(𝑝,π‘Žπ‘¦π‘;𝑝)π‘Ÿ in (4.1) and (4.2), respectively, and making use of (2.9) and (2.7), we get 𝛽𝑛=(π›Όπ‘ž,π›½π‘ž,π›Ύπ‘ž,π›Ώπ‘ž;π‘ž)𝑛(π‘ž,π›Όπ‘ž/𝛽,π›Όπ‘ž/𝛾,π›Όπ‘ž/𝛿;π‘ž)𝑛,𝛾𝑛=(π‘Žπ‘,𝑦𝑝;𝑝)∞(𝑝,π‘Žπ‘¦π‘;𝑝)βˆžβˆ’(1βˆ’π‘Žπ‘¦)(1βˆ’π‘π‘›)(π‘Ž,𝑦;𝑝)𝑛(1βˆ’π‘Ž)(1βˆ’π‘¦)(𝑝,π‘Žπ‘¦;𝑝)𝑛.(4.8) Substituting these values in (4.3), we get the following transformation for 𝛼=𝛽𝛾𝛿: Ξ¦βŽ‘βŽ’βŽ’βŽ’βŽ£π›Όπ‘ž,π›½π‘ž,π›Ύπ‘ž,π›Ώπ‘žβˆΆπ‘Ž,𝑦;π›Όπ‘žπ›½,π›Όπ‘žπ›Ύ,π›Όπ‘žπ›ΏβŽ€βŽ₯βŽ₯βŽ₯⎦+βˆΆπ‘,π‘Žπ‘¦π‘;π‘ž,𝑝;𝑝(1βˆ’π‘Žπ‘¦)⎑⎒⎒⎒⎣√(1βˆ’π‘Ž)(1βˆ’π‘¦)×Φ𝛼,π‘žβˆšπ›Ό,βˆ’π‘žβˆšπ›Ό,𝛽,𝛾,π›ΏβˆΆπ‘Ž,𝑦;βˆšπ›Ό,βˆ’π›Ό,π›Όπ‘žπ›½,π›Όπ‘žπ›Ύ,π›Όπ‘žπ›ΏβŽ€βŽ₯βŽ₯βŽ₯⎦=βˆΆπ‘,π‘Žπ‘¦;π‘ž,𝑝;π‘ž(π‘Žπ‘,𝑦𝑝;𝑝)∞(𝑝,π‘Žπ‘¦π‘;𝑝)βˆžΓ—(π›Όπ‘ž,π›½π‘ž,π›Ύπ‘ž,π›Ώπ‘ž;π‘ž)∞(π‘ž,π›Όπ‘ž/𝛽,π›Όπ‘ž/𝛾,π›Όπ‘ž/𝛿;π‘ž)∞+(1βˆ’π‘Žπ‘¦)⎑⎒⎒⎒⎣√(1βˆ’π‘Ž)(1βˆ’π‘¦)×Φ𝛼,π‘žβˆšπ›Ό,βˆ’π‘žβˆšπ›Ό,𝛽,𝛾,π›ΏβˆΆπ‘Ž,𝑦;βˆšπ›Ό,βˆ’π›Ό,π›Όπ‘žπ›½,π›Όπ‘žπ›Ύ,π›Όπ‘žπ›ΏβŽ€βŽ₯βŽ₯βŽ₯⎦,βˆΆπ‘,π‘Žπ‘¦;π‘ž,𝑝;π‘π‘ž(4.9) which on simplification gives result (3.3).

Proof of Result (3.4). Taking π›Όπ‘Ÿ=(π‘Žπ‘π‘ž;π‘π‘ž)π‘Ÿ(π‘Ž;𝑝)π‘Ÿ(𝑐;π‘ž)π‘Ÿπ‘βˆ’π‘Ÿ/((π‘Ž;π‘π‘ž)π‘Ÿ(π‘ž;π‘ž)π‘Ÿ(π‘Žπ‘/𝑐;𝑝)π‘Ÿ) and π›Ώπ‘Ÿ=(π‘₯,𝑦;𝑃)π‘Ÿπ‘ƒπ‘Ÿ/(𝑃,π‘₯𝑦𝑃;𝑃)π‘Ÿ in (4.1) and (4.2), respectively and making use of (2.10) and (2.7), we get 𝛽𝑛=(π‘Žπ‘;𝑝)𝑛(π‘π‘ž;π‘ž)π‘›π‘βˆ’π‘›(π‘ž;π‘ž)𝑛(π‘Žπ‘/𝑐;𝑝)𝑛,𝛾𝑛=(π‘₯𝑃,𝑦𝑃;𝑃)∞(𝑃,π‘₯𝑦𝑃;𝑃)βˆžβˆ’(1βˆ’π‘₯𝑦)(1βˆ’π‘ƒπ‘›)(π‘₯,𝑦;𝑃)𝑛(1βˆ’π‘₯)(1βˆ’π‘¦)(𝑃,π‘₯𝑦;𝑃)𝑛.(4.10) Putting these values in (4.3), we get the following transformation for |𝑐|>1: Φ⎑⎒⎒⎣π‘₯,π‘¦βˆΆπ‘Žπ‘βˆΆπ‘π‘ž;π‘₯π‘¦π‘ƒβˆΆπ‘Žπ‘π‘π‘ƒβˆΆπ‘ž;𝑃,𝑝,π‘ž;π‘βŽ€βŽ₯βŽ₯⎦=(1βˆ’π‘₯𝑦)⎑⎒⎒⎣(1βˆ’π‘₯)(1βˆ’π‘¦)Γ—Ξ¦π‘₯,π‘¦βˆΆπ‘Žπ‘π‘žβˆΆπ‘ŽβˆΆπ‘;π‘₯π‘¦βˆΆπ‘ŽβˆΆπ‘Žπ‘π‘π‘ƒβˆΆπ‘ž;𝑃,π‘π‘ž,𝑝,π‘ž;π‘βŽ€βŽ₯βŽ₯βŽ¦βˆ’(1βˆ’π‘₯𝑦)⎑⎒⎒⎣(1βˆ’π‘₯)(1βˆ’π‘¦)Γ—Ξ¦π‘₯,π‘¦βˆΆπ‘Žπ‘π‘žβˆΆπ‘ŽβˆΆπ‘;π‘₯π‘¦βˆΆπ‘ŽβˆΆπ‘Žπ‘π‘1βˆΆπ‘ž;𝑃,π‘π‘ž,𝑝,π‘ž;π‘βŽ€βŽ₯βŽ₯⎦,(4.11) which on simplification gives result (3.4).

Proof of Result (3.5). Taking π›Όπ‘Ÿ=(π‘Žπ‘π‘ž;π‘π‘ž)π‘Ÿ(𝑏𝑝/π‘ž;𝑝/π‘ž)π‘Ÿ(π‘Ž,𝑏;𝑝)π‘Ÿ(𝑐,π‘Ž/𝑏𝑐;π‘ž)π‘Ÿπ‘žπ‘Ÿ/((π‘Ž;π‘π‘ž)π‘Ÿ(𝑏;𝑝/π‘ž)π‘Ÿ (π‘ž,π‘Žπ‘ž/𝑏;π‘ž)π‘Ÿ(π‘Žπ‘/𝑐,𝑏𝑐𝑝;𝑝)π‘Ÿ) and π›Ώπ‘Ÿ=(π‘₯,𝑦;𝑃)π‘Ÿπ‘ƒπ‘Ÿ/(𝑃,π‘₯𝑦𝑃;𝑃)π‘Ÿ in (4.1) and (4.2), respectively, and making use of (2.11) and (2.7), we get 𝛽𝑛=(π‘Žπ‘,𝑏𝑝;𝑝)𝑛(π‘π‘ž,π‘Žπ‘ž/𝑏𝑐;π‘ž)𝑛(π‘ž,π‘Žπ‘ž/𝑏;π‘ž)𝑛(π‘Žπ‘/𝑐,𝑏𝑐𝑝;𝑝)𝑛,𝛾𝑛=(π‘₯𝑃,𝑦𝑃;𝑃)∞(𝑃,π‘₯𝑦𝑃;𝑃)βˆžβˆ’(1βˆ’π‘₯𝑦)(1βˆ’π‘ƒπ‘›)(π‘₯,𝑦;𝑃)𝑛(1βˆ’π‘₯)(1βˆ’π‘¦)(𝑃,π‘₯𝑦;𝑃)𝑛.(4.12) Putting these values in (4.3), we get the following transformation: Φ⎑⎒⎒⎣π‘₯,π‘¦βˆΆπ‘Žπ‘,π‘π‘βˆΆπ‘π‘ž,π‘Žπ‘ž;𝑏𝑐π‘₯π‘¦π‘ƒβˆΆπ‘Žπ‘π‘,π‘π‘π‘βˆΆπ‘ž,π‘Žπ‘žπ‘;⎀βŽ₯βŽ₯⎦+𝑃,𝑝,π‘ž;𝑃(1βˆ’π‘₯𝑦)⎑⎒⎒⎒⎣(1βˆ’π‘₯)(1βˆ’π‘¦)Γ—Ξ¦π‘₯,π‘¦βˆΆπ‘Žπ‘π‘žβˆΆπ‘π‘π‘žπ‘ŽβˆΆπ‘Ž,π‘βˆΆπ‘,;𝑏𝑐π‘₯π‘¦βˆΆπ‘ŽβˆΆπ‘βˆΆπ‘Žπ‘π‘,π‘π‘π‘βˆΆπ‘ž,π‘Žπ‘žπ‘;𝑝𝑃,π‘π‘ž,π‘žβŽ€βŽ₯βŽ₯βŽ₯⎦=,𝑝,π‘ž;π‘ž(π‘₯𝑃,𝑦𝑃;𝑃)∞(𝑃,π‘₯𝑦𝑃;𝑃)∞(π‘Žπ‘,𝑏𝑝;𝑝)∞(π‘ž,π‘Žπ‘ž/𝑏;π‘ž)∞(π‘π‘ž,π‘Žπ‘ž/𝑏𝑐;π‘ž)∞(π‘Žπ‘/𝑐,𝑏𝑐𝑝;𝑝)∞+(1βˆ’π‘₯𝑦)(⎑⎒⎒⎒⎣1βˆ’π‘₯)(1βˆ’π‘¦)Γ—Ξ¦π‘₯,π‘¦βˆΆπ‘Žπ‘π‘žβˆΆπ‘π‘π‘žπ‘ŽβˆΆπ‘Ž,π‘βˆΆπ‘,;𝑏𝑐π‘₯π‘¦βˆΆπ‘ŽβˆΆπ‘βˆΆπ‘Žπ‘π‘,π‘π‘π‘βˆΆπ‘ž,π‘Žπ‘žπ‘;𝑝𝑃,π‘π‘ž,π‘žβŽ€βŽ₯βŽ₯βŽ₯⎦,,𝑝,π‘ž;π‘ƒπ‘ž(4.13) which on simplification gives result (3.5).

Proof of Result (3.6). Taking π›Όπ‘Ÿ=(π‘Žπ‘‘π‘π‘ž;π‘π‘ž)π‘Ÿ(𝑏𝑝/π‘‘π‘ž;𝑝/π‘ž)π‘Ÿ(π‘Ž,𝑏;𝑝)π‘Ÿ(𝑐,π‘Žπ‘‘2/𝑏𝑐;π‘ž)π‘Ÿπ‘žπ‘Ÿ/((π‘Žπ‘‘;π‘π‘ž)π‘Ÿ(𝑏/𝑑;𝑝/π‘ž)π‘Ÿ(π‘‘π‘ž,π‘Žπ‘‘π‘ž/𝑏;π‘ž)π‘Ÿ(π‘Žπ‘‘π‘/𝑐,𝑏𝑐𝑝/𝑑;𝑝)π‘Ÿ) and π›Ώπ‘Ÿ=(π‘₯,𝑦;𝑃)π‘Ÿπ‘ƒπ‘Ÿ/(𝑃,π‘₯𝑦𝑃;𝑃)π‘Ÿ in (4.1) and (4.2), respectively, and making use of (2.12) and (2.7), we get 𝛽𝑛=ξ€·(1βˆ’π‘Ž)(1βˆ’π‘)(1βˆ’π‘)1βˆ’π‘Žπ‘‘2ξ€Έ/𝑏𝑐𝑑×(1βˆ’π‘Žπ‘‘)(1βˆ’π‘/𝑑)(1βˆ’π‘/𝑑)(1βˆ’π‘Žπ‘‘/𝑏𝑐)(π‘Žπ‘,𝑏𝑝;𝑝)π‘›ξ€·π‘π‘ž,π‘Žπ‘‘2ξ€Έπ‘ž/𝑏𝑐;π‘žπ‘›(π‘‘π‘ž,π‘Žπ‘‘π‘ž/𝑏;π‘ž)𝑛(π‘Žπ‘‘π‘/𝑐,𝑏𝑐𝑝/𝑑;𝑝)π‘›βˆ’(π‘βˆ’π‘Žπ‘‘)(π‘βˆ’π‘Žπ‘‘)(π‘‘βˆ’π‘π‘)(1βˆ’π‘‘)𝑑(1βˆ’π‘Ž)(1βˆ’π‘)(1βˆ’π‘)π‘π‘βˆ’π‘Žπ‘‘2𝛾𝑛=(π‘₯𝑃,𝑦𝑃;𝑃)∞(𝑃,π‘₯𝑦𝑃;𝑃)βˆžβˆ’(1βˆ’π‘₯𝑦)(1βˆ’π‘ƒπ‘›)(π‘₯,𝑦;𝑃)𝑛(1βˆ’π‘₯)(1βˆ’π‘¦)(𝑃,π‘₯𝑦;𝑃)𝑛.,(4.14) Putting these values in (4.3), we get the following transformation: (ξ€·1βˆ’π‘Ž)(1βˆ’π‘)(1βˆ’π‘)1βˆ’π‘Žπ‘‘2ξ€Έ/π‘π‘βŽ‘βŽ’βŽ’βŽ’βŽ£π‘‘(1βˆ’π‘Žπ‘‘)(1βˆ’π‘/𝑑)(1βˆ’π‘/𝑑)(1βˆ’π‘Žπ‘‘/𝑏𝑐)Γ—Ξ¦π‘₯,π‘¦βˆΆπ‘Žπ‘,π‘π‘βˆΆπ‘π‘ž,π‘Žπ‘‘2π‘ž;𝑏𝑐π‘₯π‘¦π‘ƒβˆΆπ‘Žπ‘‘π‘π‘,π‘π‘π‘π‘‘βˆΆπ‘‘π‘ž,π‘Žπ‘‘π‘žπ‘;⎀βŽ₯βŽ₯βŽ₯⎦+𝑃,𝑝,π‘ž;𝑃(1βˆ’π‘₯𝑦)⎑⎒⎒⎒⎣(1βˆ’π‘₯)(1βˆ’π‘¦)Γ—Ξ¦π‘₯,π‘¦βˆΆπ‘Žπ‘‘π‘π‘žβˆΆπ‘π‘π‘‘π‘žβˆΆπ‘Ž,π‘βˆΆπ‘,π‘Žπ‘‘2;𝑏𝑏𝑐π‘₯π‘¦βˆΆπ‘Žπ‘‘βˆΆπ‘‘βˆΆπ‘Žπ‘‘π‘π‘,π‘π‘π‘π‘‘βˆΆπ‘‘π‘ž,π‘Žπ‘‘π‘žπ‘;𝑝𝑃,π‘π‘ž,π‘žβŽ€βŽ₯βŽ₯βŽ₯⎦=ξ€·,𝑝,π‘ž;π‘ž(1βˆ’π‘Ž)(1βˆ’π‘)(1βˆ’π‘)1βˆ’π‘Žπ‘‘2ξ€Έ/𝑏𝑐×𝑑(1βˆ’π‘Žπ‘‘)(1βˆ’π‘/𝑑)(1βˆ’π‘/𝑑)(1βˆ’π‘Žπ‘‘/𝑏𝑐)(π‘₯𝑃,𝑦𝑃;𝑃)∞(π‘Žπ‘,𝑏𝑝;𝑝)βˆžξ€·π‘π‘ž,π‘Žπ‘‘2ξ€Έπ‘ž/𝑏𝑐;π‘žβˆž(𝑃,π‘₯𝑦𝑃;𝑃)∞(π‘‘π‘ž,π‘Žπ‘‘π‘ž/𝑏;π‘ž)∞(π‘Žπ‘‘π‘/𝑐,𝑏𝑐𝑝/𝑑;𝑝)∞+(1βˆ’π‘₯𝑦)⎑⎒⎒⎒⎣(1βˆ’π‘₯)(1βˆ’π‘¦)Γ—Ξ¦π‘₯,π‘¦βˆΆπ‘Žπ‘‘π‘π‘žβˆΆπ‘π‘π‘‘π‘žβˆΆπ‘Ž,π‘βˆΆπ‘,π‘Žπ‘‘2;𝑏𝑏𝑐π‘₯π‘¦βˆΆπ‘Žπ‘‘βˆΆπ‘‘βˆΆπ‘Žπ‘‘π‘π‘,π‘π‘π‘π‘‘βˆΆπ‘‘π‘ž,π‘Žπ‘‘π‘žπ‘;𝑝𝑃,π‘π‘ž,π‘žβŽ€βŽ₯βŽ₯βŽ₯⎦,,𝑝,π‘ž;π‘ƒπ‘ž(4.15) which on simplification gives result (3.6).