A Note on Four-Variable Reciprocity Theorem

Abstract

We give new proof of a four-variable reciprocity theorem using Heine's transformation, Watson's transformation, and Ramanujan's ψ1₁ -summation formula. We also obtain a generalization of Jacobi's triple product identity.

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Recommended by Teodor Bulboaca

Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore 570 006, India

Received 12 September 2009; Accepted 7 December 2009

1. Introduction

Throughout the paper, we let |q|<1 and we employ the standard notation: [figure omitted; refer to PDF] Ramanujan [1] stated several q -series identities in his "lost'' notebook. One of the beautiful identities is the two-variable reciprocity theorem.

Theorem 1.1 (see [2]).

For ab≠0 , [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

In the recent past many new proofs of (1.2) have been found. The first proof of (1.2) was given by Andrews [3] using four-free-variable identity and Jacobi's triple product identity. Further, Andrews [4] applied (1.2) in proving Euler partition identity analogues stated in [1]. Somashekara and Fathima [5] established an equivalent version of (1.2) using Ramanujan's _1ψ1 summation formula [6] and Heine's transformation [7, 8]. Berndt et al. [9] also derived (1.2) using the same above mentioned two transformations. In fact, Berndt et al. [9] in the same paper have given two more proofs of (1.2) one employing the Rogers-Fine identity [10] and the other is purely combinatorial. Using the q -binomial theorem: [figure omitted; refer to PDF] Kim et al. [11] gave a much different proof of (1.2). Guruprasad and Pradeep [12] also devised a proof of (1.2) using the q -binomial theorem. Adiga and Anitha [13] established (1.2) along the lines of Ismail's proof [14] of Ramanujan's _1ψ1 summation formula. Further, they showed that the reciprocity theorem (1.2) leads to a q -integral extension of the classical gamma function. Kang [2] constructed a proof of (1.2) along the lines of Venkatachaliengar's proof of the Ramanujan _1ψ1 summation formula [6, 15].

In [2] Kang proved the following three- and four-variable generalizations of (1.2).

For |c|<|a|<1 and |c|<|b|<1 , [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and for |c|,|d|<|a| , |b|<1 ,

[figure omitted; refer to PDF] where [figure omitted; refer to PDF] Kang [2] established (1.5) on employing Ramanujan's _1ψ1 summation formula and Jackson's transformation of _2[varphi]1 and _2[varphi]2 -series. Recently (1.5) was derived by Adiga and Guruprasad [16] using q -binomial theorem and Gauss summation formula. Somashekara and Mamta [17, 18] obtained (1.5) using the two-variable reciprocity theorem (1.2), Jackson's transformation, and again two-variable reciprocity theorem by parameter augmentation. Zhang [19] also established (1.5).

Kang [2] established (1.7) on employing Andrews's generalization of _1ψ1 summation formula, Sears's transformation of _3[varphi]2 -series, and a limiting case of Watson's transformation for a terminating very well-poised _8[varphi]7 -series [8]: [figure omitted; refer to PDF] Recently Ma [20, 21] proved a six-variable generalization and a five-variable generalization of (1.2). The main purpose of this paper is to provide a new proof of (1.7) using (1.9), Heine's transformation: [figure omitted; refer to PDF] and Ramanujan's _1ψ1 summation formula: [figure omitted; refer to PDF]

Jacobi's triple product identity states that [figure omitted; refer to PDF] Andrews [22] gave a proof of (1.12) using Euler identities. Combinatorial proofs of Jacobi's triple product identity were given by Wright [23], Cheema [24], and Sudler [25]. We can also find a proof of (1.12) in [26]. Using (1.12), Hirschhorn [27, 28] established Jacobi's two-square and four-square theorems.

Somashekara and Fathima [5] and Kim et al. [11] established [figure omitted; refer to PDF] Note that (1.13) which is equivalent to (1.2) may be considered as a two-variable generalization of (1.12). Corteel and Lovejoy [29, equation ( 1.5)] have given a bijective proof of (1.13) using representations of over partitions. All the reciprocity theorems (1.2), (1.5), and (1.7) are generalizations of Jacobi's triple product identity (1.12).

We also obtain a generalization of Jacobi's triple product identity (1.12) which is due to Kang [2].

2. Proof of (1.7)--The Four-Variable Reciprocity Theorem

On employing q -binomial theorem, we have [figure omitted; refer to PDF] On using Heine's transformation (1.10) with α=-cq , β=q , γ=-aq , z=^qm+1 , we have [figure omitted; refer to PDF] Substituting this in (2.1), we obtain [figure omitted; refer to PDF] Now,

[figure omitted; refer to PDF] Substituting (2.4) in (2.3), we obtain [figure omitted; refer to PDF] (Here, we used (1.10) with α=b/d , β=q , γ=c^q2 /a , z=dq/a .)

Changing c to -c/q , d to -d/q in (2.5), we get [figure omitted; refer to PDF] Interchanging a and b in (2.6), we have [figure omitted; refer to PDF] Subtracting (2.6) from (2.7), we deduce that [figure omitted; refer to PDF] Now change a to -b/d , b to -c/a , and z to -d/a in (1.11) to obtain [figure omitted; refer to PDF] Changing n to n+1 in the first summation of the above identity and then multiplying both sides by ^(1+d/b)-1 , we find that [figure omitted; refer to PDF] Using (1.10) with α=-bq/c , β=q , γ=-dq/a , and z=-c/a in the first summation of the above identity and then multiplying both sides by 1/b , we get [figure omitted; refer to PDF] Substituting (2.11) in (2.8), we see that

[figure omitted; refer to PDF] Now setting α=-cd/b , β=cd/ab , γ=c , δ=q , and ...=d in (1.9) and then multiplying both sides by 1/(1+d/b)(1+c/b) , we obtain [figure omitted; refer to PDF] Interchanging a and b in (2.13), we have [figure omitted; refer to PDF] Substituting (2.13) and (2.14) in (2.12), we deduce (1.7).

Theorem 2.1 (A four-variable generalization of Jacobi's triple product identity).

For |c|,|d|<|a| , |b|<1 , [figure omitted; refer to PDF]

Proof.

Employing [figure omitted; refer to PDF] in the right side of (2.12) and then multiplying both sides by b/(1+a)(1+b) , we obtain (2.15).

Acknowledgment

The authors thank the anonymous referee for several helpful comments.

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