1. Introduction

Throughout the paper, we let and we employ the standard notation: Ramanujan [1] stated several -series identities in his “lost’’ notebook. One of the beautiful identities is the two-variable reciprocity theorem.

Theorem 1.1 (see [2]). For , where

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 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 -binomial theorem: 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 -binomial theorem. Adiga and Anitha [13] established (1.2) along the lines of Ismail's proof [14] of Ramanujan's summation formula. Further, they showed that the reciprocity theorem (1.2) leads to a -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 summation formula [6, 15].

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

For and , where and for , ,

where Kang [2] established (1.5) on employing Ramanujan's summation formula and Jackson's transformation of and -series. Recently (1.5) was derived by Adiga and Guruprasad [16] using -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 summation formula, Sears's transformation of -series, and a limiting case of Watson's transformation for a terminating very well-poised -series [8]: 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: and Ramanujan's summation formula:

Jacobi's triple product identity states that 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 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 -binomial theorem, we have On using Heine's transformation (1.10) with , , , , we have Substituting this in (2.1), we obtain Now,

Substituting (2.4) in (2.3), we obtain (Here, we used (1.10) with , , , .)

Changing to , to in (2.5), we get Interchanging and in (2.6), we have Subtracting (2.6) from (2.7), we deduce that Now change to , to , and to in (1.11) to obtain Changing to in the first summation of the above identity and then multiplying both sides by , we find that Using (1.10) with , , , and in the first summation of the above identity and then multiplying both sides by , we get Substituting (2.11) in (2.8), we see that

Now setting , , , , and in (1.9) and then multiplying both sides by , we obtain Interchanging and in (2.13), we have 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 , ,

Proof. Employing in the right side of (2.12) and then multiplying both sides by , we obtain (2.15).

Acknowledgment

The authors thank the anonymous referee for several helpful comments.