A Note on Four-Variable Reciprocity Theorem
We give new proof of a four-variable reciprocity theorem using Heine's transformation, Watson's transformation, and Ramanujan's -summation formula. We also obtain a generalization of Jacobi's triple product identity.
Throughout the paper, we let and we employ the standard notation: Ramanujan  stated several -series identities in his “lost’’ notebook. One of the beautiful identities is the two-variable reciprocity theorem.
Theorem 1.1 (see ). 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  using four-free-variable identity and Jacobi's triple product identity. Further, Andrews  applied (1.2) in proving Euler partition identity analogues stated in . Somashekara and Fathima  established an equivalent version of (1.2) using Ramanujan's summation formula  and Heine's transformation [7, 8]. Berndt et al.  also derived (1.2) using the same above mentioned two transformations. In fact, Berndt et al.  in the same paper have given two more proofs of (1.2) one employing the Rogers-Fine identity  and the other is purely combinatorial. Using the -binomial theorem: Kim et al.  gave a much different proof of (1.2). Guruprasad and Pradeep  also devised a proof of (1.2) using the -binomial theorem. Adiga and Anitha  established (1.2) along the lines of Ismail's proof  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  constructed a proof of (1.2) along the lines of Venkatachaliengar's proof of the Ramanujan summation formula [6, 15].
For and , where and for , ,
where Kang  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  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  also established (1.5).
Kang  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 : 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  gave a proof of (1.12) using Euler identities. Combinatorial proofs of Jacobi's triple product identity were given by Wright , Cheema , and Sudler . We can also find a proof of (1.12) in . Using (1.12), Hirschhorn [27, 28] established Jacobi's two-square and four-square theorems.
Somashekara and Fathima  and Kim et al.  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).
2. Proof of (1.7)—The Four-Variable Reciprocity Theorem
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
Theorem 2.1 (A four-variable generalization of Jacobi's triple product identity). For , ,
The authors thank the anonymous referee for several helpful comments.
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Springer, Berlin, Germany, 1988.View at: MathSciNet
B. C. Berndt, Ramanujan's Notebooks. Part III, Springer, New York, NY, USA, 1991.View at: MathSciNet
G. E. Andrews, R. Askey, and R. Roy, Special Functions, vol. 71 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 1999.View at: MathSciNet
G. Gasper and M. Rahman, Basic Hypergeometric Series, vol. 96 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2nd edition, 2004.View at: MathSciNet
N. J. Fine, Basic Hypergeometric Series and Applications, vol. 27 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 1988.View at: MathSciNet
D. D. Somashekara and D. Mamta, “A note on three variable reciprocity theorem,” to appear in International Journal of Pure and Applied Mathematics, 2009.View at: Google Scholar
D. D. Somashekara and D. Mamta, “On the three variable reciprocity theorem and its applications,” to appear in The Australian Journal of Mathematical Analysis and Applications.View at: Google Scholar
Z. Zhang, “An identity related to Ramanujan's and its applications,” to appear in Indian Journal of Pure and Applied Mathematics.View at: Google Scholar
X. R. Ma, “A five-variable generalization of Ramanujan's reciprocity theorem and its applications,” to appear.View at: Google Scholar
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, New York, NY, USA, 5th edition, 1979.View at: MathSciNet