A Family of Theta-Function Identities Based upon Combinatorial Partition Identities Related to Jacobi’s Triple-Product Identity

Abstract

The authors establish a set of six new theta-function identities involving multivariable R-functions which are based upon a number of q-product identities and Jacobi’s celebrated triple-product identity. These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper, we consider and relate the multivariable R-functions to several interesting q-identities such as (for example) a number of q-product identities and Jacobi’s celebrated triple-product identity. Various recent developments on the subject-matter of this article as well as some of its potential application areas are also briefly indicated. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities and present a presumably open problem.

A family of theta-function identities based upon Rα,Rβ and Rm-functions related to Jacobi’s triple-product identity

“…In this section, we establish a set of two new relationships involving R α , R β and R m -functions, q-product identities and continued-fraction identities. Also answer an open problem of Srivastava et al [28], which state as find inter-relationships between R α , R β and R m (m ∈ N), q-product identities and continued-fraction identities. In this article we establish the relationships between R α , R β and R m (m ∈ N), q-product identities.…”

“…Recently, Srivastava et al [28] coined an open problem, which state as find an inter-relationships between R α , R β and R m (m ∈ N), q-product identities and continued-fraction identities. In this article we establish the relationships between R α , R β and R m (m ∈ N), q-product identities.…”

Relations between Rα, Rβ and Rm functions related to Jacobi’s triple-product identity and the family of theta-function identities

“…These theta-function identities depict the inter-relationships that exist among theta-function identities and combinatorial partition-theoretic identities. Here, in this paper we answer to an open question of Srivastava et al [33], and estabish relations in terms of R α , R β and R m (for m = 1, 2, 3); and q-products identities. Finally, we choose to further emphasize upon some close connections with combinatorial partition-theoretic identities.…”

“…But, recently valuable progress in this direction has been made by Andrews et al (see [3]); and established three results for the double series associated with Schur's partitions, Göllnitz-Gordon partitions and Göllnitz partitions in terms of multivariate R-functions. Further, Srivastava et al (see [33]) has generalized multivariate R-functions in terms of R α , R β and R m .…”

“…Recently, Srivastava et al [33] coined an open problem, which state as find an inter-relationships between R α , R α and R m (m ∈ N ), q-product identities and continued-fraction identities. The purpose of this article is to establish relationships between R α , R α and R m (m ∈ N ) and the q-product identities.…”