Some Theta-Function Identities Related to Jacobi’s Triple-Product Identity

Abstract

Abstract. The main object of this paper is to present some q-identities involving some of the theta functions of Jacobi and Ramanujan. These q-identities reveal certain relationships among three of the theta-type functions which arise from the celebrated Jacobi's triple-product identity in a remarkably simple way. The results presented in this paper are motivated by some recent works by Chaudhary et al.

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

“…Proof. First of all, in order to prove the assertion (29) of Theorem 3, we apply the identity (9) (with q replaced by q 3 , q 5 q 15 ) under the given precondition of result (23). Thus, by using ( 20) and ( 21), and, after some simplifications, we get the values for P and Q as follows:…”

“…The list of citations, which we have included in this article, is believed to be potentially useful for indicating some of the directions for further research and related developments on the subject-matter which we have dealt with here. In particular, the recent works by Adiga et al (see [28,39]), Cao et al [40], Chaudhary et al (see [13,21,22]), Hahn et al [25], and Srivastava et al (see [23,29,33,[41][42][43][44][45]), and by Yee [35] and Yi [31], are worth mentioning here.…”

“…Ever since the year 2015, several new advancements and generalizations of the existing results were made in regard to combinatorial partition-theoretic identities (see, for example, [8,[15][16][17][18][19][20][21][22][23][24]). In particular, Chaudhary et al generalized several known results on character formulas (see [22]), Roger-Ramanujan type identities (see [19]), Eisenstein series, the Ramanujan-Göllnitz-Gordon continued fraction (see [20]), the 3-dissection property (see [18]), Ramanujan's modular equations of degrees 3, 7, and 9 (see [16,17]), and so on, by using combinatorial partition-theoretic identities.…”

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

“…The list of citations, which we have included in this article, is believed to be potentially useful for indicating some of the directions for further researches and related developments on the subject-matter which we have dealt with here. In particular, the recent works by Adiga et al (see [1]- [3]), Cao et al [9], Chaudhary et al (see [10] to [16]), Hahn et al [17], and Srivastava et al (see [26], [31]- [33]) are worth mentioning here.…”

Theta function identities of level 6 and their application to partitions

“…al. [11] proved two q-identities which provide relationship between f (−q), ϕ(q) and ψ(q), by using well known Jacobi's triple product identity. Note that, if q = e 2πiτ then f (−q) = e −πiτ/12 η(τ), where η(τ) denotes the classical Dedekind η-function for any complex number τ and Im(τ) > 0.…”