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

: 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. identities; combinatorial partition-theoretic identities; Schur’s, the Göllnitz-Gordon’s and the Göllnitz’s partition identities; Schur’s second partition theorem


Introduction and Definitions
Throughout this article, we denote by N, Z, and C the set of positive integers, the set of integers and the set of complex numbers, respectively. We also let In what follows, we shall make use of the following q-notations for the details of which we refer the reader to a recent monograph on q-calculus by Ernst [1] and also to the earlier works on the subject by Slater [2] (Chapter 3, Section 3.2.1), and by Srivastava et al. [3] (pp. 346 et seq.) and [4] (Chapter 6) . The q-shifted factorial (a; q) n is defined (for |q| < 1) by (a; q) n := where a, q ∈ C, and it is assumed tacitly that a = q −m (m ∈ N 0 ). We also write (a; q) ∞ := (1 − aq k−1 ) (a, q ∈ C; |q| < 1).
It should be noted that, when a = 0 and |q| 1, the infinite product in Equation (2) diverges. Thus, whenever (a; q) ∞ is involved in a given formula, the constraint |q| < 1 will be tacitly assumed to be satisfied.

Theorem 1. (Euler's Pentagonal Number
Theorem) The number of partitions of a given positive integer n into distinct parts is equal to the number of partitions of n into odd parts.
We also recall the Rogers-Ramanujan continued fraction R(q) given by Here, G(q) and H(q), which are associated with the widely-investigated Roger-Ramanujan identities, are defined as follows: and and the functions f(a, b) and f (−q) are given by Equations (5) and (10), respectively.
For a detailed historical account of (and for various related developments stemming from) the Rogers-Ramanujan continued fraction (12) as well as the Rogers-Ramanujan identities (13) and (14), the interested reader may refer to the monumental work [7] (p. 77 et seq.) (see also [4,8]).
Here, in this paper, our main objective is to establish a set of six new theta-function identities which depict the inter-relationships that exist between the multivariable R-functions, q-product identities, and partition-theoretic identities.
Each of the following preliminary results will be needed for the demonstration of our main results in this paper (see [26] (pp. 1749-1750 and 1752-1754)): IV. If

A Set of Main Results
In this section, we state and prove a set of six new theta-function identities which depict inter-relationships among q-product identities and the multivariate R-functions. Theorem 3. Each of the following relationships holds true: and Equations (29) and (30) give inter-relationships between R 1 , R 3 , R 5 and R 15 .
Equation (33) gives inter-relationships between R 2 , R 6 , and R α . Furthermore, it is asserted that Equation (34) gives inter-relationships between R 1 , R 3 and R α . It is assumed that each member of the assertions (29) to (34) exists.
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: and Now, upon substituting from these last results (35) and (36) into (23), if we rearrange the terms and use some algebraic manipulations, we are led to the first assertion (29) of Theorem 3.
Secondly, we prove the second relationship (30) of Theorem 3. Indeed, if we first apply the identity (9) (with q replaced by q 3 , q 5 and q 15 ) under the given precondition of the assertion (24), and then make use of (20) and (21), after some simplifications, the following values for P and Q would follow: and Now, upon substituting from these last results (37) and (38) into (24), if we rearrange the terms and use some algebraic manipulations, we obtain the second assertion (30) of Theorem 3.
Thirdly, we prove the third relationship (31) of Theorem 3. For this purpose, we first apply the identity (9) (with q replaced by q 3 , q 7 and q 21 ) under the given precondition of (25), and then use (20) and (21). We thus find for the values of P and Q that and which, in view of (25) and after some rearrangements of the terms and the resulting algebraic manipulations, yields the third assertion (31) of Theorem 3.
Fourthly, we prove the identity (32) by applying the identity (9) (with the parameter q replaced by q 2 , q 3 and q 6 ) under the given precondition of (26), we further use the assertions (20) and (21). Then, upon simplifications, we get the values for P and Q as follows: and Now, after using (41) and (42) in (26), if we rearrange the terms and and apply some algebraic manipulations, we get required result (32) asserted by Theorem 3.
We next prove the fifth identity (33). We apply the identity (9) (with the parameter q replaced by −q, −q 3 , q 2 and q 6 ) under the given precondition of (27). We then further use the results (20) and (21). After simplification, we find the values for P and Q as follows: and Now, after using (43) and (44) in (27), we rearrange the terms and apply some algebraic manipulations. We are thus led to the required result (33).
Finally, we proceed to prove the last identity (34) asserted by Theorem 3. We make use of the identity (9) (with the parameter q replaced by −q, −q 3 and q 3 ) under the given precondition of (28). Then, by applying the identities (20) and (21), we obtain the values for P and Q as follows: and Thus, upon using (45) and (46) in (28), we rearrange the terms and apply some algebraic simplifications. This leads us to the required result (34), thereby completing the proof of Theorem 3.

Remark 2.
Even though the results of Theorem 3 are apparently considerably involved, each of the asserted theta-function identities does have the potential for other applications in analytic number theory and partition theory (see, for example, [30,31]) as well as in real and complex analysis, especially in connection with a significant number of wide-spread problems dealing with various basic (or q-) series and basic (or q-) operators (see, for example, [32,33]).
Each of the theta-function identities (29) to (34), which are asserted by Theorem 3, obviously depict the inter-relationships that exist between q-product identities and the multivariate R-functions. Some corollaries and consequences of Theorem 3 may be worth pursuing for further research in the direction of the developments which we have presented in this article.

Connections with Combinatorial Partition-Theoretic Identities
Various extensions and generalizations of partition-theoretic identities and other q-identities, which we have investigated in this paper, as well as their connections with combinatorial partition-theoretic identities, can be found in several recent works (see, for example, [31,34,35]). The demonstrations in some of these recent developments are also based upon their combinatorial interpretations and generating functions (see also [25]).
As far as the connections with many different partition-theoretic identities are concerned, the existing literature is full of interesting findings and observations on the subject. In fact, in the year 2015, valuable progress in this direction was made by Andrews et al. [14], who established a number of interesting results including those for the q-series, q-products, and q-hypergeometric functions, which are associated closely with Schur's partitions, the Göllnitz-Gordon's partitions, and the Göllnitz's partitions in terms of multivariate R-functions. With a view to making our presentation to be self-sufficient, we choose to recall here some relevant parts of the developments in the remarkable investigation by Andrews et al. (see, for details, [14]).

Concluding Remarks and Observations
The present investigation was motivated by several recent developments dealing essentially with theta-function identities and combinatorial partition-theoretic identities. Here, in this article, we have established a family of six presumably new theta-function identities which depict the inter-relationships that exist among q-product identities and combinatorial partition-theoretic identities. We have also considered several closely-related identities such as (for example) q-product identities and Jacobi's triple-product identities. In addition, with a view to further motivating research involving theta-function identities and combinatorial partition-theoretic identities, we have chosen to indicate rather briefly a number of recent developments on the subject-matter of this article.