Some new modular relations for the Rogers–Ramanujan type functions of order eleven with applications to partitions

Abstract

In this paper, we establish several modular relations for the Rogers-Ramanujan type functions of order eleven which are analogous to Ramanujan's forty identities for RogersRamanujan functions. Furthermore, we give interesting partition-theoretic interpretation of some of the modular relations which are derived in this paper.

On Ramanujan's general theta function and a generalization of the Borweins' cubic theta functions

“…For a proof of Lemma 2.4, see [3]. Our functions a(α, β, γ, z 1 , z 2 ; q), b(α, β, γ, z 1 , z 2 ; q) and c(α, β, γ, z 1 , z 2 ; q) with α, γ > 0 satisfy the following basic properties: …”

Theta function identities of level 6 and their application to partitions

“…al. [1] have established several modular relations for the Rogers−Ramanujan type functions of order eleven which analogous to Ramanujan's forty identities for Rogers−Ramanujan functions and also they established certain interesting partition-theoretic interpretation of some of the modular relations. H. M. Srivastava and M. P. Chaudhary [10] established a set of four new results which depict the interrelationships between q-product identities, continued fraction identities and combinatorial partition identities.…”

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

“…We also recall here that, in Chapter 16 of his celebrated Notebooks, Srinivasa Ramanujan (1887Ramanujan ( -1920 defined the general theta function f(a, b) as follows (see, for example, [10] and [11]; see also [1] and [13]):…”

“…Besides, the subject of q-analysis, which is popularly known as the quantum analysis, has its roots in such important areas as (for example) Mathematical Physics, Analytic Number Theory, and the Theory of Partitions. Motivated essentially by the potential for applications of q-series and q-products, we investigate here the following three most interesting functions which are related closely to such entities as Jacobi's theta functions in the equations (1) to (4), Ramanujan's general theta function in (5) and Jacobi's triple-product identity in (7) or (8) (see also [1] and [13]):…”

“…For more details and further results, the interested reader may be referred to the works presented in [1], [2], [3], [6], [7], [8] and [13].…”