Combinatorial Properties of Rogers-Ramanujan Type Identities Arising from Hall-Littlewood Polynomials

Abstract

Here we consider the q-series coming from the Hall-Littlewood polynomials, $$\begin{array}{l}{R_{v} (a, b; q) = {\sum_{\mathop {\lambda}\limits_{\lambda_1 \leq a}}} q^{c | \lambda |} P_{2\lambda}(1, q, q^{2}, \ldots ; q^{2b+d}).}\end{array}$$ These series were defined by Griffin, Ono, and Warnaar in their work on the framework of the Rogers-Ramanujan identities. We devise a recursive method for computing the coefficients of these series when they arise within the Rogers-Ramanujan framework. Furthermore, we study the congruence properties of certain quotients and products of these series, generalizing the famous Ramanujan congruence $$p(5n+4) \equiv 0\quad ({\rm mod}\, 5).$$