Open Access

Andrews-Gordon style identities

Research in Number Theory20151:13

DOI: 10.1007/s40993-015-0014-6

Received: 16 June 2015

Accepted: 30 June 2015

Published: 8 October 2015

Abstract

In a recent paper, Griffin, Ono and Warnaar present a framework for Rogers-Ramanujan type identities using extended Hall-Littlewood polynomials P λ (x 1,x 2,…;q) to arrive at expressions of the form
$$\sum\limits_{\lambda : \lambda_{1} \leq m} q^{a|\lambda|}P_{2\lambda}(1,q,q^{2},\ldots ; q^{n}) = \text{{\textquotedblleft}Infinite product modular function{\textquotedblright}} $$

for a=1,2 and any positive integers m and n. A recent paper of Rains and Warnaar presents further Rogers-Ramanujan type identities involving sums of terms q |λ|/2 P λ (1,q,q 2,…;q n ). It is natural to attempt to reformulate these various identities to match the well-known Andrews-Gordon identities they generalize. Here, we find combinatorial formulas to replace the Hall-Littlewood polynomials and arrive at such expressions.