A solution to a problem of Frechette and Locus

Abstract

In a recent paper, Frechette and Locus examined and found expressions for the infinite product $D_m\left(q\right):={\prod }_{t=1}^{\infty }\left(1-q^{mt}\right)∕\left(1-q^t\right)$ in terms of products of $q$-series of the Rogers–Ramanujan type coming from Hall–Littlewood polynomials, when $m\equiv 0,1,2\left(mod4\right)$. These $q$-series were originally discovered in 2014 by Griffin, Ono, and Warnaar in their work on the framework of the Rogers–Ramanujan identities. Extending this framework, Rains and Warnaar also recently discovered more $q$-series and their corresponding infinite products. Frechette and Locus left open the case where $m\equiv 3\left(mod4\right)$. Here we find such an expression for the infinite products for $m\equiv 3\left(mod4\right)$ by making use of the new $q$-series obtained by Rains and Warnaar.