$\vec M$ Versions of Andrews-Gordon Identities Revisited

Abstract

In this paper, we revisit the work of Berkovich and Paule on variants of the Andrews-Gordon identities. We find a generalization of their principal polynomial identity and, as a consequence, obtain new $\vec M$ versions of the Andrews-Gordon identities. More precisely, for non-negative integers $M_1\ge M_2\ge M_3\ge\ldots\ge M_\nu$, we show that a broad class of multi-sums of the form $$ \sum\limits_{\mathbf n} \frac{ q^{ N_1^2+\cdots+N_\nu^2 - M_1N_1 - M_2N_2 - \cdots - M_\nu N_\nu } }{ (q)_{n_1}(q)_{n_2}\cdots(q)_{n_\nu} } $$ can be expressed as a sum of products. Above, we use standard notations for $q$-Pochhammer symbols and $N_i = \sum\limits_{k=i}^{\nu}n_k$ for $1\le i\le \nu$.