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

Source 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 $M_1\ge M_2\ge M_3\ge\ldots\ge M_ν$, we show that a broad class of multi-sums of the form $$\sum\limits_{\mathbf n} \frac{ q^{ N_1^2+\cdots+N_ν^2 - M_1N_1 - M_2N_2 - \cdots - M_νN_ν} }{ (q)_{n_1}(q)_{n_2}\cdots(q)_{n_ν} }$$ can be expressed as a sum of products. Above, we use standard notations for $q$-Pochhammer symbols and $N_i = \sum\limits_{k=i}^νn_k$ for $1\le i\le ν$.