$\vec{M}$ 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 $M_1\ge M_2\ge M_3\ge \dots \ge M_{\nu }$, we show that a broad class of multi-sums of the form $$\sum _{\mathbf{n}}\frac{q^{N_1^2+\cdots +N_{\nu }^2-M_1{N}_1-M_2{N}_2-\cdots -M_{\nu }{N}_{\nu }}}{(q{)}_{n_1}(q{)}_{n_2}\cdots (q{)}_{n_{\nu }}}$$ can be expressed as a sum of products. Above, we use standard notations for $q$-Pochhammer symbols and $N_i=\sum _{k=i}^{\nu }{n}_k$ for $1\le i\le \nu$.

13 pages. Comments are welcome!