Multiple Rogers-Ramanujan type identities for inert quadratic orders

Abstract

We compute the Quot and finitized Coh zeta functions of the inert quadratic orders ${\mathbb{F}}_q[[T]]+T^m{\mathbb{F}}_{q^2}[[T]]$ for every $m\ge 1$ in terms of a $2m$-fold multisum, and then show this multisum equals an $m$-fold Bressoud sum. This proves a recent conjecture of the second author, rounding up the line of exploration in the series of work by the authors and Jiang. The equality between the $2m$-fold multisum and the $m$-fold Bressoud sum is built upon generalizing the multisum by introducing a ``ghost'' parameter $a$ to its summands. We then show that such an $a$-generalization is surprisingly $a$-independent by purely $q$-theoretic techniques. Finally, we propose a refined multisum that interpolates two versions of Quot zeta functions for all three types of quadratic orders.

41 pages