Multiple Rogers-Ramanujan type identities for inert quadratic orders

2025/11/12 by Shane Chern, Yifeng Huang, Chern, Shane +1
Mathematics · #05A15 #11P84 #11S45 #14D20 #33D15 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2511.09452

openalex publication_date 2025/11/12 · openalex created_date 2025/11/14 · openalex updated_date 2026/07/28

Abstract

We compute the Quot and finitized Coh zeta functions of the inert quadratic orders \mathbbF_q[[T]]+T^m\mathbbF_q²[[T]] for every m≥ 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.

Related

• Coh zeta functions for inert quadratic orders
• Multiple Rogers--Ramanujan type identities for torus links
• The Rogers-Ramanujan dissection of a theta function
• Partial-sum analogues of the Rogers-Ramanujan identities
• q-Bessel Functions and Rogers-Ramanujan Type Identities