A $q$-multisum identity arising from finite chain ring probabilities

TL;DR

This paper establishes a new identity linking a $q$-multisum to theta function products, motivated by probability calculations over modules in finite chain rings, advancing the mathematical understanding of these structures.

Contribution

It introduces a novel identity connecting $q$-multisums with theta functions, derived from probability computations in finite chain ring modules.

Findings

Proves a general identity between $q$-multisum and theta function products.

Links combinatorial identities to probability in algebraic structures.

Provides a new tool for analyzing modules over finite chain rings.

Abstract

In this note, we prove a general identity between a $q$-multisum $B_N(q)$ and a sum of $N^2$ products of quotients of theta functions. The $q$-multisum $B_N(q)$ recently arose in the computation of a probability involving modules over finite chain rings.