Integer Matrix Exact Covering Systems and Product Identities for Theta Functions

TL;DR

This paper establishes a correspondence between product identities for theta functions and integer matrix exact covering systems, providing a general theorem that unifies and extends many known identities, including new results.

Contribution

It introduces a novel framework linking theta function identities to integer matrix covering systems, enabling derivation of new and existing identities systematically.

Findings

Proves a general theorem expressing products of theta functions as linear combinations of other products.

Shows many classical identities are special cases of the main theorem.

Derives new identities for products of three and four theta functions.

Abstract

In this paper, we prove that there is a natural correspondence between product identities for theta functions and integer matrix exact covering systems. We show that since $\mathbb{Z}^n$ can be taken as the disjoint union of a lattice generated by $n$ linearly independent vectors in $\mathbb{Z}^n$ and a finite number of its translates, certain products of theta functions can be written as linear combinations of other products of theta functions. We firstly give a general theorem to write a product of $n$ theta functions as a linear combination of other products of theta functions. Many known identities for products of theta functions are shown to be special cases of our main theorem. Several entries in Ramanujan's notebooks as well as new identities are proved as applications, including theorems for products of three and four theta functions that have not been obtained by other methods.