Quadratic Forms, Exact Covering Systems, and Product Identities for Theta Functions

In this paper, we establish a connection between integral quadratic forms and exact covering systems (ECS) and present a structural framework for a class of product identities involving Ramanujan’s theta functions. This approach yields infinitely many such identities. As applications, we provide a unified interpretation for twenty-two of Ramanujan’s forty identities for the Rogers-Ramanujan functions. Many identities analogous to the forty identities can be naturally explained from this perspective. In addition, we discuss ternary quadratic forms and derive new identities involving products of three or more theta functions. We conclude by unifying several previous approaches and providing a summary.