Proofs of some conjectural Hurwitz type identities for three squares due to Cooper and Hirschhorn

Abstract

Let rs(n) denote the number of representations of n as the sum of s squares of integers. Hurwitz provided eleven cases in which the generating function of r3(an+b) is a simple infinite product. In 2004, Cooper and Hirschhorn proved eleven identities of Hurwitz, another twelve of the same sort and eighty infinite families of similar results by utilizing some q-series techniques. Moreover, they further conjectured eighty-two infinite families of Hurwitz type identities for r3(n). In this paper, we confirm these conjectural infinite families of Hurwitz type identities by using the theory of modular forms.