Ramanujan’s theta function identities and the relations between sums of squares and sums of triangular numbers

Abstract

<jats:p> For positive integers [Formula: see text], [Formula: see text] and [Formula: see text], let [Formula: see text] denote the number of representations of a nonnegative integer [Formula: see text] as [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are nonnegative integers, and let [Formula: see text] denote the number of representations of [Formula: see text] as [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are integers. Recently, Sun proved a number of relations between [Formula: see text] and [Formula: see text] along with numerous conjectures on such relations. In this work, we confirm several conjectures of Sun by using Ramanujan’s theta function identities. </jats:p>