The Product-to-sum Expressing with a Divisor Function σ3(n)

A formula expressing an infinite product as an infinite sum is called a product-to-sum identity.In this paper we try to consider a special product-to-sum asqr∞∏n=1(1−qn)a(1−q2n)b(1−q4n)c=u+∞∑n=1{xσ3(n) +yσ3(n2) +zσ3(n4)}qnand so for integersr,u,a,b,c,x,y, andzwe deduce all solutions of (r,u,a,b,c,x,y,z) withr≥0.