Some infinite product identities

Abstract

In this paper we derive the power series expansions of four infinite products of the form \[ ∏ n ∈ S 1 ( 1 − x n ) ∏ n ∈ S 2 ( 1 + x n ) , \prod \limits _{n \in {S_1}} {(1 - {x^n})\;\prod \limits _{n \in {S_2}} {(1 + {x^n}),} } \] where the index sets S 1 {S_1} and S 2 {S_2} are specified with respect to a modulus (Theorems 1, 3, and 4). We also establish a useful formula for expanding the product of two Jacobi triple products (Theorem 2). Finally, we give nonexistence results for identities of two forms.