Some infinite product identities

Abstract

In this paper we derive the power series expansions of four infinite products of the form J](l-x") J](l + x"), nSSi neS2 where the index sets Si and S2 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.

A generalization of a modular identity of Rogers

“…This of course can be duplicated for any other pair whose sum is 64. We now prove the aforementioned formula of Blecksmith, Brillhart, and Gerst [4]. The reformulation we give here can be found in [1, p. 73].…”

“…As applications we provide new modular equations as theta function identities and new identities for the Rogers-Ramanujan functions. We also obtain as a special case a formula of Blecksmith, Brillhart, and Gerst [4] that provides a representation for a product of two fairly general theta functions as a certain sum of products of pairs of theta functions. This formula, in turn, generalizes formulas of Schröter [1, pp.…”

Some more identities of the Rogers-Ramanujan type

“…The first of these is an identity of Blecksmith, Brillhart and Gerst [11] (a proof is also given in [13]):…”

Septic analogues of the Rogers–Ramanujan functions

“…65-72] pro-viding a representation for a product of two theta functions as a sum of m products of pairs of theta functions, under certain conditions. An elegant generalization of Schröter's work has been discovered by R. Blecksmith, J. Brillhart, and I. Gerst [4,Theorem 2]. We translate their formula into Ramanujan's notation.…”