Some infinite product identities

Blecksmith, Richard; Brillhart, John; Gerst, Irving

doi:10.1090/S0025-5718-1988-0942157-2

Some infinite product identities
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by Richard Blecksmith, John Brillhart and Irving Gerst PDF

Math. Comp. 51 (1988), 301-314 Request permission

Abstract:

In this paper we derive the power series expansions of four infinite products of the form \[ \prod \limits _{n \in {S_1}} {(1 - {x^n})\;\prod \limits _{n \in {S_2}} {(1 + {x^n}),} } \] where the index sets ${S_1}$ and ${S_2}$ 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.

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An Introduction to the Theory of Numbers