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On the $ {\rm mod}\,2$ reciprocation of infinite modular-part products and the parity of certain partition functions

Authors: Richard Blecksmith, John Brillhart and Irving Gerst
Journal: Math. Comp. 54 (1990), 345-376
MSC: Primary 05A17; Secondary 05A30
MathSciNet review: 995206
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Abstract: An infinite, modular-part (MP) product is defined to be a product of the form $ {\Pi _{n \in S}}(1 - {x^n})$, where $ S = \{ {n \in {{\mathbf{Z}}^ + }:n \equiv {r_1}, \ldots ,{r_t}\;\pmod m} \}$. Some products of this kind have a $ \bmod \, 2$ reciprocal that is also an MP product, while others do not. A complete method is first developed which determines if a given MP product has an MP reciprocal modulo 2 and finds it if it does. Next, a graph-theoretic interpretation of this method is made from which a streamlined algorithm is derived for deciding whether the given MP product is such a reciprocal. This algorithm is then applied to the single-variable Jacobi triple product and the quintuple product to determine the cases when these products have an MP reciprocal $ \pmod 2$. When this occurs--and this occurs in infinitely many cases--the parity of the associated partition function can readily be found. A discussion is also made of the probability that a given MP product with modulus m has an MP reciprocal $ \pmod 2$.

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Keywords: $ \operatorname{Mod} 2$ reciprocation, infinite modular-part products, Jacobi triple product, quintuple product
Article copyright: © Copyright 1990 American Mathematical Society

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