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Mathematics of Computation

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Parity results for certain partition functions and identities similar to theta function identities

Authors: Richard Blecksmith, John Brillhart and Irving Gerst
Journal: Math. Comp. 48 (1987), 29-38
MSC: Primary 11P76
MathSciNet review: 866096
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Abstract: In this paper we give a collection of parity results for partition functions of the form

$\displaystyle \prod\limits_{n \in S} {{{(1 - {x^n})}^{ - 1}} \equiv \sum\limits_{ - \infty }^\infty {{x^{e(n)}}\;\pmod 2} } $


$\displaystyle \prod\limits_{n \in S} {{{(1 - {x^n})}^{ - 1}} \equiv \sum\limits_{ - \infty }^\infty {({x^{e(n)}} + {x^{f(n)}})\;\pmod 2} } $

for various sets of positive integers S, which are specified with respect to a modulus, and quadratic polynomials $ e(n)$ and $ f(n)$. Several identities similar to theta function identities, such as

$\displaystyle \prod\limits_{\mathop {n = 1}\limits_{n \nequiv \pm (4,6,8,10)\;\... ...\mathop \sum \limits_{n = 1}^\infty {{( - 1)}^n}({x^{{n^2}}} + {x^{2{n^2}}}),} $

and its associated congruence

$\displaystyle \prod\limits_{\mathop {n = 1}\limits_{n \nequiv 0, \pm 2, \pm 12,... ...quiv 1 + \sum\limits_{n = 1}^\infty {({x^{{n^2}}} + {x^{2{n^2}}})\;} \pmod 2,} $

are also proved.

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Keywords: Partition function parity, theta function analogues
Article copyright: © Copyright 1987 American Mathematical Society