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

DOI:
https://doi.org/10.1090/S0025-5718-1987-0866096-X

MathSciNet review:
866096

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Abstract: In this paper we give a collection of parity results for partition functions of the form \[
\prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty x^{e(n)} \pmod 2 \] and \[
\prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty (x^{e(n)} + x^{f(n)})\pmod 2 \] 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 \[
\prod _{\substack {n = 1\\n \nequiv \pm (4,6,8,10)\pmod {32}}}^\infty (1 - x^n) = 1 + \sum _{n = 1}^\infty (-1)^n (x^{n^2} + x^{2 n^2}), \] and its associated congruence \[
\prod _{\substack {n = 1\\n \nequiv 0, \pm 2, \pm 12, \pm 14,16 \pmod {32}}}^\infty (1 - x^n)^{-1} \equiv 1 + \sum _{n = 1}^\infty (x^{n^2} + x^{2 n^2}) \pmod 2, \] are also proved.

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Keywords:
Partition function parity,
theta function analogues

Article copyright:
© Copyright 1987
American Mathematical Society