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Parity results for certain partition functions and identities similar to theta function identities. (English) Zbl 0617.10010
The authors present a collection of parity results for partition functions of the form \[ \prod_{n\in S}(1-x^ n)^{- 1}=\sum^{\infty}_{-\infty}x^{e(n)} (\bmod 2)\text{ and } \prod_{n\in S}(1-x^ n)^{-1}=\sum^{\infty}_{- \infty}(x^{e(n)}+x^{f(n)}) (\bmod 2) \] for various sets of positive integers S which are specified with respect to a modulus and quadratic polynomials e(n) and f(n). These were obtained using methods of their earlier paper and (mod 2) study of the generating functions on computers.
Several identities similar to theta function identities are also proved. All possible residue classes for moduli up to 16 as well as symmetrically placed classes for moduli up to 30 were tried. Typical examples are \[ \prod^{\infty}_{n=1}(1-x^ n)^{-1}=1+\sum^{\infty}_{n=1}(- 1)^ n(x^{n^ 2}+x^{2n^ 2}),\quad n\neq \pm (4,6,8,10) (\bmod 32) \] and the associated congruence \[ \prod^{\infty}_{n=1}(1-x^ n)^{-1}=1+\sum^{\infty}_{n=1}(x^{n^ 2}+x^{2n^ 2}) (\bmod 2),\quad n\neq \pm 0,\pm 2,\pm 12,\pm 14,16 (\bmod 32). \]
Reviewer: M.Cheema

MSC:
11P81 Elementary theory of partitions
05A19 Combinatorial identities, bijective combinatorics
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