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

1987; American Mathematical Society; Volume: 48; Issue: 177 Linguagem: Inglês

10.1090/s0025-5718-1987-0866096-x

1088-6842

Richard Blecksmith, John Brillhart, Irving Gerst,

Analytic Number Theory Research

In this paper we give a collection of parity results for partition functions of the form \[ ∏ n ∈ S ( 1 − x n ) − 1 ≡ ∑ − ∞ ∞ x e ( n ) ( mod 2 ) \prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty x^{e(n)} \pmod 2 \] and \[ ∏ n ∈ S ( 1 − x n ) − 1 ≡ ∑ − ∞ ∞ ( x e ( n ) + x f ( n ) ) ( mod 2 ) \prod _{n \in S} (1 - x^n)^{-1} \equiv \sum _{-\infty }^\infty (x^{e(n)} + x^{f(n)})\pmod 2 \] for various sets of positive integers S , which are specified with respect to a modulus, and quadratic polynomials e ( n ) e(n) and f ( n ) f(n) . Several identities similar to theta function identities, such as \[ ∏ n = 1 n ≢ ± ( 4 , 6 , 8 , 10 ) ( mod 32 ) ∞ ( 1 − x n ) = 1 + ∑ n = 1 ∞ ( − 1 ) n ( x n 2 + x 2 n 2 ) , \prod _{\substack {n = 1\\n \nequiv \pm (4,6,8,10)\pmod {32}}}^\infty (1 - x^n) = 1 + \sum _{n = 1}^\infty (-1)^n (x^{n^2} + x^{2 n^2}), \] and its associated congruence \[ ∏ n = 1 n ≢ 0 , ± 2 , ± 12 , ± 14 , 16 ( mod 32 ) ∞ ( 1 − x n ) − 1 ≡ 1 + ∑ n = 1 ∞ ( x n 2 + x 2 n 2 ) ( mod 2 ) , \prod _{\substack {n = 1\\n \nequiv 0, \pm 2, \pm 12, \pm 14,16 \pmod {32}}}^\infty (1 - x^n)^{-1} \equiv 1 + \sum _{n = 1}^\infty (x^{n^2} + x^{2 n^2}) \pmod 2, \] are also proved.