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On a certain (mod $ 2$) identity and a method of proof by expansion

Authors: Richard Blecksmith, John Brillhart and Irving Gerst
Journal: Math. Comp. 56 (1991), 775-794
MSC: Primary 11P83
MathSciNet review: 1068825
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Abstract: We prove the congruence

$\displaystyle \prod\limits_{\begin{array}{*{20}{c}} {n = 1} \\ {n \nequiv 7\;\p... ...mits_{ - \infty }^\infty {({x^{n(3n + 2)}} + {x^{7n(3n + 2) + 2}})\;\pmod 2} } $

by first establishing a related equation, which reduces to the congruence modulo 2. The method of proof (called "expanding zero") is based on a formula of the authors for expanding the product of two triple products. A second proof of the result more fully explicates the various aspects of the method. A parity result for an associated partition function is also included.

References [Enhancements On Off] (What's this?)

  • [1] R. Blecksmith, J. Brillhart, and I. Gerst, Parity results for certain partition functions and identities similar to theta function identities, Math. Comp. 48 (1987), 29-38. MR 866096 (87k:11113)
  • [2] -, Some infinite product identities, Math. Comp. 51 (1988), 301-314. MR 942157 (89f:05017)
  • [3] -, On the $ \bmod\, 2$ reciprocation of infinite modular-part products and the parity of certain partition functions, Math. Comp. 54 (1990), 345-376. MR 995206 (90g:05025)
  • [4] -, On a certain $ \pmod 2$ identity, Abstracts Amer. Math. Soc. 11 (1990), 145.

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Keywords: Jacobi triple product, quintuple product, $ \bmod\, 2$ identity, expansion formula
Article copyright: © Copyright 1991 American Mathematical Society

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