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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1991-1068825-2
PII: S 0025-5718(1991)1068825-2
Keywords: Jacobi triple product, quintuple product, $ \bmod\, 2$ identity, expansion formula
Article copyright: © Copyright 1991 American Mathematical Society