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
DOI:
https://doi.org/10.1090/S0025-5718-1991-1068825-2
MathSciNet review:
1068825
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Abstract | References | Similar Articles | Additional Information
Abstract: We prove the congruence \[ \prod \limits _{\begin {array}{*{20}{c}} {n = 1} \\ {n \nequiv 7\;\pmod {14}} \\ \end {array} }^\infty {(1 - {x^n}) \equiv \sum \limits _{ - \infty }^\infty {({x^{n(3n + 2)}} + {x^{7n(3n + 2) + 2}})\;\pmod 2} } \] 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.
- Richard Blecksmith, John Brillhart, and Irving Gerst, Parity results for certain partition functions and identities similar to theta function identities, Math. Comp. 48 (1987), no. 177, 29–38. MR 866096, DOI https://doi.org/10.1090/S0025-5718-1987-0866096-X
- Richard Blecksmith, John Brillhart, and Irving Gerst, Some infinite product identities, Math. Comp. 51 (1988), no. 183, 301–314. MR 942157, DOI https://doi.org/10.1090/S0025-5718-1988-0942157-2
- Richard Blecksmith, John Brillhart, and Irving Gerst, On the ${\rm mod}\,2$ reciprocation of infinite modular-part products and the parity of certain partition functions, Math. Comp. 54 (1990), no. 189, 345–376. MR 995206, DOI https://doi.org/10.1090/S0025-5718-1990-0995206-9 ---, On a certain $\pmod 2$ identity, Abstracts Amer. Math. Soc. 11 (1990), 145.
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Keywords:
Jacobi triple product,
quintuple product,
<IMG WIDTH="60" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\bmod 2$"> identity,
expansion formula
Article copyright:
© Copyright 1991
American Mathematical Society