On a certain (mod $2$) identity and a method of proof by expansion
HTML articles powered by AMS MathViewer
- by Richard Blecksmith, John Brillhart and Irving Gerst PDF
- Math. Comp. 56 (1991), 775-794 Request permission
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.References
- 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 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 10.1090/S0025-5718-1988-0942157-2
- Richard Blecksmith, John Brillhart, and Irving Gerst, On the $\textrm {mod}\,2$ reciprocation of infinite modular-part products and the parity of certain partition functions, Math. Comp. 54 (1990), no. 189, 345–376. MR 995206, DOI 10.1090/S0025-5718-1990-0995206-9 —, On a certain $\pmod 2$ identity, Abstracts Amer. Math. Soc. 11 (1990), 145.
Additional Information
- © Copyright 1991 American Mathematical Society
- Journal: Math. Comp. 56 (1991), 775-794
- MSC: Primary 11P83
- DOI: https://doi.org/10.1090/S0025-5718-1991-1068825-2
- MathSciNet review: 1068825