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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Sieved partition functions and $q$-binomial coefficients
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by Frank Garvan and Dennis Stanton PDF
Math. Comp. 55 (1990), 299-311 Request permission

Abstract:

The q-binomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved q-binomial coefficient is the sum of those terms whose exponents are congruent to r modulo t. In this paper explicit polynomial identities in ${q^t}$ are given for sieved q-binomial coefficients. As a limiting case, generating functions for the sieved partition function are found as multidimensional theta functions. A striking corollary of this representation is the proof of Ramanujanโ€™s congruences $\bmod 5, 7$, and 11 by exhibiting symmetry groups of orders 5, 7, and 11 of explicit quadratic forms. We also verify the Subbarao conjecture for $t = 3$, $t = 5$, and $t = 10$.
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Additional Information
  • © Copyright 1990 American Mathematical Society
  • Journal: Math. Comp. 55 (1990), 299-311
  • MSC: Primary 11P68; Secondary 05A19, 05A30, 11B65
  • DOI: https://doi.org/10.1090/S0025-5718-1990-1023761-1
  • MathSciNet review: 1023761