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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

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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


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:
  • MathSciNet review: 1023761