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Mathematics of Computation

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Sieved partition functions and $q$-binomial coefficients

Authors: Frank Garvan and Dennis Stanton
Journal: Math. Comp. 55 (1990), 299-311
MSC: Primary 11P68; Secondary 05A19, 05A30, 11B65
MathSciNet review: 1023761
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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 Ramanujan’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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Keywords: <I>q</I>-binomial coefficient, partitions
Article copyright: © Copyright 1990 American Mathematical Society