Mathematics of Computation

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

 

 

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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DOI: http://dx.doi.org/10.1090/S0025-5718-1990-1023761-1
Keywords: q-binomial coefficient, partitions
Article copyright: © Copyright 1990 American Mathematical Society