Sieved partition functions and -binomial coefficients

Authors:
Frank Garvan and Dennis Stanton

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

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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 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 , and 11 by exhibiting symmetry groups of orders 5, 7, and 11 of explicit quadratic forms. We also verify the Subbarao conjecture for , , and .

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1990-1023761-1

Keywords:
*q*-binomial coefficient,
partitions

Article copyright:
© Copyright 1990
American Mathematical Society