Sieved partition functions and binomial coefficients
Authors:
Frank Garvan and Dennis Stanton
Journal:
Math. Comp. 55 (1990), 299311
MSC:
Primary 11P68; Secondary 05A19, 05A30, 11B65
MathSciNet review:
1023761
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Abstract: The qbinomial coefficient is a polynomial in q. Given an integer t and a residue class r modulo t, a sieved qbinomial 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 qbinomial 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:
http://dx.doi.org/10.1090/S00255718199010237611
PII:
S 00255718(1990)10237611
Keywords:
qbinomial coefficient,
partitions
Article copyright:
© Copyright 1990
American Mathematical Society
