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Sieved partition functions and $q$-binomial coefficients

1990
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Mathematics of Computation
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The ^-binomial coefficient is a polynomial in q . Given an integer t and a residue class r modulo ;, a sieved ^-binomial coefficient is the sum of those terms whose exponents are congruent to r modulo /. In this paper explicit polynomial identities in q are given for sieved ij-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

doi:10.1090/s0025-5718-1990-1023761-1
fatcat:u6fayrk5arcc7jzy3flydxpvsm