Sieved partition functions and $q$-binomial coefficients

Frank Garvan, Dennis Stanton
1990 Mathematics of Computation  
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
more » ... of Ramanujan's congruences mod 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 ; = 10 .
doi:10.1090/s0025-5718-1990-1023761-1 fatcat:u6fayrk5arcc7jzy3flydxpvsm