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Sieved partition functions and q-binomial coefficients. (English) Zbl 0702.05008

The q-binomial coefficient \[ \left[ \begin{matrix} N\\ k\end{matrix} \right]_ q=\frac{(1-q^ N)...(1-q^{N-k+1})}{(1-q)...(1-q^ k)}=\sum_{i\geq 0}a_ iq^ i \] is a polynomial in q with integer coefficients. Given an integer t and a residue class r mod t, a sieved q-binomial coefficient is the sum of those terms whose exponents are congruent to r mod t. In this article, explicit polynomial identities in \(q^ t\) are given for these coefficients. Generating functions for the sieved partition function are found to be multidimensional theta functions. A corollary of this representation is the proof of Ramanujan’s congruences mod 5, 7 and 11 by exhibiting symmetry groups of orders 5, 7 and 11 of explicit quadratic forms. Subbarao conjectured that for \(0\leq r\leq t-1\), \(p(tn+r)\) is even infinitely often and odd infinitely often. This conjectue is verified for \(t=3\), 5 and 10.
Reviewer: M.Cheema

MSC:

05A19 Combinatorial identities, bijective combinatorics
05A17 Combinatorial aspects of partitions of integers
11P81 Elementary theory of partitions
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