Factors of Alternating Sums of Products of Binomial and q-Binomial Coefficients

Abstract

In this paper we study the factors of some alternating sums of products of binomial and q-binomial coefficients. We prove that for all positive integers n_1,...,n_m, n_{m+1}=n_1, and 0\leq j\leq m-1, {n_1+n_{m}\brack n_1}^{-1}\sum_{k=-n_1}^{n_1}(-1)^kq^{jk^2+{k\choose 2}} \prod_{i=1}^m {n_i+n_{i+1}\brack n_i+k}\in \N[q], which generalizes a result of Calkin [Acta Arith. 86 (1998), 17--26]. Moreover, we show that for all positive integers n, r and j, {2n\brack n}^{-1}{2j\brack j} \sum_{k=j}^n(-1)^{n-k}q^{A}\frac{1-q^{2k+1}}{1-q^{n+k+1}} {2n\brack n-k}{k+j\brack k-j}^r\in N[q], where A=(r-1){n\choose 2}+r{j+1\choose 2}+{k\choose 2}-rjk, which solves a problem raised by Zudilin [Electron. J. Combin. 11 (2004), #R22].