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Factors of sums and alternating sums of products of \(q\)-binomial coefficients and powers of \(q\)-integers. (English) Zbl 1405.05017

Summary: We prove that, for all positive integers \(n_1, \dots, n_m\), \(n_{m+1} = n_1\), and non-negative integers \(j\) and \(r\) with \(j \leq m\), the following two expressions
\[ \frac{1}{[n_1+n_m+1]} {n_1+n_{m} \brack n_1}^{-1} \sum_{k=0}^{n_1} q^{j(k^2+k) - (2r+1)k} [2k+1]^{2r+1} \prod_{i=1}^m {n_i+n_{i+1}+1 \brack n_i-k}, \]
\[ \frac{1}{[n_1+n_m+1]} {n_1+n_{m} \brack n_1}^{-1} \sum_{k=0}^{n_1} (-1)^k q^{\binom{k}{2} + j(k^2+k) - 2rk} [2k+1]^{2r+1} \prod_{i=1}^m {n_i+n_{i+1}+1 \brack n_i-k}, \]
are Laurent polynomials in \(q\) with integer coefficients, where \([n] = 1+q+\cdots+q^{n-1}\) and \({n \brack k} = \prod_{i=1}^k (1-q^{n-i+1})/(1-q^i)\). This gives a \(q\)-analogue of some divisibility results of sums and alternating sums involving binomial coefficients and powers of integers obtained by V. J. W. Guo and J. Zeng [Int. J. Number Theory 7, No. 7, 1959–1976 (2011; Zbl 1235.05009)]. We also confirm some related conjectures of Guo and Zeng [loc. cit.] by establishing their \(q\)-analogues. Several conjectural congruences for sums involving products of \(q\)-ballot numbers \(\left( {2n \brack n-k} - {2n \brack n-k-1} \right)\) are proposed in the last section of this paper.

MSC:

05A30 \(q\)-calculus and related topics
11B65 Binomial coefficients; factorials; \(q\)-identities

Citations:

Zbl 1235.05009
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