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February, 2019 Factors of Sums and Alternating Sums of Products of $q$-binomial Coefficients and Powers of $q$-integers
Victor J. W. Guo, Su-Dan Wang
Taiwanese J. Math. 23(1): 11-27 (February, 2019). DOI: 10.11650/tjm/180601

Abstract

We prove that, for all positive integers $n_1, \ldots, n_m$, $n_{m+1} = n_1$, and non-negative integers $j$ and $r$ with $j \leq m$, the following two expressions \begin{gather*} \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} \end{gather*} 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 Guo and Zeng. We also confirm some related conjectures of Guo and Zeng 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.

Citation

Victor J. W. Guo. Su-Dan Wang. "Factors of Sums and Alternating Sums of Products of $q$-binomial Coefficients and Powers of $q$-integers." Taiwanese J. Math. 23 (1) 11 - 27, February, 2019. https://doi.org/10.11650/tjm/180601

Information

Received: 8 March 2018; Accepted: 4 June 2018; Published: February, 2019
First available in Project Euclid: 9 June 2018

zbMATH: 1405.05017
MathSciNet: MR3909988
Digital Object Identifier: 10.11650/tjm/180601

Subjects:
Primary: 05A30, 11B65, 65Q05

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

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