Factors of sums involving q-binomial coefficients and powers of q-integers
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DOI: 10.1080/10236198.2017.1355366
Abstract: Abstract We show that, for all positive integers , , and any non-negative integers j and r with , the expression is a Laurent polynomial in q with integer coefficients, where and . This gives… read more here.
Keywords: coefficients powers; sums involving; factors sums; involving binomial ... See more keywords
Factors of Sums and Alternating Sums of Products of $q$-binomial Coefficients and Powers of $q$-integers
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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 $j\leqslant m$, the following two expressions \begin{align*} &\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},\\[5pt] &\frac{1}{[n_1+n_m+1]}{n_1+n_{m}\brack n_1}^{-1}\sum_{k=0}^{n_1}(-1)^k q^{{k\choose 2}+j(k^2+k)-2rk}[2k+1]^{2r+1}\prod_{i=1}^m {n_i+n_{i+1}+1\brack… read more here.
Keywords: binomial coefficients; sums alternating; powers integers; alternating sums ... See more keywords