## 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

Zbl 1235.05009
Full Text:

### References:

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