Factors of Sums and Alternating Sums of Products of q-binomial Coefficients and Powers of q-integers

We prove that, for all positive integers n(1), ..., n(m), n(m+1) = n(1), and non-negative integers j and r with j <= m, the following two expressions 1/[n(1) + n(m) + 1] [(n1 + nm)(n1)](-)(1) Sigma(n1)(k=0)q(j(k2 + k)-(2r + 1)k)[2k + 1](2r+1)Pi(m)(i=1)[(ni + ni+1+1)(ni-k)], 1/[n(1) + n(m) + 1] [(n1 + nm)(n1)](-)(1 )Sigma(n1)(k=0) (-1)k q((2k) + j(k2 + k) -2rk)[2k +1](2r + 1)Pi(m)(i=1)[(ni + ni+1+1)(ni-k)], are Laurent polynomials in q with integer coefficients, where [n] = 1 + q + ... + q(n-1) and [(n)(k)] = Pi(k)(i=1) (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 ([(2n)(n-k)] - [(2n)(n - k - 1)]) are proposed in the last section of this paper.