Factors of sums involving q-binomial coefficients and powers of q-integers

We show that, for all positive integers n(1),..., n(m), n(m+ 1) = n(1), andany non-negative integers j and r with j <= m, the expression 1/[n(1)] similar to n1 + nm n1 similar to -1 n1 similar to k= 1 [2k][k] 2rqjk2 -(r+ 1) k m similar to i= 1 similar to ni + ni+ 1 ni + k similar to is a Laurent polynomial in q with integer coefficients, where [n] = 1+ q+ ... + qn-1 and similar to nk similar to = similar to ki = 1 (1-qn-i+1)/(1-qi). This gives a q-analogue of a divisibility result on the Catalan triangle obtained by the first author and Zeng, and also confirms a conjecture of the first author and Zeng. We further propose several related conjectures.