On a generalization of a congruence related to q-Narayana numbers

In this note, we study factors of some alternating sums of products of q -binomial coefficients related to q -Narayana numbers. Let n k denote the q -binomial coefficients. We prove that for all positive integers n 1 , … , n m , n m + 1 = n 1 , and j = 0 or 2 m - 1 , the alternating sum n 1 + n m + 1 n 1 - 1 ∑ k = - n 1 n 1 ( - 1 ) k q j k 2 + k 2 ∏ i = 1 m n i + n i + 1 + 1 n i + k n i + n i + 1 + 1 n i + k + 1 is a polynomial in q with integer coefficients, and it has non-negative coefficients if m is odd. This partially confirms a conjecture of Guo and Jiang.