Let \([n]!=\prod _{i=1}^n(1+q+\cdots +q^{i-1})\) denote the q-factorials and let \({n\brack k}=[n]!/([k]![n-k]!)\) be the q-binomial coefficients, where \(1/[k]!=0\) if k is a negative integer. Let \(m_1,\ldots ,m_r,m_{r+1}=m_1\) and \(n_1,\ldots ,n_s\) , \(n_{s+1}=n_1\) be positive integers with \(r,s\geqslant 1\) . We prove that the alternating sum \(\begin{aligned}&\frac{[m_1]![n_1]![m_r+n_s+1]!}{[m_1+m_r+1]![n_1+n_s]!}\sum _{k=-n_1}^{n_1}(-1)^k q^{ak^2+(2r-1){k\atopwithdelims ()2}}\\&\quad \times \prod _{i=1}^{r}{m_i+m_{i+1}+1\brack m_i+k}\cdot \prod _{j=1}^s {n_j+n_{j+1}\brack n_j+k} \end{aligned}\) is a polynomial in q with non-negative integer coefficients for \(0\leqslant a\leqslant s\) . We also propose some related conjectures.
