Scholar - SciOpen

Using the $q$ -WZ (Wilf-Zeilberger) pairs we give divisibility properties of certain polynomials. These results may deemed generalizations of some $q$ -congruences obtained by Guo earlier, or $q$ -analogues of some congruences of Sun. For example, we prove that, for $n ⩾ 1$ and $0 ⩽ j ⩽ n$ , the following two polynomials

$\begin{aligned} \sum_{k = j}^{n} (- 1)^{k} [3 k - 2 j + 1] [\binom{2 k - 2 j}{k}] \frac{(q; q^{2})_{k} (q; q^{2})_{k - j} (- q; q)_{n}^{3}}{(q; q)_{k} (q^{2}; q^{2})_{k - j}}, \\ \sum_{k = j}^{n} (- 1)^{n - k} q^{(k - j)^{2}} [4 k + 1] \frac{(q; q^{2})_{k}^{2} (q; q^{2})_{k + j} (- q; q)_{n}^{6}}{(q^{2}; q^{2})_{k}^{2} (q^{2}; q^{2})_{k - j} (q; q^{2})_{j}^{2}} . \end{aligned}$

are divisible by $(1 + q^{n})^{2} [2 n + 1] [\binom{2 n}{n}]$ . Here $[m] = 1 + q + \dots + q^{m - 1}, (a; q)_{m} = (1 - a) (1 - a q) \dots (1 - a q^{m - 1})$ , and $[\binom{m}{k}] = (q^{m - k + 1}; q)_{k} / (q; q)_{k}$ .