The $q$-WZ pairs and divisibility properties of certain polynomials

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

$$\begin{array}{rl}& \sum _{k=j}^n(-1{)}^k[3k-2j+1][\genfrac{}{}{0ex}{}{2k-2j}{k}]\frac{(q;q^2{)}_k(q;q^2{)}_{k-j}(-q;q{)}_n^3}{(q;q{)}_k(q^2;q^2{)}_{k-j}},\\ & \sum _{k=j}^n(-1{)}^{n-k}{q}^{(k-j{)}^2}[4k+1]\frac{(q;q^2{)}_k^2(q;q^2{)}_{k+j}(-q;q{)}_n^6}{(q^2;q^2{)}_k^2(q^2;q^2{)}_{k-j}(q;q^2{)}_j^2}.\end{array}$$

are divisible by $(1+q^n{)}^2[2n+1][\genfrac{}{}{0ex}{}{2n}{n}]$. Here $[m]=1+q+\cdots +q^{m-1},(a;q{)}_m=(1-a)(1-aq)\cdots (1-a{q}^{m-1})$, and $[\genfrac{}{}{0ex}{}{m}{k}]=(q^{m-k+1};q{)}_k/(q;q{)}_k$.