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\geqslant 1 $ and $ 0\leqslant j\leqslant n $, the following two polynomials
$ \begin{align*} &\sum\limits_{k = j}^{n} (-1)^{k}[3k-2j+1]{2k-2j\brack 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\limits_{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{align*} $
are divisible by $ (1+q^n)^2[2n+1]{2n\brack n} $. Here $ [m] = 1+q+\cdots+q^{m-1}, (a; q)_m = (1-a)(1-aq)\cdots (1-aq^{m-1}) $, and $ {m\brack k} = (q^{m-k+1};q)_k/(q; q)_k $.
Citation: Su-Dan Wang. The $ q $-WZ pairs and divisibility properties of certain polynomials[J]. AIMS Mathematics, 2022, 7(3): 4115-4124. doi: 10.3934/math.2022227
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\geqslant 1 $ and $ 0\leqslant j\leqslant n $, the following two polynomials
$ \begin{align*} &\sum\limits_{k = j}^{n} (-1)^{k}[3k-2j+1]{2k-2j\brack 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\limits_{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{align*} $
are divisible by $ (1+q^n)^2[2n+1]{2n\brack n} $. Here $ [m] = 1+q+\cdots+q^{m-1}, (a; q)_m = (1-a)(1-aq)\cdots (1-aq^{m-1}) $, and $ {m\brack k} = (q^{m-k+1};q)_k/(q; q)_k $.
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Su-Dan Wang. The $ q $-WZ pairs and divisibility properties of certain polynomials[J]. AIMS Mathematics, 2022, 7(3): 4115-4124. doi: 10.3934/math.2022227
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