In 2022, Z.-W. Sun defined \(\begin{aligned} w_k^{(\alpha )}{(x)}=\sum _{j=1}^{k}w(k,j)^{\alpha }x^{j-1}, \end{aligned}\) where \(k,\alpha \) are positive integers and \(w(k,j)=\frac{1}{j}\left( {\begin{array}{c}k-1\\ j-1\end{array}}\right) \left( {\begin{array}{c}k+j\\ j-1\end{array}}\right) \) . Let \((x)_{0}=1\) and \((x)_{n}=x(x+1)\cdots (x+n-1)\) for all \(n\ge 1\) . In this paper, it is proved by q-congruences that for any positive integers \({\alpha ,\beta , m,n,r}\) , we have \(\begin{aligned}&\frac{(2,n)}{n(n+1)(n+2)}\sum _{k=1}^{n}k^r(k+1)^r(2k+1)w_{k}^{(\alpha )}(x)^{m}\in \mathbb {Z}[x], \\&\frac{(2,n)}{n(n+1)(n+2)}\sum _{k=1}^{n}(-1)^{k}k^r(k+1)^r(2k+1) w_{k}^{(\alpha )}(x)^{m}\in {\mathbb {Z}}[x], \end{aligned}\) and \(\begin{aligned} \frac{2}{[n,n+1,\cdots ,n+2\beta +1]}\sum _{k=1}^{n}(k)_{\beta }^r(k+\beta +1)_{\beta }^r(k+\beta ) \prod _{i=0}^{2\beta -1}w_{k+i}^{(\alpha )}(x)^m\in {\mathbb {Z}}[x], \end{aligned}\) where \([n,n+1,\cdots ,n+2\beta +1]\) is the least common multiple of n, \(n+1\) , \(\cdots \) , \(n+2\beta +1\) . Taking \(r=\beta =1\) above will confirm some of Z.-W. Sun’s conjectures.
