$q$-Congruences for Z.-W. Sun's generalized polynomials $w^{(\alpha)}_k(x)$

Abstract

In 2022, Z.-W. Sun defined \begin{equation*} w_k^{(\alpha)}{(x)}=\sum_{j=1}^{k}w(k,j)^{\alpha}x^{j-1}, \end{equation*} where $k,\alpha$ are positive integers and $w(k,j)=\frac{1}{j}\binom{k-1}{j-1}\binom{k+j}{j-1}$. Let $(x)_{0}=1$ and $(x)_{n}=x(x+1)\cdots(x+n-1)$ for all $n\geq 1$. In this paper, it is proved by $q$-congruences that for any positive integers ${\alpha,\beta, m,n,r}$, we have \begin{equation*} \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], \end{equation*} \begin{equation*} \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{equation*} and \begin{equation*} \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{equation*} 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.