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

Abstract

In 2022, Z.-W. Sun defined \begin{equation*} w_k^{(α)}{(x)}=\sum_{j=1}^{k}w(k,j)^αx^{j-1}, \end{equation*} where $k,\alpha$ are positive integers and $w(k,j)=\frac{1}{j}\left(\genfrac{}{}{0ex}{}{k-1}{j-1}\right)\left(\genfrac{}{}{0ex}{}{k+j}{j-1}\right)$. 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{equation*} \frac{(2,n)}{n(n+1)(n+2)}\sum_{k=1}^{n}k^r(k+1)^r(2k+1)w_{k}^{(α)}(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}^{(α)}(x)^{m}\in\mathbb{Z}[x], \end{equation*} and \begin{equation*} \frac{2}{[n,n+1,\cdots,n+2β+1]}\sum_{k=1}^{n}(k)_β^r(k+β+1)_β^r(k+β) \prod_{i=0}^{2β-1}w_{k+i}^{(α)}(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.