Arithmetic properties of generalized Delannoy polynomials and Schr\"oder polynomials

Abstract

Let $n$ be any nonnegative integer and \[ D_n^{(h)}(x)=\sum_{k=0}^{n}\binom{n+k}{2k}^{h}\binom{2k}{k}^{h}{x}^{k} \text{ and } S_{n}^{(h)}(x)=\sum_{k=0}^{n}\binom{n+k}{2k}^{h}C_{k}^{h}{x}^{k} \] be the generalized Delannoy polynomials and Schr\"oder polynomials respectively. Here $C_k$ is the Catalan number and $h$ is a positive integer. In this paper, we prove that $$\begin{align*} & \frac{(2,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}k^a(k+1)^a(2k+1)D_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x],\\ &\frac{(2,hm-1,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}(-1)^{k}k^a(k+1)^a(2k+1)D_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x],\\ &\frac{(2,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}k^a(k+1)^a(2k+1)S_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x],\\ &\frac{(2,m-1,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}(-1)^{k}k^a(k+1)^a(2k+1)S_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x]. \end{align*}$$ Taking $a=1$ will confirm some of Z.-W. Sun's conjectures.