Arithmetic properties of generalized Delannoy polynomials and Schröder 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 Schröder polynomials respectively. Here $C_k$ is the Catalan number and $h$ is a positive integer. In this paper, we prove that $$\begin{array}{rl}& \frac{(2,n)}{n(n+1)(n+2)}\sum _{k=1}^n{k}^a(k+1{)}^a(2k+1)D_k^{(h)}(x{)}^m\in \mathbb{Z}[x],\\ & \frac{(2,hm-1,n)}{n(n+1)(n+2)}\sum _{k=1}^n(-1{)}^k{k}^a(k+1{)}^a(2k+1)D_k^{(h)}(x{)}^m\in \mathbb{Z}[x],\\ & \frac{(2,n)}{n(n+1)(n+2)}\sum _{k=1}^n{k}^a(k+1{)}^a(2k+1)S_k^{(h)}(x{)}^m\in \mathbb{Z}[x],\\ & \frac{(2,m-1,n)}{n(n+1)(n+2)}\sum _{k=1}^n(-1{)}^k{k}^a(k+1{)}^a(2k+1)S_k^{(h)}(x{)}^m\in \mathbb{Z}[x].\end{array}$$ Taking $a=1$ will confirm some of Z.-W. Sun's conjectures.

18 pages