Arithmetic properties of generalized Delannoy polynomials and Schröder polynomials
Lin-Yue Li, Rong-Hua Wang
TL;DR
The paper advances the arithmetic theory of Schmidt-type polynomials by proving that carefully weighted sums of powers of the generalized Delannoy polynomials $D_n^{(h)}(x)$ and generalized Schröder polynomials $S_n^{(h)}(x)$ yield integer-coefficient polynomials in $x$, thus confirming and extending Sun’s conjectures. The authors develop binomial-convolution reductions to unify products of binomial terms into single-binomial sums, and apply Gosper-type telescoping identities and parity arguments to establish divisibility results with scaling factors such as $\frac{(2,n)}{n(n+1)(n+2)}$ or $\frac{(2,m-1,n)}{n(n+1)(n+2)}$. The results cover both $D_k^{(h)}(x)^m$ and $S_k^{(h)}(x)^m$, with distinct treatments for $h>1$ (where stronger divisibility often holds) and $h=1$ (requiring more delicate combinatorial arguments). Overall, the work generalizes prior arithmetic properties of Delannoy and Schröder polynomials and provides a cohesive framework for Schmidt-type polynomials in this arithmetic context.
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 $$\begin{align*} & \frac{(2,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}k^a(k+1)^a(2k+1)D_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x],\\ &\frac{(2,hm-1,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}(-1)^{k}k^a(k+1)^a(2k+1)D_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x],\\ &\frac{(2,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}k^a(k+1)^a(2k+1)S_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x],\\ &\frac{(2,m-1,n)}{n(n+1)(n+2)} \sum_{k=1}^{n}(-1)^{k}k^a(k+1)^a(2k+1)S_{k}^{(h)}(x)^{m}\in\mathbb{Z}[x]. \end{align*}$$ Taking $a=1$ will confirm some of Z.-W. Sun's conjectures.