Congruences for sums of Delannoy numbers and polynomials

Abstract

In this paper, we apply the power-partible reduction to study arithmetic properties of sums involving Delannoy numbers $D_k$ and polynomials $D_k(z)$. Let $v\in\bN$ and $p$ be an odd prime. It is proved that, for any $z\in\bZ\setminus\{0,-1\}$, there exist $c_v\in z^{-v}\bZ[z]$ and $\tilde{c}_v\in (z+1)^{-v}\bZ[z]$, both free of $p$ and can be determined mechanically, such that \begin{equation*} \sum_{k=0}^{p-1}(2k+1)^{2v}D_k(z)\equiv c_v \left(\frac{-z}{p}\right) \pmod {p} \end{equation*} if $\gcd(p,z)=1$ and \begin{equation*} \sum_{k=0}^{p-1}(-1)^k(2k+1)^{2v}D_k(z)\equiv \tilde{c}_v \left(\frac{z+1}{p}\right) \pmod {p} \end{equation*} if $\gcd(p,z+1)=1$. Here $(-)$ denotes the Legendre symbol. When $n$ is a power of $2$, we find there exist odd integers $\rho_v$ and even integers $\tilde{\rho}_v$, both independent of $n$ and can be determined mechanically, such that \[ \sum_{k=0}^{n-1}(2k+1)^{2v+1}D_k\equiv \rho_v n \pmod {n^3} \] and \[ \sum_{k=0}^{n-1}(-1)^k(2k+1)^{2v+1}D_k\equiv \tilde{\rho}_v n^2 \pmod {n^3}. \] The case $v=1$ in the last congruence confirms a conjecture of Guo and Zeng in 2012.