Some congruences involving powers of Delannoy polynomials

TL;DR

This paper proves new congruences involving powers of Delannoy polynomials, confirming conjectures by Z.-W. Sun and establishing relationships between sums of these polynomials and Legendre symbols.

Contribution

It establishes novel supercongruences for sums of powers of Delannoy polynomials, confirming two of Sun's conjectures and proposing a conjecture for higher modulus.

Findings

Confirmed conjectures of Sun on supercongruences involving Delannoy polynomials.

Derived explicit congruences modulo p^2 for sums involving D_k(x)^3 and D_k(x)^4.

Proposed a conjecture extending a congruence to modulo p^3.

Abstract

The Delannoy polynomial $D_n(x)$ is defined by $$ D_n(x)=\sum_{k=0}^{n}{n\choose k}{n+k\choose k}x^k. $$ We prove that, if $x$ is an integer and $p$ is a prime not dividing $x(x+1)$, then \begin{align*} \sum_{k=0}^{p-1}(2k+1)D_k(x)^3 &\equiv p\left(\frac{-4x-3}{p}\right) \pmod{p^2}, \\ \sum_{k=0}^{p-1}(2k+1)D_k(x)^4 &\equiv p \pmod{p^2}, \\ \sum_{k=0}^{p-1}(-1)^k(2k+1)D_k(x)^3 &\equiv p\left(\frac{4x+1}{p}\right) \pmod{p^2}, \end{align*} where $\big(\frac{\cdot}{p}\big)$ denotes the Legendre symbol. The first two congruences confirm a conjecture of Z.-W. Sun [Sci. China 57 (2014), 1375--1400]. The third congruence confirms a special case of another conjecture of Z.-W. Sun [J. Number Theory 132 (2012), 2673--2699]. We also prove that, for any integer $x$ and odd prime $p$, there holds \begin{align*} \sum_{k=0}^{p-1}(-1)^k(2k+1)D_k(x)^4 &\equiv p\sum_{k=0}^{\frac{p-1}{2}} (-1)^k {2k\choose k}^2(x^2+x)^k(2x+1)^{2k} \pmod{p^2}, \end{align*} and conjecture that it still holds modulo $p^3$.