Proof of a conjecture involving Sun polynomials

TL;DR

This paper proves a conjecture by Z.-W. Sun regarding the integrality of a sum involving Sun polynomials and provides multiple proofs, including a key combinatorial congruence, advancing understanding of Sun polynomial properties.

Contribution

It confirms a recent conjecture on Sun polynomials and offers three different proofs, including a novel combinatorial congruence, addressing open questions in the field.

Findings

Confirmed the integrality conjecture for Sun polynomials sums.

Established a new combinatorial congruence involving binomial coefficients.

Provided multiple proofs, enriching the theoretical framework.

Abstract

The Sun polynomials $g_n(x)$ are defined by \begin{align*} g_n(x)=\sum_{k=0}^n{n\choose k}^2{2k\choose k}x^k. \end{align*} We prove that, for any positive integer $n$, there hold \begin{align*} &\frac{1}{n}\sum_{k=0}^{n-1}(4k+3)g_k(x) \in\mathbb{Z}[x],\quad\text{and}\\ &\sum_{k=0}^{n-1}(8k^2+12k+5)g_k(-1)\equiv 0\pmod{n}. \end{align*} The first one confirms a recent conjecture of Z.-W. Sun, while the second one partially answers another conjecture of Z.-W. Sun. We give three different proofs of the former. One of them depends on the following congruence: $$ {m+n-2\choose m-1}{n\choose m}{2n\choose n}\equiv 0\pmod{m+n}\quad\text{for $m,n\geqslant 1$.} $$