Proof of a conjecture involving Sun polynomials
arXiv:1511.04005
Abstract
The Sun polynomials 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 , 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:
15 pages