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28 March 2024
 
  » arxiv » 1511.4005

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Proof of a conjecture involving Sun polynomials
Victor J. W. Guo ; Guo-Shuai Mao ; Hao Pan ;
Date 27 Oct 2015
AbstractThe Sun polynomials $g_n(x)$ are defined by egin{align*} g_n(x)=sum_{k=0}^n{nchoose k}^2{2kchoose k}x^k. end{align*} We prove that, for any positive integer $n$, there hold egin{align*} &frac{1}{n}sum_{k=0}^{n-1}(4k+3)g_k(x) inmathbb{Z}[x],quad ext{and}\ &sum_{k=0}^{n-1}(8k^2+12k+5)g_k(-1)equiv 0pmod{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. Our proof depends on the following congruence: $$ {m+n-2choose m-1}{nchoose m}{2nchoose n}equiv 0pmod{m+n}quad ext{for $m,ngeqslant 1$.} $$
Source arXiv, 1511.4005
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