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Proof of a conjecture involving Sun polynomials | Victor J. W. Guo
; Guo-Shuai Mao
; Hao Pan
; | Date: |
27 Oct 2015 | Abstract: | The 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 | Services: | Forum | Review | PDF | Favorites |
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