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Guo","submitted_at":"2015-10-27T01:28:23Z","abstract_excerpt":"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. 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W. Guo","submitted_at":"2015-10-27T01:28:23Z","abstract_excerpt":"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. 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