用新的Bernoulli数恒等式证明幂和多项式的对称性
A Proof of Symmetry of the Power Sum Polynomials using a Novel Bernoulli Numbers Identity
摘要：
寻找自然数的幂和公式的问题已经引起了数学家们许多世纪的兴趣。其中Faulhaber的著名公式表示的权力总和多项式的系数涉及伯努利数。本文给出了一个初等证明：第一个$n$自然数的$p$次幂和可表示为$n$中的一个$p+1$次多项式.我们还证明了一个新的身份涉及伯努利数，并用它来显示这个多项式的对称性。
英文摘要：
The problem of finding formulas for sums of powers of natural numbers has been of interest to mathematicians for many centuries. Among these is Faulhaber's well-known formula expressing the power sums as polynomials whose coefficients involve Bernoulli numbers. In this paper we give an elementary proof that the sum of $p$-th powers of the first $n$ natural numbers can be expressed as a polynomial in $n$ of degree $p+1$. We also prove a novel identity involving Bernoulli numbers and use it to show symmetry of this polynomial.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
