Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.6

A Proof of Symmetry of the Power Sum Polynomials Using a Novel Bernoulli Number Identity

Nicholas J. Newsome, Maria S. Nogin, and Adnan H. Sabuwala
Department of Mathematics
California State University, Fresno
Fresno, CA 93740


The problem of finding formulas for sums of powers of natural numbers has been of interest to mathematicians for many centuries. Among these is Faulhabers 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 the symmetry of this polynomial.

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(Concerned with sequences A027641 A027642.)

Received February 15 2017; revised versions received February 22 2017; March 1 2017; June 2 2017. Published in Journal of Integer Sequences, June 25 2017.

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