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A Proof of Symmetry of the Power Sum Polynomials Using a Novel Bernoulli Number Identity
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Nicholas J. Newsome, Maria S. Nogin, and Adnan H. Sabuwala

Department of Mathematics

California State University, Fresno

Fresno, CA 93740

USA

**Abstract:**

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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