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.
Full version: pdf,
(Concerned with sequences
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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