Euler polynomials and alternating sums of powers of integers

Abstract

In this note, we consider the alternating sum of the mth powers of the first n positive integers . In 1989, Gessel and Viennot showed that, for even m = 2k, can be expressed as a polynomial of degree k in the triangular number without constant term. Here, we offer an alternative demonstration of this result that can be made suitable for first-year undergraduate students by using some basic properties of the Euler numbers and polynomials. We also give the corresponding closed formula for in the case of odd powers m = 2k + 1. In addition, we express both and as polynomials in n and give their coefficients in terms of the Bernoulli numbers. Finally, we give yet another representation for and involving the Stirling numbers of the second kind.