Faulhaber's Theorem on Power Sums
William Y.C. Chen, Amy M. Fu, and Iris F. Zhang
Abstract: We observe that the classical Faulhaber's theorem on sums of odd powers also holds for an arbitrary arithmetic progression, namely, the odd power sums of any arithmetic progression a+b, a+2b, ..., a+nb is a polynomial in na+ n(n + 1)b/2. While this assertion can be deduced from the original Fauhalber's theorem, we give an alternative formula in terms of the Bernoulli polynomials. Moreover, by utilizing the central factorial numbers as in the approach of Knuth, we derive formulas for r-fold sums of powers without resorting to the notion of r-reflexive functions. We also provide formulas for the r-fold alternating sums of powers in terms of Euler polynomials.
AMS Classification: 05A10; 11B68
Keywords: Faulhaber's theorem, power sum, alternating sum, r-fold power sum, r-fold alternating power sum, Bernoulli polynomial, Euler polynomial