Faulhaber’s theorem on power sums

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+nba+b,a+2b,…,a+nb is a polynomial in na+n(n+1)b/2na+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 rr-fold sums of powers without resorting to the notion of rr-reflective functions. We also provide formulas for the rr-fold alternating sums of powers in terms of Euler polynomials.