The Generalization of Faulhaber's Formula to Sums of Arbitrary Complex Powers

Abstract

In this paper we present a generalization of Faulhaber's formula to sums of arbitrary complex powers m ∈ C. These summation formulas for sums of the form ⌊x⌋ k=1 k m and n k=1 k m , where x ∈ R + and n ∈ N, are based on a series acceleration involving Stirling numbers of the first kind. While it is well-known that the corresponding expressions obtained from the Euler-Maclaurin summation formula diverge, our summation formulas are all very rapidly convergent.