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 $m\in \mathbb{C}$. These summation formulas for sums of the form ${\sum }_{k=1}^{\lfloor x\rfloor }{k}^m$ and ${\sum }_{k=1}^n{k}^m$, where $x\in {\mathbb{R}}^{+}$ and $n\in \mathbb{N}$, 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.

References in corpus (4)

• Error bounds for the asymptotic expansion of the Hurwitz zeta function
• Rapidly Convergent Summation Formulas involving Stirling Series
• Quantum surface holonomies for loop quantum gravity and their application to black hole horizons
• Extension of Summation Formulas involving Stirling series