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

TL;DR

This paper extends Faulhaber's formula to complex powers, providing rapidly convergent summation formulas for sums of powers that are based on Stirling numbers, overcoming divergence issues of traditional methods.

Contribution

It introduces a generalized Faulhaber's formula for complex powers using Stirling numbers, resulting in rapidly convergent summation formulas.

Findings

Derived formulas for sums of complex powers

Formulas are rapidly convergent

Overcome divergence of Euler-Maclaurin-based expressions

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.