Journal of Integer Sequences, Vol. 25 (2022), Article 22.3.7

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

Raphael Schumacher
Department of Mathematics
ETH Zurich
Rämistrasse 101
8092 Zurich


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.

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(Concerned with sequences A000142 A008275 A027641 A027642.)

Received March 11 2021; revised versions received March 14 2021; December 14 2021; March 6 2022; March 8 2022. Published in Journal of Integer Sequences, March 25 2022.

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