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In this paper we present a generalization of Faulhaber's formula to sums ofarbitrary complex powers $m\in\mathbb{C}$. These summation formulas for sums ofthe 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 accelerationinvolving Stirling numbers of the first kind. While it is well-known that thecorresponding expressions obtained from the Euler-Maclaurin summation formuladiverge, our summation formulas are all very rapidly convergent.

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