The Faulhaber Formula Analytic Continuation

TL;DR

This paper extends classical formulas like Faulhaber's and Hurwitz zeta to the entire complex plane, providing exact expressions and unifying various special functions through analytic continuation.

Contribution

It introduces exact analytic continuations of Faulhaber's formula, generalized harmonic progressions, and Hurwitz zeta function across the whole complex plane, advancing theoretical understanding.

Findings

Exact expression for Faulhaber's formula in the complex plane

Generalized harmonic progressions valid everywhere in the complex plane

Extended Hurwitz zeta function formula to the entire complex plane

Abstract

We extend the Faulhaber formula to the whole complex plane, obtaining an expression that fully resembles the Euler-Maclaurin summation formula, only it's exact. Thereafter, an expression for the generalized harmonic progressions valid in the whole complex plane is also derived. Lastly, we extend a formula for the Hurwitz zeta function valid at the negative integers, $\zeta(-k,b)$, to the whole complex plane, following a similar procedure.