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The Faulhaber formula analytic continuation

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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.

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  • Risomar Sousa
Risomar Sousa, Jose Lerch's Φ and the Polylogarithm at the Positive Integers, eprint arXiv:2006.08406, 2020.
Jose The Lerch's Φ Analytic Continuation
  • Risomar Sousa
Risomar Sousa, Jose The Lerch's Φ Analytic Continuation, eprint arXiv:2101.06835, 2021.