Number Theory


Generalized Harmonic Numbers Revisited

Authors: Jose R. Sousa

This paper presents new formulae for the harmonic numbers of order $k$, $H_{k}(n)$, and for the partial sums of two Fourier series associated with them, denoted here by $C^m_{k}(n)$ and $S^m_{k}(n)$. I believe this new formula for $H_{k}(n)$ is an improvement over the digamma function, $\psi$, because it's simpler and it stems from Faulhaber's formula, which provides a closed-form for the sum of powers of the first $n$ positive integers. We demonstrate how to create an exact power series for the harmonic numbers, a new integral representation for $\zeta(2k+1)$ and a new generating function for $\zeta(2k+1)$, among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature.

Comments: 28 Pages. This is a cleaned up and improved version, the original is on arXiv

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Submission history

[v1] 2019-12-27 22:12:45

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