Generalized Harmonic Numbers

Abstract

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 $C_k^m(n)$ and $S_k^m(n)$. I believe this new formula for $H_k(n)$ is an improvement over the digamma function, $\ddot{I}ˆ$, 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 $\hat{I}\P (2k+1)$ and a new generating function for $\hat{I}\P (2k+1)$, among many other original results. The approaches and formulae discussed here are entirely different from solutions available in the literature.

35 pages. Set the grammar to academic standards and fixed cut-off formulas