Generalized Harmonic Numbers

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, $\psi$, 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.