Generalized Harmonic Numbers

TL;DR

This paper introduces new, simpler formulae for harmonic numbers of order k, along with novel series, integral representations, and generating functions for related zeta values, offering fresh approaches distinct from existing literature.

Contribution

It provides new formulae and methods for harmonic numbers and zeta function values, improving upon previous techniques by deriving from Faulhaber's formula and offering exact series and representations.

Findings

New formulae for harmonic numbers of order k

A novel integral representation for ζ(2k+1)

A new generating function for ζ(2k+1)

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 $C^m_{k}(n)$ 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.