Generalized Harmonic Numbers: Identities and Properties

TL;DR

This paper introduces generalized harmonic numbers, derives new identities involving them, and extends classical identities, enhancing understanding of their properties through elementary techniques.

Contribution

It defines generalized harmonic numbers and generalizes Gould's identity, providing new identities and connections with Fibonacci numbers and Laguerre polynomials.

Findings

Derived new identities for generalized harmonic numbers.

Extended Gould's identity to include generalized harmonic numbers.

Connected generalized harmonic numbers with Fibonacci numbers and Laguerre polynomials.

Abstract

This paper builds on the research initiated by Boyadzhiev, but introduces generalized harmonic numbers, \[ H_n(\alpha)= \sum_{k=1}^n \frac{\alpha^{k}}{k}, \] which enable the derivation of new identities as well as the reformulation of existing ones. We also generalize Gould's identity, allowing classical harmonic numbers to be replaced by their generalized counterparts. Our results contribute to a deeper understanding of the structural properties of these numbers and highlight the effectiveness of elementary techniques in uncovering new mathematical phenomena. In particular, we recover several known identities for generalized harmonic numbers and establish new ones, including identities involving generalized harmonic numbers together with Fibonacci numbers, Laguerre polynomials, and related sequences.