New Finite and Infinite Summation Identities Involving the Generalized Harmonic Numbers

TL;DR

This paper introduces new summation identities involving generalized harmonic numbers, deriving novel finite and infinite sum formulas, including non-linear and alternating sums, expanding the mathematical understanding of these special numbers.

Contribution

It presents a general summation identity and applies it to derive several new formulas involving generalized harmonic numbers, including non-linear and alternating sums.

Findings

New infinite summation formulas involving generalized harmonic numbers

Discovery of non-linear summation identities

Derivation of alternating sum formulas

Abstract

We state and prove a general summation identity. The identity is then applied to derive various summation formulas involving the generalized harmonic numbers and related quantities. Interesting results, mostly new, are obtained for both finite and infinite sums. The high points of this paper are perhaps the discovery of several previously unknown infinite summation results involving {\em non-linear} generalized harmonic number terms and the derivation of interesting alternating summation formulas involving these numbers