Generalizations and variants of Knuth's old sum

TL;DR

This paper generalizes Knuth's old sum using hypergeometric series and complex parameters, introduces new binomial-harmonic identities, and applies Fourier-Legendre theory and Abel's lemma to derive related identities involving harmonic numbers.

Contribution

It presents novel hypergeometric series proofs and extends Knuth's sum with complex parameters, enriching the theoretical framework of harmonic sum identities.

Findings

Generalized Knuth's sum with complex parameters

Derived new binomial-harmonic sum identities

Connected harmonic numbers with Fourier-Legendre theory

Abstract

We extend the Reed Dawson identity for Knuth's old sum with a complex parameter, and we offer two separate hypergeometric series-based proofs of this generalization, and we apply this generalization to introduce binomial-harmonic sum identities. We also provide another ${}_{2}F_{1}(2)$-generalization of the Reed Dawson identity involving a free parameter. We then apply Fourier-Legendre theory to obtain an identity involving odd harmonic numbers that resembles the formula for Knuth's old sum, and the modified Abel lemma on summation by parts is also applied.