Generalizations and variants of Knuth's old sum

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.