Generalizations and variants of Knuth's old sum

2022/05/12 by Rathie, Arjun K., Campbell, John M.
#33C20 #Combinatorics (math.CO) #Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2205.05905

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 ₂F₁(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.

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