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arXiv · 2205.05905

Generalizations and variants of Knuth's old sum

Arjun K. Rathie·John M. Campbell
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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.

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Arjun K. Rathie, John M. Campbell. 2022-05-12. Generalizations and variants of Knuth's old sum. https://arxiv.org/abs/2205.05905

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