On Mneimneh's binomial sum involving harmonic numbers

Abstract

Quite recently, Mneimneh introduced the remarkable result whereby∑k=0nHk(nk)pk(1−p)n−k=∑i=1n1−(1−p)ii for 0≤p≤1 as the main result of a 2023 Discrete Mathematics article, where Hk=1+12+⋯+1k denotes the kth harmonic number. Mneimneh's formula, as above, generalizes a result due to Paule and Schneider and to Spivey, and naturally arises as an expected value of a sum involving permutations chosen uniformly at random. In contrast to Mneimneh's probabilistic methods employed to prove the above formula, we generalize the above result so as to not require the given bounds on p, according to our two much simplified proofs of this generalized version of Mneimneh's formula introduced in this article. Our new and simplified proofs demonstrate how our previously introduced methods concerning harmonic sums may be applied systematically to prove and generalize identities as above. Our first proof is based on a way of extending Zeilberger's algorithm so as to be applicable to non-hypergeometric summands; our second proof is based on a beta-type integral formula. We also consider how Mneimneh's identity may be proved and generalized using the Mathematica package Sigma.