Generalized Mneimneh sums and their application to multiple polylogarithms

Abstract

The purpose of this paper is the study of the binomial sum$\sum _{k=1}^n\left(\begin{array}{c}n\\ k\end{array}\right)\cdot H_k^{\left(s\right)}\left(a\right)\cdot p^k\cdot {(1-p)}^{n-k},$ where $H_k^{\left(s\right)}\left(a\right)={\sum }_{i=1}^k{a}^i/i^s$ denotes the parameterized analogue of the k-th harmonic number of order s. For $a=s=1$, these binomial sums were investigated by Mneimneh, who gave a probabilistic interpretation related to hiring problems. We present a generalization of Mneimneh's summation formula and establish several new identities and a connection of these sums with specific multiple polylogarithms, called unit Euler sums, based upon the Toeplitz limit theorem.