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General Mneimneh-type Binomial Sum involving Harmonic Numbers

Ende Pan, Ce Xu

Abstract

Recently, Mneimneh proved the remarkable identity \begin{align*} \sum_{k=0}^n H_k\binom{n}{k} p^k(1-p)^{n-k}=\sum_{i=1}^n \frac{1-(1-p)^i}{i}\quad (p\in [0,1]) \end{align*} as the main result of a 2023 \emph{Discrete Mathematics} paper, where $H_k:=\sum\nolimits_{i=1}^k 1/i$ is the classical $k$-th harmonic number. Thereafter, Campbell provided several other proofs of Mneimneh's formula as above in a note published in \emph{Discrete Mathematics} in 2023. Moreover, Campbell also considered how Mneimneh's identity may be proved and generalized using the \emph{Mathematica package Sigma}. In particular, he found the generalized Mneimneh's identity \begin{align*} \sum_{k=0}^n x^k y^{n-k} \binom{n}{k}H_k =(x+y)^n \left(H_n-\sum_{i=1}^n \frac{y^i (x+y)^{-i}}{i}\right). \end{align*} In this paper, we will prove a more generalization of Mneimneh's identity involving Bell numbers and some Mneimneh-type identities involving (alternating) harmonic numbers by using a few results of our previous papers.

General Mneimneh-type Binomial Sum involving Harmonic Numbers

Abstract

Recently, Mneimneh proved the remarkable identity \begin{align*} \sum_{k=0}^n H_k\binom{n}{k} p^k(1-p)^{n-k}=\sum_{i=1}^n \frac{1-(1-p)^i}{i}\quad (p\in [0,1]) \end{align*} as the main result of a 2023 \emph{Discrete Mathematics} paper, where is the classical -th harmonic number. Thereafter, Campbell provided several other proofs of Mneimneh's formula as above in a note published in \emph{Discrete Mathematics} in 2023. Moreover, Campbell also considered how Mneimneh's identity may be proved and generalized using the \emph{Mathematica package Sigma}. In particular, he found the generalized Mneimneh's identity \begin{align*} \sum_{k=0}^n x^k y^{n-k} \binom{n}{k}H_k =(x+y)^n \left(H_n-\sum_{i=1}^n \frac{y^i (x+y)^{-i}}{i}\right). \end{align*} In this paper, we will prove a more generalization of Mneimneh's identity involving Bell numbers and some Mneimneh-type identities involving (alternating) harmonic numbers by using a few results of our previous papers.
Paper Structure (8 sections, 7 theorems, 52 equations)

This paper contains 8 sections, 7 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and lemmas
  3. (unsigned) Stirling number of the first kind
  4. Bell polynomials
  5. Some Lemmas
  6. Proof of Theorem \ref{['thm:Mneimneh-type']}
  7. Proofs of Theorem \ref{['thm:Mneimneh-type-G']} and Corollary \ref{['cor:Mneimneh-type-order2']}
  8. Conclusion

Key Result

Theorem 1.1

For any reals $x,y$ with $x/(x+y)\geq 0$ and $n,p\in \mathbb{N}$, we have where $s(n,k)$ and $Y_k(n)$ stand for the (unsigned) Stirling numbers of the first kind and Bell numbers (see Section Intr-STT-Bell), respectively.

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Conjecture 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3