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Mneimneh-type Binomial Sums of Multiple Harmonic-type Sums

Ende Pan, Ce Xu

Abstract

In this paper, we establish some expressions of Mneimneh-type binomial sums involving multiple harmonic-type sums in terms of finite sums of Stirling numbers, Bell numbers and some related variables. In particular, we present some new formulas of Mneimneh-type binomial sums involving generalized (alternating) harmonic numbers. Further, we establish a new identity relating the multiple zeta star values $ζ^\star(m+2,\{1\}_{r-1})$ and specific multiple polylogarithms by applying the Toeplitz principle. Furthermore, we present some interesting consequences and illustrative examples.

Mneimneh-type Binomial Sums of Multiple Harmonic-type Sums

Abstract

In this paper, we establish some expressions of Mneimneh-type binomial sums involving multiple harmonic-type sums in terms of finite sums of Stirling numbers, Bell numbers and some related variables. In particular, we present some new formulas of Mneimneh-type binomial sums involving generalized (alternating) harmonic numbers. Further, we establish a new identity relating the multiple zeta star values and specific multiple polylogarithms by applying the Toeplitz principle. Furthermore, we present some interesting consequences and illustrative examples.
Paper Structure (10 sections, 12 theorems, 64 equations)

This paper contains 10 sections, 12 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. (unsigned) Stirling number of the first kind
  4. Bell polynomials
  5. Proofs of Main Results
  6. Lemmas and Propositions
  7. Proof of Theorems \ref{['thm:Mneimneh-type-mhtss']}, \ref{['thm:Mneimneh-type-mhtss-2']} and \ref{['thm:Mneimneh-type-mhtss-3']}
  8. Corollaries and Examples
  9. Proof of Theorem \ref{['thm:Mneimneh-type-mhtss-4']}
  10. A Conjecture

Key Result

Theorem 1.1

For any reals $x,y$ and $z\in (-\infty,1]$ with $x/(x+y)\geq 0$ and $n,p\in \mathbb{N}$, we have

Theorems & Definitions (17)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • ...and 7 more