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Integral Representations for Multiple Apéry-Like Series

Jorge Antonio González Layja

Abstract

By elementary methods, we obtain integral representations for several families of multiple Apéry-like series. These involve multiple harmonic sums in addition to the characteristic central binomial factors. The derivations are based on systematic applications of integration by parts together with two key Fourier expansions. As applications, we recover known evaluations and derive new identities.

Integral Representations for Multiple Apéry-Like Series

Abstract

By elementary methods, we obtain integral representations for several families of multiple Apéry-like series. These involve multiple harmonic sums in addition to the characteristic central binomial factors. The derivations are based on systematic applications of integration by parts together with two key Fourier expansions. As applications, we recover known evaluations and derive new identities.
Paper Structure (3 sections, 11 theorems, 58 equations)

This paper contains 3 sections, 11 theorems, 58 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Lemmas
  3. Main Results

Key Result

Lemma 2.1

The following identities hold: where $O_n^{(m)}\coloneqq\sum _{k=1}^n\frac{1}{(2k-1)^m}$ and $\overline{O}_n^{(m)}\coloneqq\sum _{k=1}^n\frac{(-1)^{k-1}}{(2k-1)^m}$ denote the odd harmonic numbers of order $m$ and their alternating analogues, respectively.

Theorems & Definitions (22)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • proof
  • ...and 12 more