Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Combinatorial sums, series and integrals involving odd harmonic numbers

Kunle Adegoke, Robert Frontczak, Taras Goy

Abstract

We present several types of ordinary generating functions involving central binomial coefficients, harmonic numbers, and odd harmonic numbers. Our results complement those of Boyadzhiev from 2012 and Chen from 2016. Based on these generating functions we evaluate several infinite series in closed form. In addition, we offer some combinatorial sum identities involving Catalan numbers, harmonic numbers and odd harmonic numbers. Finally, we analyze a special log-integral with Fibonacci numbers and odd harmonic numbers.

Combinatorial sums, series and integrals involving odd harmonic numbers

Abstract

We present several types of ordinary generating functions involving central binomial coefficients, harmonic numbers, and odd harmonic numbers. Our results complement those of Boyadzhiev from 2012 and Chen from 2016. Based on these generating functions we evaluate several infinite series in closed form. In addition, we offer some combinatorial sum identities involving Catalan numbers, harmonic numbers and odd harmonic numbers. Finally, we analyze a special log-integral with Fibonacci numbers and odd harmonic numbers.
Paper Structure (7 sections, 31 theorems, 114 equations)

This paper contains 7 sections, 31 theorems, 114 equations.

Table of Contents

  1. Introduction and motivation
  2. An integral and some consequences
  3. Two additional combinatorial identities derived from the integral
  4. Some Fibonacci series identities
  5. Several other generating functions and series
  6. A family of related integrals
  7. Concluding comments

Key Result

Lemma 1

For all real $a>0$ and integers $n\geq 0$,

Theorems & Definitions (58)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Corollary 2
  • proof
  • Remark 1
  • Corollary 3
  • ...and 48 more