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On Series Involving Cubed Catalan Numbers

Kunle Adegoke

Abstract

Using generalized binomial coefficient identities and some results of John Dougall, we derive some families of series involving the cubes of Catalan numbers. We also establish a family of series containing fourth powers of Catalan numbers. Finally, we find a generalization of the Bauer series for $1/π$ and obtain some Ramanujan-like series for $1/π^2$ and~$1/π^3$.

On Series Involving Cubed Catalan Numbers

Abstract

Using generalized binomial coefficient identities and some results of John Dougall, we derive some families of series involving the cubes of Catalan numbers. We also establish a family of series containing fourth powers of Catalan numbers. Finally, we find a generalization of the Bauer series for and obtain some Ramanujan-like series for and~.

Paper Structure

This paper contains 6 sections, 20 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Required results
  3. Series involving cubed Catalan numbers
  4. Series involving cubed Catalan numbers and odd harmonic numbers
  5. Series involving fourth powers of Catalan numbers
  6. Ramanujan-like series and related series

Key Result

Lemma 1

If $k$ and $r$ are non-negative integers, then $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (49)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 39 more