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An integral representation of Catalan numbers using the Féaux formula

Jean-Christophe Pain

Abstract

We present an integral expression of the Catalan numbers, based on Féaux' integral representation of $\log\left[Γ(x)\right]$, $Γ$ being the usual Gamma function. The obtained formula may be the starting point of the derivation of new relations involving central binomial coefficients or Catalan numbers.

An integral representation of Catalan numbers using the Féaux formula

Abstract

We present an integral expression of the Catalan numbers, based on Féaux' integral representation of , being the usual Gamma function. The obtained formula may be the starting point of the derivation of new relations involving central binomial coefficients or Catalan numbers.

Paper Structure

This paper contains 4 sections, 1 theorem, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Integral representation deduced from the Féaux formula
  3. Tracks for deriving further new integral representations
  4. Conclusion

Key Result

Theorem 1

Let $n$ be a positive integer and $C_n$ the $n^{th}$ Catalan number. Then the integral representation holds true.

Figures (1)

  • Figure 1: Two monotonic lattice paths in the case $n=4$. They can be represented by listing the Catalan elements by column height: [0,1,1,3] (left) and [0,0,2,2] (right). There are 14 diagrams of the kinds of these two ones, which is precisely the fourth Catalan number $C_4$.

Theorems & Definitions (2)

  • Theorem 1
  • proof