An integral representation of Catalan numbers using the F\'eaux formula

TL;DR

This paper derives an integral formula for Catalan numbers using Féaux's integral representation of the logarithm of the Gamma function, potentially enabling new relations with binomial coefficients.

Contribution

It introduces a novel integral representation of Catalan numbers based on advanced Gamma function techniques, expanding analytical tools for combinatorial sequences.

Findings

Provides a new integral formula for Catalan numbers.

Lays groundwork for deriving new relations involving binomial coefficients.

Potential applications in combinatorics and special functions.

Abstract

We present an integral expression of the Catalan numbers, based on F\'eaux' integral representation of $\log\left[\Gamma(x)\right]$, $\Gamma$ 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.