An integral representation of Catalan numbers using Malmst\'en's formula

TL;DR

This paper presents a new integral formula for Catalan numbers derived from Malmstén's integral representation of the logarithm of the Gamma function, potentially leading to new summation identities.

Contribution

It introduces an integral representation of Catalan numbers based on Malmstén's formula, offering a novel analytical approach.

Findings

New integral expression for Catalan numbers

Potential for deriving new summations involving Catalan numbers

Connection to Malmstén's integral representation of the Gamma function

Abstract

In this article, we propose an integral expression of the Catalan numbers, based on Malmst\'en's definite-integral representation of $\ln\left[\Gamma(x)\right]$, $\Gamma$ being the usual Gamma function. The obtained expression is likely to yield new summations involving Catalan numbers or central binomial coefficients.