Integral representations of Catalan numbers using Touchard-like identities

TL;DR

This paper derives new integral formulas for Catalan numbers by applying Touchard identities and binomial theorem techniques, enabling potential generalizations for other combinatorial sequences.

Contribution

It introduces a novel method combining Touchard identities with integral representations to generate new formulas for Catalan numbers.

Findings

New integral representations for Catalan numbers.

Method can be generalized to other sequences with suitable forms.

Demonstrates the utility of combining identities with integral techniques.

Abstract

In this article, we use the Touchard identity in order to obtain new integral representations for Catalan numbers. The main idea consists in combining the identity with a known integral representation and resorting to the binomial theorem. The same procedure is applied to a variant of the Touchard identity proposed by Callan a few years ago. The method presented here can be generalized to derive additional integral representations from known ones, provided that the latter have a well-suited form and lend themselves to an analytical summation under the integral sign.