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arXiv · 2408.01201

Integral representations of Catalan numbers using Touchard-like identities

Jean-Christophe Pain
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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.

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Jean-Christophe Pain. 2024-08-02. Integral representations of Catalan numbers using Touchard-like identities. https://arxiv.org/abs/2408.01201

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