An analytic generalization of the Catalan numbers and its integral representation

TL;DR

This paper generalizes Catalan numbers analytically, derives an integral representation using complex analysis, and suggests future research directions in combinatorial number theory.

Contribution

It introduces an analytic generalization of Catalan numbers and provides an integral representation using Cauchy's integral formula.

Findings

Established an integral representation of the generalized Catalan numbers

Connected combinatorial number theory with complex analysis techniques

Outlined potential avenues for further research

Abstract

In the paper, the authors analytically generalize the Catalan numbers in combinatorial number theory, establish an integral representation of the analytic generalization of the Catalan numbers by virtue of Cauchy's integral formula in the theory of complex functions, and point out potential directions to further study.