A new approach to Catalan numbers using differential equations

TL;DR

This paper introduces differential equations linked to Catalan numbers' generating functions, leading to new identities and insights into their combinatorial and arithmetic properties.

Contribution

It presents a novel differential equation framework for Catalan numbers and derives new explicit identities and combinatorial formulas.

Findings

New differential equations for Catalan numbers

Explicit identities for Catalan and higher-order Catalan numbers

Additional combinatorial and arithmetic identities

Abstract

In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are `inverses' to each other in some sense. From these differential equations, we obtain some new and explicit identities for Catalan and higher-order Catalan numbers. In addition, by other means than differential equations we also derive some interesting identities involving Catalan numbers which are of arithmetic and combinatorial nature.