A new approach to Catalan numbers using differential equations

In this paper, we introduce two differential equations arising from the generating function of the Catalan numbers which are ‘inverses’ to each other in a certain 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.