Function series, Catalan numbers and random walks on trees

Abstract

The delight of finding unexpected connections is one of the rewards of studying mathematics. In this paper, we link several superficially unrelated entities. We use the generalized Catalan numbers of parameter k as the coefficients of a power series in a certain polynomial of degree k. These series converge and yield continuous–but not differentiable–functions on intervals. These functions turn out to determine the transience of various random walks on trees. They also satisfy functional equations that, by means of the Lagrange-Bürmann inversion formula, lead back to the Catalan numbers.