Universal asymptotic properties of positive functional equations with one catalytic variable

TL;DR

This paper demonstrates that positive functional equations with one catalytic variable exhibit universal asymptotic singularity behaviors, with linear cases showing square root singularities and nonlinear cases showing 3/2 singularities.

Contribution

It establishes a universal asymptotic behavior for solutions of positive catalytic functional equations, distinguishing between linear and nonlinear cases.

Findings

Linear catalytic equations have square root singularities.

Nonlinear catalytic equations typically have 3/2 singularities.

The results apply under certain positivity assumptions.

Abstract

Functional equations with one catalytic appear in several combinatorial applications, most notably in the enumeration of lattice paths and in the enumeration of planar maps. The main purpose of this paper is to show that under certain positivity assumptions the dominant singularity of the solutions have a universal behavior. We have to distinguish between linear catalytic equations, where a dominating square root singularity appears, and non-linear catalytic equations, where we - usually - have a singularity of type 3/2.