Multiparameter Fuss--Catalan numbers with application to algebraic equations

TL;DR

This paper consolidates and clarifies the theory of multiparameter Fuss--Catalan numbers, and applies this formalism to derive a new criterion for the convergence of series solutions to algebraic equations.

Contribution

It provides a unified presentation of multiparameter Fuss--Catalan numbers and introduces a new necessary and sufficient condition for the convergence domain of algebraic series solutions.

Findings

Unified notation and identities for multiparameter Fuss--Catalan numbers

New formula for the convergence domain of algebraic series solutions

Extension and correction of previous convergence results

Abstract

We present an exposition on the Fuss--Catalan numbers, which are a generalization of the well known Catalan numbers. The literature on the subject is scattered (especially for the case of multiple independent parameters, as will be explained in the text), with overlapping definitions by different authors and duplication of proofs. This paper collects the main theorems and identities, with a consistent notation. Contact is made with the works of numerous authors, including the early works of Lambert and Euler. We demonstrate the application of the formalism to solve algebraic equations by infinite series. Our main result in this context is a new necessary and sufficient formula for the domain of absolute convergence of the series solutions of algebraic equations, which corrects and extends previous work in the field. Some historical material is placed in an Appendix.