Multiparameter Fuss--Catalan numbers with application to algebraic equations

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.