Asymptotics for generating functions of the Fuss-Catalan numbers

Abstract

We consider a certain class of polynomials with coefficients in Z_M, all of which admit a unique zero. We prove that the zero of each of those can be given by a (multiple) sum involving the coefficients and a vectorial generalization of the Fuss-Catalan numbers. We also consider the sequence of the partial sums of the generating function of the d -Fuss- Catalan numbers. Using the holonomy of this sequence, we study its asymptotic behaviour. The main difference from the known case d = 2 is, in that one, we have a “closed” expression for the generating function.