Asymptotics for Generalized Riordan Arrays

Abstract

A Riordan array is an infinite complex matrix (ars) of a certain type (see below for exact definitions). The Riordan array formalism has been much used recently to study combinatorial questions in analysis of algorithms and other areas. Most work has been concerned with “exact” results. In this article we discuss asymptotics of such arrays. We apply general machinery for deriving asymptotics of bivariate generating functions, following the research programme begun in [PW02, PW04]. Asymptotic expansions of special cases of Riordan arrays have been discussed by several authors [Drm94, Gar95]. The main purposes of this article are to show how the work in [PW02] immediately yields strong results for (a generalization of) Riordan arrays, and to use this case as an introduction to the much more general results in [PW02, PW04], the computations being simpler to understand. In addition we try to simplify and automate the process of extracting asymptotics as far as possible.