Asymptotic expansions for Taylor coefficients of the composition of two functions

Article type: Research Article

Authors: Hilberdink, Titus

Affiliations: Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX, England, UK. E-mail: [email protected]

Abstract: We give an asymptotic expansion for the Taylor coefficients of L(P(z)) where L(z) is analytic in the open unit disc whose Taylor coefficients vary ‘smoothly’ and P(z) is a probability generating function. We show how this result applies to a variety of problems, amongst them obtaining the asymptotics of Bernoulli transforms and weighted renewal sequences.

Keywords: asymptotic expansions, Taylor coefficients, weighted renewal sequences

DOI: 10.3233/ASY-2008-0926

Journal: Asymptotic Analysis, vol. 63, no. 3, pp. 125-142, 2009