The asymptotic behaviour of algebraic approximants

Abstract

We study the convergence of algebraic approximants to a function represented by a power series. We consider, for an arbitrary but fixed degree, approximant sequences which can be generated recursively by use of Sergeyev's algorithm. For the exponential function, a logarithmic function and a power of a binomial, we find explicit formulae for the coefficients that appear in a resulting linear recurrence relation. We assume that the error equation may be linearized for small errors. Analysis then yields the generic dominant term in the asymptotic behaviour of the error when a large number of terms of the series are used. Extensive numerical results confirm the behaviour. Finally, we compare this behaviour with that for the closely related method of Drazin & Tourigny, in which the degree of the approximants grows without bound.