Remainder Pade Approximants for the Exponential Function

Abstract

Abstract. Following earlier research of ours, we propose a new method for obtaining the complete Padé table of the exponential function. It is based on an explicit construction of certain Padé approximants not for the usual power series for exp at 0 but for a formal power series related in a simple way to the remainder term of the power series for exp. This surprising and non trivial coincidence is proved more generally for type II simultaneous Padé approximants for a family (exp(a j z)) j=1,...,r with distinct complex a's and we recover Hermite's classical formulae. The proof uses certain discrete multiple orthogonal polynomials recently introduced by Arvesú, Coussement and van Assche, which generalise the classical Charlier orthogonal polynomials.

On the q-Charlier Multiple Orthogonal Polynomials

“…Furthermore, in [14] the authors pointed out a possible relationship of these polynomials to the orthogonal functions appearing in two speed totally asymmetric simple exclusion process (TASEP) [6]. Our new q-family of multiple orthogonal polynomials is likely to be of relevance in TASEP-like models as well as in q-extensions of the mathematical problems addressed in [13,18].…”

Asymptotics for the ratio and the zeros of multiple Charlier polynomials

“…Multiple Charlier polynomials satisfy a number of (higher order) difference equations (Lee [13] and Van Assche [21]). They appear in remainder Padé approximation for the exponential function [19], as common eigenstates of a set of r non-Hermitian oscillator Hamiltonians [15], and we believe that they are related to the orthogonal functions appearing in two speed TASEP (totally asymmetric simple exclusion process) [3].…”

A favard type theorem for Hurwitz polynomials

“…Such functions are important in the study of the stability of linear delay differential equations. For more details on stable entire functions and their approximation, see [17], [22], [26], [2] and references therein.…”