Hermite-padé approximation and location of singularities of systems of analytic functions

Abstract

We consider row sequences of (type II) Hermite-Padé approximations with common denominator associated with a vector f of formal power expansions about the origin. In terms of the asymptotic behavior of the sequence of common denominators, we describe some analytic properties of f and restate some conjectures corresponding to questions once posed by A. A. Gonchar for row sequences of Padé approximants. We obtain extensions of the Poincaré and Perron theorems for higher order recurrence relations and apply them to obtain an inverse type theorem for row sequences of (type II) Hermite-Padé approximation of a vector of formal power series. We also give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of multipoint Hermite-Padé approximants.