On Padé Approximants of Meromorphic Functions of Markov Type

The paper is devoted to the asymptotic properties of diagonal Padé approximants for Markov-type meromorphic functions. The main result is strong asymptotic formulas for the denominators of diagonal Padé approximants for Markov-type meromorphic functions $f=\widehat\sigma+r$ under additional constraints on the measure $\sigma$ ($r$ is a rational function). On the basis of these formulas, it is proved that, in a sufficiently small neighborhood of a pole of multiplicity $m$ of such a meromorphic function $f$, all poles of the diagonal Padé approximants $f_n$ are simple and asymptotically located at the vertices of a regular $m$-gon.