Hermite-Padé approximants to the Weyl function and its derivative for discrete measures

Abstract

Hermite-Padé approximants of the second kind to the Weyl function and its derivatives are investigated. The Weyl function is constructed from the orthogonal Meixner polynomials. The limiting distribution of the zeros of the common denominators of these approximants, which are multiple orthogonal polynomials for a discrete measure, is found. It is proved that the limit measure is the unique solution of the equilibrium problem in the theory of the logarithmic potential with an Angelesco matrix. The effect of pushing some zeros off the real axis to some curve in the complex plane is discovered. An explicit form of the limit measure in terms of algebraic functions is given. Bibliography: 10 titles.