The paper is devoted to the asymptotic properties of diagonal Pad, approximants for Markov-type meromorphic functions. The main result is strong asymptotic formulas for the denominators of diagonal Pad, approximants for Markov-type meromorphic functions f + 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 Pad, approximants f (n) are simple and asymptotically located at the vertices of a regular m-gon.

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 = \(\hat \sigma \) + r under additional constraints on the measure σ (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.

Leighton's well-known conjecture about singular points of a meromorphic function defined by its expansion in a general -fraction is discussed. A theorem proved in the paper yields, in particular, this conjecture for an arbitrary non-decreasing sequence of exponents .

The Hermite-Pade approximants are studied for systems of Markov functions (introduced in this paper) with structure described by a graph. Results of an asymptotic nature are stated in terms of certain equilibrium problems of potential theory concerning vector potentials.

The paper provides an overview of the author’s contribution to the theory of constructive rational approximations of analytic functions. The results presented are related to the convergence theory of Padé approximants and of more general rational interpolation processes, which significantly expand the classical theory’s framework of continuous fractions, to inverse problems in the theory of Padé approximants, to the application of multipoint Padé approximants (solutions of Cauchy-Jacobi interpolation problem) in explorations connected with the rate of Chebyshev rational approximation of analytic functions and to the asymptotic properties of Padé-Hermite approximation for systems of Markov type functions.