On the distribution of the zeros of the Hermite-Padé polynomials for a quadruple of functions

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

“…The purpose of the present paper is to put forward and discuss, on an example of a pair of functions of the form (1), a new approach to the study of the limit distribution of the zeros for Hermite-Padé polynomials of type I as defined by (3). As it is our intention to apply, in subsequent studies, this approach to fairly general classes of analytic functions (see the result announced in [50] and Remark 1 below), we shall first give the notation to be used below (in this respect, see [46], [32], [49]).…”

Hermite--Padé polynomials and analytic continuation: new approach and some results

“…The new approach proposed below to efficient solution of the analytic continuation problem for power series beyond its disc of convergence will be demonstrated (from the theoretical as well as the numerical standpoints) on an example of some "model" class of multivalued analytic functions based on the use of the inverse of the Zhukovskii function (see representation (1) below). This class was first introduced in [45] (see also the papers [46], [48]), where it was denoted by Z . Below, this notation will be retained, but when required the parameters in representation (1) will be refined.…”

“…In this section we shall discuss some possible of applications of the above theoretical results in numerical mathematics for approximate solution of problems based on the analytic continuation of the original data (for example, the analytic continuation with respect to the small parameter). We give three examples based on functions of the form (46), in which the quantities A and B have the same imaginary part and opposite real parts; i.e.,…”