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Title: The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system

The distribution of the zeros of the Hermite-Padé polynomials of the first kind for a pair of functions with an arbitrary even number of common branch points lying on the real axis is investigated under the assumption that this pair of functions forms a generalized complex Nikishin system. It is proved (Theorem 1) that the zeros have a limiting distribution, which coincides with the equilibrium measure of a certain compact set having the S-property in a harmonic external field. The existence problem for S-compact sets is solved in Theorem 2. The main idea of the proof of Theorem 1 consists in replacing a vector equilibrium problem in potential theory by a scalar problem with an external field and then using the general Gonchar-Rakhmanov method, which was worked out in the solution of the '1/9'-conjecture. The relation of the result obtained here to some results and conjectures due to Nuttall is discussed. Bibliography: 51 titles.
Authors:
;  [1]
  1. Steklov Mathematical Institute of the Russian Academy of Sciences (Russian Federation)
Publication Date:
OSTI Identifier:
22365971
Resource Type:
Journal Article
Resource Relation:
Journal Name: Sbornik. Mathematics; Journal Volume: 204; Journal Issue: 9; Other Information: Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; EQUILIBRIUM; MATHEMATICAL SOLUTIONS; PADE APPROXIMATION; POLYNOMIALS; POTENTIALS; SCALARS; VECTORS