Geodesic
The distribution of the zeros of the Hermite-Pad\'e polynomials for a~pair of functions forming a~Nikishin system
Sbornik. Mathematics, Tome 204 (2013) no. 9, pp. 1347-1390

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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 $\mathscr S$-property in a harmonic external field. The existence problem for $\mathscr S$-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.
Keywords: Hermite-Padé polynomials, distribution of zeros, stationary compact set, Nuttall condenser.
Mots-clés : orthogonal polynomials 
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E. A. Rakhmanov; S. P. Suetin. The distribution of the zeros of the Hermite-Pad\'e polynomials for a~pair of functions forming a~Nikishin system. Sbornik. Mathematics, Tome 204 (2013) no. 9, pp. 1347-1390. http://geodesic.mathdoc.fr/item/SM_2013_204_9_a5/