Scalar equilibrium problem and the limit distribution of the zeros of Hermite-Pad\'e polynomials of type II

TL;DR

This paper proves the existence of the limit distribution of zeros of Hermite-Padé polynomials of type II for a Nikishin system using scalar equilibrium problems, connecting theoretical results with numerical experiments.

Contribution

It introduces a scalar equilibrium problem approach on a Riemann surface to analyze zero distributions of Hermite-Padé polynomials for Nikishin systems, extending previous work and numerical validation.

Findings

Limit distribution of zeros exists for the considered polynomials.

Results align with Stahl's earlier findings and numerical experiments.

The scalar equilibrium problem approach is effective for such analyses.

Abstract

The existence of the limit distribution of the zeros of Hermite-Pad\'e polynomials of type II for a pair of functions forming a Nikishin system is proved using the scalar equilibrium problem posed on the two-sheeted Riemann surface. The relation of the results obtained here to some results of H. Stahl (1988) is discussed. Results of numerical experiments are presented. The results of the present paper and those obtained in the earlier paper of the second author [28], [32], [33] are shown to be in good accordance with both H. Stahl's results and with results of numerical experiments.