Maximum Principle and Asymptotic Properties of Hermite--Pad\'e Polynomials

Abstract

In the paper, we discuss how it would be possible to succeed in Stahl's novel approach, 1987--1988, to explore Hermite--Pad\'e polynomials based on Riemann surface properties. In particular, we explore the limit zero distribution of type I Hermite--Pad\'e polynomials $Q_{n,0},Q_{n,1},Q_{n,2}$, $\operatorname{deg}{Q_{n,j}}\leq{n}$, for a collection of three analytic elements $[1,f_\infty,f^2_\infty]$. The element $f_\infty$ is an element of a function $f$ from the class $\mathbb C(z,w)$ where $w$ is supposed to be from the class $Z_{\pm1/2}([-1,1])$ of multivalued analytic functions generated by the inverse Zhukovskii function with the exponents from the set $\{\pm1/2\}$. The Riemann surface corresponding to $f\in\mathbb C(z,w)$ is a four-sheeted Riemann surface $\mathfrak R_4(w)$ and all branch points of $f$ are of the first order (i.e., all branch points are of square root type). Since the algebraic function $f\in\mathbb C(z,w)$ is of fourth order and we consider the triple of the analytic elements $[1,f_\infty,f^2_\infty]$ but not the quadruple $[1,f_\infty,f^2_\infty,f^3_\infty]$ ones, the result is new and does not follow from the known results. As in previous paper arXiv: 2108.00339 and following to Stahl's ideas, 1987--1988, we do not use the orthogonality relations at all. The proof is based on the maximum principle only.