2021/09/21 by Sergey Pavlovich Suetin, Suetin, Sergey P.
Mathematics · #30 #31 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Complex Variables (math.CV) #FOS: Mathematics #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2109.10144
openalex publication_date 2021/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In the paper, we discuss how it would be possible to succeed in Stahl's novel approach, 1987--1988, to explore Hermite--Padé polynomials based on Riemann surface properties. In particular, we explore the limit zero distribution of type I Hermite--Padé polynomials Qn,0,Qn,1,Qn,2, degQn,j≤n, for a collection of three analytic elements [1,f_∞,f2_∞]. The element f_∞ is an element of a function f from the class \mathbb C(z,w) where w is supposed to be from the class Z±1/2([-1,1]) of multivalued analytic functions generated by the inverse Zhukovskii function with the exponents from the set \±1/2\. The Riemann surface corresponding to f∈\mathbb C(z,w) is a four-sheeted Riemann surface \mathfrak R4(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∈\mathbb C(z,w) is of fourth order and we consider the triple of the analytic elements [1,f_∞,f2_∞] but not the quadruple [1,f_∞,f2_∞,f3_∞] 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.