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Seminar on Complex Analysis (Gonchar Seminar)
January 16, 2017 17:00–19:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)

On Nuttall's partition for four sheeted Riemann surfaces of some class of multivalued analytic functions

S. P. Suetin

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: During the talk we will discuss the problem of limit zero distribution of type I Hermite–Padé polynomials for the collection of four functions $[1,f,f^2,f^3]$ where $f$ is from class ${\mathscr Z}(\Delta)$, $\Delta=[-1,1]$, of multivalued analytic functions, i.e., the functions given by the following representation
$$f(z):=\prod_{j=1}^m(A_j-\frac1{\varphi(z)})^{\alpha_j}. \label{3}$$
Here $\varphi(z)=z+\sqrt{z^2-1}$, $z\in\overline{\mathbb C}\setminus\Delta$, is the inverse to Joukowsky function and we suppose that $\sqrt{z^2-1}/z\to1$ as $z\to\infty$. We suppose also that $m\geq2$, and all the $A_j\in{\mathbb C}$ are pairwise distinct, $|A_j|>1$, the exponents $\alpha_j\in{\mathbb C}\setminus{\mathbb Z}$, and $\sum_{j=1}^m\alpha_j=0$. As ussual, the type I Hermite–Padé polynomials $Q_{n,j}\not\equiv0$, $\operatorname{deg}{Q_{n,j}}\leq{n}$, $j=0,1,2,3$, of degree $n\in{\mathbb N}$ are defined from relation
$$(Q_{n,0}+Q_{n,1}f+Q_{n,2}f^2+Q_{n,3}f^3)(z) =O(\frac1{z^{3n+3}}),\quad z\to\infty. \label{4}$$