

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

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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
\begin{equation}
f(z):=\prod_{j=1}^m(A_j\frac1{\varphi(z)})^{\alpha_j}.
\label{3}
\end{equation}
Here $\varphi(z)=z+\sqrt{z^21}$, $z\in\overline{\mathbb C}\setminus\Delta$, is the inverse to Joukowsky function and we suppose that $\sqrt{z^21}/z\to1$ 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
\begin{equation}
(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}
\end{equation}

