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

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\left(A_j-\frac1{\varphi(z)}\right)^{\alpha_j}. \label{3} \end{equation}
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
\begin{equation} (Q_{n,0}+Q_{n,1}f+Q_{n,2}f^2+Q_{n,3}f^3)(z) =O\left(\frac1{z^{3n+3}}\right),\quad z\to\infty. \label{4} \end{equation}