Hermite–Padé polynomials for meromorphic functions on a compact Riemann surface

Let $\mathfrak R$ be a compact Riemann surface and $\pi\colon \mathfrak R \to\widehat{\mathbb C}$ be a $(m+1)$-fold branched covering of the Riemann sphere $\widehat{\mathbb C}$, $m\ge 1$. Suppose that $f_1$, $f_2$, …, $f_m$ are meromorphic functions on the Riemann surface $\mathfrak R$ such that the functions $1$, $f_1$, $f_2$, …, $f_m$ are independent over the field $\mathbb C(z)$ of rational functions on $\widehat{\mathbb C}$. Fix a point $\circ\in\mathfrak R$ that is not critical for the projection $\pi$. Without loss of generality we can suppose that $\circ\in\pi^{-1}(\infty)$ and denote $\pmb\infty^{(0)}:=\circ$. If we choose a small enough neighborhood of $\pmb\infty^{(0)}$, then the restriction $\pi_0$ of the projection $\pi$ to this neighborhood is biholomorphic. For $j=1,\dots,m$ set $f_{j,\infty}(z):=f_j(\pi_0^{-1}(z))$ in the neighborhood of $\infty\in\widehat{\mathbb C}$. For convenience we suppose that the germs $f_{j,\infty}$ are holomorphic at $\infty$.
The Hermite–Padé polynomials of the first kind $Q_{n,0}$, …, $Q_{n,m}$ of order $n\in\mathbb N$ for the tuple of germs $[1,f_{1,\infty},\dots, f_{m,\infty}]$ at the point $\infty\in\widehat{\mathbb C}$ are defined as the polynomials of degree not greater than $n$ such that at least one $Q_{n,j}\not\equiv0$ and the following asymptotic relation at $\infty$ holds true:
$$ Q_{n,0}(z)+\sum\limits_{j=1}^m Q_{n,j}(z)f_{j,\infty}(z)=O\left(\dfrac 1 {z^{m(n+1)}}\right)\text{ as } z\to\infty. $$

In the talk we discuss asymptotic behaviour of the ratios $\frac{Q_{n,j}(z)}{Q_{n,k}(z)}$, $k,j=0,\dots,m$ as $n\to\infty$. Our research uses the approach of J. Nuttall that is based on a special “Nuttall's partition” of the Riemann surface $\mathfrak R$ into sheets. In particular, our results allow us to asymptotically reconstruct the values of a meromorphic function $f$ on $\mathfrak R$ on $m$ Nuttall's sheets (all except one) from the initial germ of $f$ at $\pmb\infty^{(0)}$ as roots of some algebraic equation of degree $m$. For this one should take $f_j:=f^j$, $j=1,\dots,m$.
The talk is based on the joint work with E.M.Chirka, A.V.Komlov, and S.P.Suetin.