Reconstruction of the values of algebraic function via polynomial Hermite–Pade $m$-system

For an arbitrary tuple of $m+1$ analytic germs $[f_0, f_1,\dots, f_m]$ at some point $x_0$ we introduce the polynomial Hermite–Pade $m$-system. For each $n\in\mathbb N$ this system consists of $m$ tuples of polynomials. These tuples are numerated by the number $k=1,\dots,m$. The $k$-th tuple consists of ${m+1}\choose k$ polynomials, which are called "$k$-th polynomials of Hermite–Pade $m$-system" of order $n$. We show, that for the case, when the germs $f_j=f^j$, where $f$ is a germ of some algebraic function of order $m+1$, the ratio of some $k$-th polynomials of Hermite–Pade $m$-system converges (as $n\to\infty$) to the sum of the values of $f$ on first $k$ sheets of so-called Nuttall partition of its Riemann surface into sheets.

Note that the well known Hermite–Pade polynomials of types 1 and 2 are $m$-th and 1-st polynomials of Hermite–Pade $m$-system, respectively.