Polynomial Hermite-Padé $m$-system for meromorphic functions on a compact Riemann surface

Abstract

For an arbitrary tuple of $m+1$ germs of analytic functions at a fixed point, we introduce the so-called polynomial Hermite-Padé $m$-system (of order $n$, $n\in \mathbb{N}$), which consists of $m$ tuples of polynomials; these tuples, which are indexed by a natural number $k\in [1,\dots ,m]$, are called the $k$th polynomials of the Hermite-Padé $m$-system. We study the weak asymptotics of the polynomials of the Hermite-Padé $m$-system constructed at the point $\infty$ from the tuple of germs $[1,f_{1,\infty },\dots$, $f_{m,\infty }]$ of the functions $1,f_1,\dots ,f_m$ that are meromorphic on some $(m+1)$-sheeted branched covering $\pi :\mathfrak{R}\to \hat{\mathbb{C}}$ of the Riemann sphere $\hat{\mathbb{C}}$ of a compact Riemann surface $\mathfrak{R}$. In particular, under some additional condition on $\pi$, we find the limit distribution of the zeros and the asymptotics of the ratios of the $k$th polynomials for all $k\in [1,\dots ,m]$. It turns out that in the case, where $f_j=f^j$ for some meromorphic function $f$ on $\mathfrak{R}$, the ratios of some $k$th polynomials of such Hermite-Padé $m$-system converge to the sum of the values of the function $f$ on the first $k$ sheets of the Nuttall partition of the Riemann surface $\mathfrak{R}$ into sheets.