The polynomial Hermite-Pade m-system for meromorphic functions on a compact Riemann surface

Given a tuple of m + 1 germs of arbitrary analytic functions at a fixed point, we introduce the polynomial Hermite-Pade m-system, which includes the Hermite-Pade polynomials of types I and II. In the generic case we find the weak asymptotics of the polynomials of the Hermite-Pade m-system constructed from the tuple of germs of functions 1, f(1), ... , f(m) that are meromorphic on an (m + 1)-sheeted compact Riemann surface R. We show that if f(j) = f(j) for some meromorphic function f on R, then with the help of the ratios of polynomials of the Hermite-Pade m-system we recover the values of f on all sheets of the Nuttall partition of R, apart from the last sheet.