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 ∈ N), which consists of m tuples of polynomials; these tuples, which are indexed by a natural number k ∈ [1, . . . , m], are called the kth 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 ∞ from the tuple of germs [1, f 1,∞ , . . . , f m,∞ ] of the functions 1, f 1 , . . . , f m that are meromorphic on some (m + 1)-sheeted branched covering π : R → C of the Riemann sphere C of a compact Riemann surface R. In particular, under some additional condition on π, we find the limit distribution of the zeros and the asymptotics of the ratios of the kth polynomials for all k ∈ [1, . . . , m]. It turns out that in the case, where f j = f j for some meromorphic function f on R, the ratios of some kth 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 R into sheets. Contents 1 Introduction 2 Statements of the main results 3 Reconstruction of the values of a function meromorphic on R from its germ via the polynomial Hermite-Padé m-system 4 The Riemann surface R [k] and the definition of the kth polynomials of the Hermite-Padé m-system in terms of this surface 5 Proofs of Theorems 1 and 2 6 The condition of connectedness of the Riemann surface R [k]

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Abstract. We obtain a strong asymptotic formula for the leading coefficient α n (n) of a degree n polynomial q n (z; n) orthonormal on a system of intervals on the real line with respect to a varying weight. The weight depends on n as e −2nQ (x) , where Q(x) is a polynomial and corresponds to the "hard-edge case". The formula in Theorem 1 is quite similar to Widom's classical formula for a weight independent of n. In some sense, Widom's formulas are still true for a varying weight and are thus universal. As a consequence of the asymptotic formula we have that α n (n)e −nw Q oscillates as n → ∞ and, in a typical case, fills an interval (here w Q is the equilibrium constant in the external field Q).

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