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An asymptotic formula for polynomials orthonormal with respect to a varying weight. (English. Russian original) Zbl 1277.42031
Trans. Mosc. Math. Soc. 2012, 139-159 (2012); translation from Tr. Mosk. Mat. O.-va 73, No. 2, 175-200 (2012).
Authors’ abstract: We obtain a strong asymptotic formula for the leading coefficient \(\alpha_n(n)\) 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 \(\alpha_n(n)e^{-nw_Q}\) oscillates as \(n\to\infty\) and, in a typical case, fills an interval (here \(w_Q\) is the equilibrium constant in the external field \(Q\)).

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C50 Orthogonal polynomials and functions in several variables expressible in terms of special functions in one variable
33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
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