## Widom’s formula for the leading coefficient of a polynomial which is orthonormal with respect to a varying weight.(English. Russian original)Zbl 1281.42025

Russ. Math. Surv. 67, No. 1, 183-185 (2012); translation from Usp. Mat. Nauk 67, No. 1, 183-184 (2012).
Summary: 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
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