Geodesic
On the asymptotics of the ratio of orthogonal polynomials
Sbornik. Mathematics, Tome 32 (1977) no. 2, pp. 199-213 Cet article a éte moissonné depuis la source Math-Net.Ru

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Conditions are obtained for the existence of “exterior” asymptotics for orthogonal polynomials. In particular, it is shown that if $\rho'>0$ almost everywhere on the interval $[-1,1]$ ($\rho(x)$ is a nondecreasing function on $[-1,1]$), then for the corresponding orthonormal polynomials the relation $\frac{P_{n+1}(z)}{P_n(z)}\rightrightarrows z+\sqrt{z^2-1}$ holds on compact subsets of $\mathbf C\setminus[-1,1]$. The branch of the square root is chosen so that $|z+\sqrt{z^2-1}\,|>1$ in the region described. Bibliography: 6 titles. 
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E. A. Rakhmanov. On the asymptotics of the ratio of orthogonal polynomials. Sbornik. Mathematics, Tome 32 (1977) no. 2, pp. 199-213. http://geodesic.mathdoc.fr/item/SM_1977_32_2_a1/

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[2] Ya. L. Geronimus, Mnogochleny ortogonalnye na okruzhnosti i otrezke, Fizmatgiz, Moskva, 1958 | Zbl

[3] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, izd-vo «Nauka», Moskva, 1972 | MR

[4] A. A. Gonchar, “O skhodimosti approksimatsii Pade dlya nekotorykh klassov meromorfnykh funktsii”, Matem. sb., 97 (139) (1975), 607–629 | Zbl

[5] P. P. Korovkin, “Emkost mnozhestva i polinomy, minimiziruyuschie integral”, Uchenye zapiski kaliningradskogo pedinstituta, 1958, no. 5, 34–52

[6] P. P. Korovkin, “Asimptoticheskoe predstavlenie polinomov, minimiziruyuschikh integral”, Issledovaniya po sovremennym problemam konstruktivnoi teorii funktsii, Fizmatgiz, Moskva, 1961 | MR