Geodesic
On the asymptotics of the ratio of orthogonal polynomials
Sbornik. Mathematics, Tome 32 (1977) no. 2, pp. 199-213 
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/
@article{SM_1977_32_2_a1,
     author = {E. A. Rakhmanov},
     title = {On the asymptotics of the ratio of orthogonal polynomials},
     journal = {Sbornik. Mathematics},
     pages = {199--213},
     year = {1977},
     volume = {32},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1977_32_2_a1/}
}
TY  - JOUR
AU  - E. A. Rakhmanov
TI  - On the asymptotics of the ratio of orthogonal polynomials
JO  - Sbornik. Mathematics
PY  - 1977
SP  - 199
EP  - 213
VL  - 32
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1977_32_2_a1/
LA  - en
ID  - SM_1977_32_2_a1
ER  - 
%0 Journal Article
%A E. A. Rakhmanov
%T On the asymptotics of the ratio of orthogonal polynomials
%J Sbornik. Mathematics
%D 1977
%P 199-213
%V 32
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1977_32_2_a1/
%G en
%F SM_1977_32_2_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] G. Sege, Ortogonalnye mnogochleny, Fizmatgiz, Moskva, 1962

[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