Geodesic
On the asymptotics of the ratio of orthogonal polynomials. II
Sbornik. Mathematics, Tome 46 (1983) no. 1, pp. 105-117
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Let $\mu$ be a positive measure on the circumference $\Gamma=\{z:|z|=1\}$ and let $\mu'=\dfrac{d\mu}{d\theta}>0$ almost everywhere on $\Gamma$. Let $\Phi_n(z)=z^n+\cdots$ be the orthogonal polynomials corresponding to $\mu$, and let $a_n=-\overline{\Phi_{n+1}(0)}$ be their parameters. Then $\lim\limits_{n\to\infty}a_n=0$. Bibliography: 5 titles. 
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E. A. Rakhmanov. On the asymptotics of the ratio of orthogonal polynomials. II. Sbornik. Mathematics, Tome 46 (1983) no. 1, pp. 105-117. http://geodesic.mathdoc.fr/item/SM_1983_46_1_a4/

[1] Rakhmanov E. A., “Ob asimptotike otnosheniya ortogonalnykh mnogochlenov”, Matem. sb., 103(145) (1977), 237–252 | Zbl

[2] Sëge G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[3] Geronimus Ya. L., “Dopolneniya”, Segë G. Ortogonalnye mnogochleny, Fizmatgiz, M., 1962, 414–494

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

[5] Golinskii B. L., “O svyazi mezhdu poryadkom ubyvaniya parametrov ortogonalnykh mnogochlenov i svoistvami sootvetstvuyuschei funktsii raspredeleniya”, Izv. AN Arm.SSR, XV:2 (1980), 127–144 | MR