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 Mat. Sb. (N.S.), 1977, Volume 103(145), Number 2(6), Pages 237–252 (Mi msb2806)

On the asymptotics of the ratio of orthogonal polynomials

E. A. Rakhmanov

Abstract: 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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English version:
Mathematics of the USSR-Sbornik, 1977, 32:2, 199–213

Bibliographic databases:

UDC: 517.5
MSC: 30A84, 33A65

Citation: E. A. Rakhmanov, “On the asymptotics of the ratio of orthogonal polynomials”, Mat. Sb. (N.S.), 103(145):2(6) (1977), 237–252; Math. USSR-Sb., 32:2 (1977), 199–213

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\Bibitem{Rak77} \by E.~A.~Rakhmanov \paper On the asymptotics of the ratio of orthogonal polynomials \jour Mat. Sb. (N.S.) \yr 1977 \vol 103(145) \issue 2(6) \pages 237--252 \mathnet{http://mi.mathnet.ru/msb2806} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=445212} \zmath{https://zbmath.org/?q=an:0373.30034} \transl \jour Math. USSR-Sb. \yr 1977 \vol 32 \issue 2 \pages 199--213 \crossref{https://doi.org/10.1070/SM1977v032n02ABEH002377} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1977GH22400002} 

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