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

This article is cited in 109 scientific papers (total in 109 papers)

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.

Full text: PDF file (1137 kB)
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English version:
Mathematics of the USSR-Sbornik, 1977, 32:2, 199–213

Bibliographic databases:

UDC: 517.5
MSC: 30A84, 33A65
Received: 13.10.1976

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

Citation in format AMSBIB
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\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
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\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}
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