Abstract:
Conditions are obtained for the existence of “exterior” asymptotics for orthogonal polynomials. In particular, it is shown that if ρ′>0 almost everywhere on the interval [−1,1] (ρ(x) is a nondecreasing function on [−1,1]), then for the corresponding orthonormal polynomials the relation Pn+1(z)Pn(z)⇉z+√z2−1 holds on compact subsets of C∖[−1,1]. The branch of the square root is chosen so that |z+√z2−1|>1 in the region described.
Bibliography: 6 titles.
\Bibitem{Rak77}
\by E.~A.~Rakhmanov
\paper On the asymptotics of the ratio of orthogonal polynomials
\jour Math. USSR-Sb.
\yr 1977
\vol 32
\issue 2
\pages 199--213
\mathnet{http://mi.mathnet.ru/eng/sm2806}
\crossref{https://doi.org/10.1070/SM1977v032n02ABEH002377}
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