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 Trudy Inst. Mat. i Mekh. UrO RAN, 2008, Volume 14, Number 3, Pages 38–42 (Mi timm38)

Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences

Abstract: Let $\{p_n(t)\}_{n=0}^\infty$ be a system of algebraic polynomials orthonormal on the segment $[-1,1]$ with a weight $p(t)$; let $\{x_{n,\nu}^{(p)}\}_{\nu=1}^n$ be zeros of a polynomial $p_n(t)$ ($x_{n,\nu}^{(p)}=\cos\theta_{n,\nu}^{(p)}$; $0<\theta_{n,1}^{(p)}<\theta_{n,2}^{(p)}<…<\theta_{n,n}^{(p)}<\pi$). It is known that, for a wide class of weights $p(t)$ containing the Jacobi weight, the quantities $\theta_{n,1}^{(p)}$ and $1-x_{n,1}^{(p)}$ coincide in order with $n^{-1}$ and $n^{-2}$, respectively. In the present paper, we prove that, if the weight $p(t)$ has the form $p(t)=4(1-t^2)^{-1}\{\ln^2[(1+t)/(1-t)]+\pi^2\}^{-1}$, then the following asymptotic formulas are valid as $n\to\infty$:
$$\theta_{n,1}^{(p)}=\frac{\sqrt2}{n\sqrt{\ln(n+1)}}[1+O(\frac1{\ln(n+1)})],\quad x_{n,1}^{(p)}=1-\frac1{n^2\ln(n+1)}+O(\frac1{\ln(n+1)}).$$

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2009, 264, suppl. 1, S39–S43

Bibliographic databases:

UDC: 517.5

Citation: V. M. Badkov, “Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight”, Trudy Inst. Mat. i Mekh. UrO RAN, 14, no. 3, 2008, 38–42; Proc. Steklov Inst. Math. (Suppl.), 264, suppl. 1 (2009), S39–S43

Citation in format AMSBIB
\Bibitem{Bad08} \by V.~M.~Badkov \paper Asymptotic behavior of the maximal zero of a~polynomial orthogonal on a~segment with a~nonclassical weight \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2008 \vol 14 \issue 3 \pages 38--42 \mathnet{http://mi.mathnet.ru/timm38} \elib{http://elibrary.ru/item.asp?id=11929743} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2009 \vol 264 \issue , suppl. 1 \pages S39--S43 \crossref{https://doi.org/10.1134/S0081543809050034} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000265511100003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-65349139672} 

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This publication is cited in the following articles:
1. V. M. Badkov, “Pointwise estimates of polynomials orthogonal on a circle with respect to a weight not belonging to the spaces $L^r$ ($r>1$).”, Proc. Steklov Inst. Math. (Suppl.), 265, suppl. 1 (2009), S64–S77
2. V. M. Badkov, “Some properties of Jacobi polynomials orthogonal on a circle”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S49–S58
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