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2 papers
Asymptotic behavior of the maximal zero of a polynomial orthogonal on a segment with a nonclassical weight
V. M. Badkov 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
$p_n(t)$ (
$x_{n,\nu}^{(p)}=\cos\theta_{n,\nu}^{(p)}$;
$0<\theta_{n,1}^{(p)}<\theta_{n,2}^{(p)}<\dots<\theta_{n,n}^{(p)}<\pi$). 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
$n\to\infty$:
$$
\theta_{n,1}^{(p)}=\frac{\sqrt2}{n\sqrt{\ln(n+1)}}\biggl[1+O\biggl(\frac1{\ln(n+1)}\biggr)\biggr],\quad x_{n,1}^{(p)}=1-\frac1{n^2\ln(n+1)}+O\biggl(\frac1{\ln(n+1)}\biggr).
$$
UDC:
517.5
Received: 29.04.2008