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

V. M. Badkov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 14 (2008) no. 3, pp. 38-42

Résumé

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 $n\to\infty$: $$ \theta_{n,1}^{(p)}=\frac{\sqrt2}{n\sqrt{\ln(n+1)}}\biggl[1+O\biggl(\frac1{\ln(n+1)}\biggr)\biggr],\quad x_{n,1}^{(p)}=1-\frac1{n^2\ln(n+1)}+O\biggl(\frac1{\ln(n+1)}\biggr). $$