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

Abstract

Let {pn(t)}n=0t8 be a system of algebraic polynomials orthonormal on the segment [−1, 1] with a weight p(t); let {xn,ν(p)}ν=1n be zeros of a polynomial pn(t) (xx,ν(p) = cosθn,ν(p); 0 < θn,1(p) < θn,2(p) < ... < θn,n(p) < π). It is known that, for a wide class of weights p(t) containing the Jacobi weight, the quantities θn,1(p) and 1 − xn,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 − t2)−1{ln2[(1 + t)/(1 − t)] + π2}−1, then the following asymptotic formulas are valid as n → ∞: $$ \theta _{n,1}^{(p)} = \frac{{\sqrt 2 }} {{n\sqrt {\ln (n + 1)} }}\left[ {1 + {\rm O}\left( {\frac{1} {{\ln (n + 1)}}} \right)} \right],x_{n,1}^{(p)} = 1 - \left( {\frac{1} {{n^2 \ln (n + 1)}}} \right) + O\left( {\frac{1} {{n^2 \ln ^2 (n + 1)}}} \right). $$