Asymptotic behavior and zero distribution of polynomials orthogonal with respect to Bessel functions

A. DEAÑO; A. B. J. KUIJLAARS; PABLO ROMÁN

CONSTRUCTIVE APPROXIMATION

SPRINGER

Lugar: Berlin; Año: 2014 vol. 43 p. 153 - 196

0176-4276

We consider polynomials P_n orthogonal with respect to the weight J_nu on [0,infty), where J_nu is the Bessel function of order nu. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros are complex and accumulate as n->infty near the vertical line Re z = nupi/2. We prove this fact for the case 0 leq nu leq 1/2 from strong asymptotic formulas that we derive for the polynomials P_n in the complex plane. Our main tool is the Riemann-Hilbert problem for orthogonal polynomials, suitably modified to cover the present situation, and the Deift-Zhou steepest descent method. A major part of the work is devoted to the construction of a local parametrix at the origin, for which we give an existence proof that only works for nu leq 1/2.