Asymptotic Behavior and Zero Distribution of Polynomials Orthogonal with Respect to Bessel Functions

Abstract

We consider polynomials Pn orthogonal with respect to the weight Jν on [0,∞), where Jν is the Bessel function of order ν. Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros of Pn are complex and accumulate as n→∞ near the vertical line Rez=νπ2. We prove this fact for the case 0≤ν≤1/2 from strong asymptotic formulas that we derive for the polynomials Pn 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 ν≤1/2.