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

作者：Deano Alfredo^*; Kuijlaars Arno B J; Roman Pablo

来源：Constructive Approximation, 2016, 43(1): 153-196.

DOI：10.1007/s00365-015-9300-8

• Orthogonal polynomials
• Riemann-Hilbert problems
• Asymptotic representations in the complex domain
• Limiting zero distribution
• Bessel functions

We consider polynomials P-n orthogonal with respect to the weight J(nu) on [0, infinity), 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 of P-n are complex and accumulate as n -> infinity near the vertical line Re z = nu pi/2. We prove this fact for the case 0 <= nu <= 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 = 1/2.

• 出版日期2016-2