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

We consider polynomials PnPn orthogonal with respect to the weight JnuJnu on [0,∞)[0,∞) , where JnuJnu is the Bessel function of order nunu . Asheim and Huybrechs considered these polynomials in connection with complex Gaussian quadrature for oscillatory integrals. They observed that the zeros of PnPn are complex and accumulate as n→∞n→∞ near the vertical line Rez=nupi2Rez=nupi2 . We prove this fact for the case 0≤nu≤1/20≤nu≤1/2 from strong asymptotic formulas that we derive for the polynomials PnPn 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≤1/2nu≤1/2.

orthogonal polynomials; riemann hilbert problems; asymptotic representations in the complex domain; limiting zero distribution; bessel functions

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