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

Abstract

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 $n\to \infty$ near the vertical line $\text{Re}z=\frac{\nu \pi }{2}$. We prove this fact for the case $0\le \nu \le 1/2$ 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 $\nu \le 1/2$.

42 pages, 5 figures

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