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 of $P_n$ are complex and accumulate as $n \rightarrow \infty$ near the vertical line ${{\mathrm{{\text {Re}}\,}}}z = \frac{\nu \pi }{2}$. 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 $\nu \le 1/2$.