We consider polynomials PnPn orthogonal with respect to the weight JνJν on [0,∞)[0,∞) , where JνJν is the Bessel function of order νν . 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=νπ2Rez=νπ2 . We prove this fact for the case 0≤ν≤1/20≤ν≤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 ν≤1/2ν≤1/2 .
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