Construction and implementation of asymptotic expansions for Jacobi--type orthogonal polynomials

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree n goes to ∞. These are defined on the interval [-1, 1] with weight function w(x) = (1-x) α (1+x) β h(x), α,β > -1 and h(x) a real, analytic and strictly positive function on [-1, 1]. This information is available in the work of Kuijlaars, McLaughlin, Van Assche and Vanlessen, where the authors use the Riemann-Hilbert formulation and the Deift-Zhou non-linear steepest descent method. We show that computing higher order terms can be simplified, leading to their efficient construction. The resulting asymptotic expansions in every region of the complex plane are implemented both symbolically and numerically, and the code is made publicly available.nrpages: 36status: publishe