Bessel-type asymptotic expansions via the Riemann–Hilbert approach

A Bessel-type asymptotic expansion is established for the monic polynomials πn(x) that are orthogonal with respect to the modified Jacobi weight , x∈(−1,1), where α, β>−1 and h(x) is real analytic and strictly positive on [−1,1]. This expansion holds uniformly in a region containing the neighbourhood of the critical value x=1. This result complements the two recent results obtained by Kuijlaars and his co-workers, one for x bounded away from (−1,1) and the other for x in (−1+δ,1−δ), δ>0. Our method is also based on the Riemann–Hilbert approach.