On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials

Abstract

In 1995, Magnus posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to Jacobi weights with an algebraic singularity combined with a jump.

We show rigorously that Magnus’ conjecture is correct even in a more general situation, when the weight above has an extra factor, which is analytic in a neighborhood of the interval of orthogonality and positive on that interval. The proof is based on the steepest descendent method of Deift and Zhou applied to the non-commutative Riemann–Hilbert problem characterizing the orthogonal polynomials. A feature of this situation is that the local analysis at has to be carried out in terms of confluent hypergeometric functions.