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

In 1995 Magnus [15] posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [−1, 1] of the form (1− x) (1 + x) |x0 − x| × { B, for x ∈ [−1, x0) , A, for x ∈ [x0, 1] , with A,B > 0, α, β, γ > −1, and x0 ∈ (−1, 1). 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 [−1, 1] and positive on the interval. The proof is based on the steepest descendent method of Deift and Zhou applied to the non-commutative RiemannHilbert problem characterizing the orthogonal polynomials. A feature of this situation is that the local analysis at x0 has to be carried out in terms of confluent hypergeometric functions.