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

Abstract

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 ) α ( 1 + x ) β | x 0 − x | γ × { B , for x ∈ [ − 1 , x 0 ) , A , for x ∈ [ x 0 , 1 ] , with A , B > 0 , α , β , γ > − 1 , and x 0 ∈ ( − 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 Riemann–Hilbert problem characterizing the orthogonal polynomials. A feature of this situation is that the local analysis at x 0 has to be carried out in terms of confluent hypergeometric functions.