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][−1,1] of the form (1−x)α(1+x)β|x0−x|γ×{B,for x∈[−1,x0),A,for x∈[x0,1], with A,B>0A,B>0, α,β,γ>−1α,β,γ>−1, and x0∈(−1,1)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][−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 x0x0 has to be carried out in terms of confluent hypergeometric functions.