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)^α (1 + x)^β | x₀ - x |^γ × {(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 Riemann-Hilbert 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.