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

In 1995 Magnus posed a conjecture about the asymptotics of the recurrence coefficients of orthogonal polynomials with respect to the weights on [-1,1] of the form $$(1-x{)}^{\alpha }(1+x{)}^{\beta }\mathrm{\mid }{x}_0-x{\mathrm{\mid }}^{\gamma }\times ajumpat{x}_0,$$ with $\alpha ,\beta ,\gamma >-1$ and $x_0\in (-1,1)$. 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.