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In 1995 Magnus posed a conjecture about the asymptotics of the recurrencecoefficients of orthogonal polynomials with respect to the weights on [-1,1] ofthe form  $$  (1-x)^\alpha (1+x)^\beta |x_0 - x|^\gamma \times a jump at x_0, $$ with$\alpha, \beta, \gamma>-1$ and $x_0 \in (-1,1)$. We show rigorously thatMagnus' conjecture is correct even in a more general situation, when the weightabove has an extra factor, which is analytic in a neighborhood of [-1,1] andpositive on the interval. The proof is based on the steepest descendent methodof Deift and Zhou applied to the non-commutative Riemann-Hilbert problemcharacterizing the orthogonal polynomials. A feature of this situation is thatthe local analysis at $x_0$ has to be carried out in terms of confluenthypergeometric functions.

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