  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
  $$
  (1-x)^\alpha (1+x)^\beta |x_0 - x|^\gamma \times a jump at x_0, $$ 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.
