Journal of Approximation Theory
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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)^@a(1+x)^@b|x"0-x|^@cx{B,for x@?[-1,x"0),A,for x@?[x"0,1], with A,B0, @a,@b,@c-1, and x"0@?(-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.