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arXiv · 0905.2753

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

A. Foulquie Moreno·A. Martinez-Finkelshtein·V. L. Sousa
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Abstract

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 $$ (1-x)^α(1+x)^β|x_0 - x|^γ\times a jump at x_0, $$ with $α, β, γ>-1$ 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.

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A. Foulquie Moreno, A. Martinez-Finkelshtein, V. L. Sousa. 2009-05-17. On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials. https://arxiv.org/abs/0905.2753

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