# On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials - Mathematics > Classical Analysis and ODEs

On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials - Mathematics > Classical Analysis and ODEs - Download this document for free, or read online. Document in PDF available to download.

Abstract: 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.

Author: A. Foulquie Moreno, A. Martinez-Finkelshtein, V. L. Sousa

Source: https://arxiv.org/