On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials
| dc.creator | Moreno, A. Foulquie | |
| dc.creator | Martinez-Finkelshtein, A. | |
| dc.creator | Sousa, V. L. | |
| dc.date | 2009-05-17 | |
| dc.date.accessioned | 2026-07-07T13:15:56Z | |
| dc.date.available | 2026-07-07T13:15:56Z | |
| dc.description | 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. | |
| dc.description | 29 pages, 4 figures | |
| dc.identifier | https://arxiv.org/abs/0905.2753 | |
| dc.identifier | http://arxiv.org/abs/0905.2753 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/230636 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Complex Variables | |
| dc.subject | 42C05; 33C15; 33C45 | |
| dc.title | On a conjecture of A. Magnus concerning the asymptotic behavior of the recurrence coefficients of the generalized Jacobi polynomials | |
| dc.type | text |