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    Strong asymptotics for Jacobi polynomials with varying nonstandard parameters.

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    0309443v2.pdf (540.5Kb)
    Identifiers
    URI: http://hdl.handle.net/10835/1638
    ISSN: 0021-7670
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    Author/s
    Kuijlaars, A. B. J.; Martínez-Finkelshtein, Andrei
    Date
    2004
    Abstract
    Strong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomials $P_n^{(\alpha_n, \beta_n)}$ is studied, assuming that $$ \lim_{n\to\infty} \frac{\alpha_n}{n}=A, \qquad \lim_{n\to\infty} \frac{\beta _n}{n}=B, $$ with $A$ and $B$ satisfying $ A > -1$, $ B>-1$, $A+B < -1$. The asymptotic analysis is based on the non-Hermitian orthogonality of these polynomials, and uses the Deift/Zhou steepest descent analysis for matrix Riemann-Hilbert problems. As a corollary, asymptotic zero behavior is derived. We show that in a generic case the zeros distribute on the set of critical trajectories $\Gamma$ of a certain quadratic differential according to the equilibrium measure on $\Gamma$ in an external field. However, when either $\alpha_n$, $\beta_n$ or $\alpha_n+\beta_n$ are geometrically close to $\Z$, part of the zeros accumulate along a different trajectory of the same quadratic differential.
    Palabra/s clave
    Análisis asintótico
    Polinomios de Jacobi
    Análisis de algoritmos
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    • Artículos de revista Dpto. Matemáticas [119]

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