Strong asymptotics for Jacobi polynomials with varying nonstandard parameters

By Arno B. J. Kuijlaars, Andrei Martínez-Finkelshtein. 2003

72 citations recorded by the source corpus.

AbstractStrong asymptotics on the whole complex plane of a sequence of monic Jacobi polynomialsPnαnβn are studied, assuming that1 $$\mathop {\lim }\limits_{n \to \infty } \frac{{\alpha _n }}{n} = A, \mathop {\lim }\limits_{n \to \infty } \frac{{\beta _n }}{n} = B,$$ withA andB satisfyingA>−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 Γ of a certain quadratic differential according to the equilibrium measure on Γ in an external field. However, when either αnβn or αn+βn are geometrically close to ℤ, part of the zeros accumulate along a different trajectory of the same quadratic differential.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials (2011, methodology): “It has also been noticed (see [7], [33], [3], [28], [34], [31], [13], [11]) that the method works for the non-Hermitian orthogonality in the complex plane with respect to complex-valued weights.”
Orthogonality of Jacobi polynomials with general parameters (2003, methodology): “In [31] the case O E Q S`8TB with O 4 Q ö`8TB was considered, which corresponds to the multiple orthogonality on three contours, as given in Theorem 6.”
Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials (2009, background): “[4], [24], [47], [49], [50], [64]; this is necessarily a very partial list).”