A Riemann–Hilbert problem for skew-orthogonal polynomials

We find a local ( d + 1 ) × ( d + 1 ) Riemann–Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d. Our Riemann–Hilbert problem is similar to a local d × d Riemann–Hilbert...

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Bibliographic Details
Published in:Journal of computational and applied mathematics Vol. 215; no. 1; pp. 230 - 241
Main Author: Pierce, Virgil U.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 15.05.2008
Elsevier
Subjects:
Exact sciences and technology
Fourier analysis
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations
Riemann–Hilbert problems
Sciences and techniques of general use
Skew-orthogonal polynomials
Special functions
Skew-orthogonal polynomials
Riemann–Hilbert problems
Polynomial function
Partition
Numerical analysis
Asymptotic expansion
Riemann problem
Riemann-Hilbert problems
Applied mathematics
Asymptotic approximation
Orthogonal polynomial
ISSN:0377-0427, 1879-1778
Online Access:Get full text
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Summary:We find a local ( d + 1 ) × ( d + 1 ) Riemann–Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of orthogonal ensembles of random matrices with a potential function of degree d. Our Riemann–Hilbert problem is similar to a local d × d Riemann–Hilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polynomials. This gives more motivation for finding methods to compute asymptotics of high order Riemann–Hilbert problems, and brings us closer to finding full asymptotic expansions of the skew-orthogonal polynomials.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2007.04.006