orthogonal matrix polynomials
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1: 18.36 Miscellaneous Polynomials
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§18.36(iv) Orthogonal Matrix Polynomials
►These are matrix-valued polynomials that are orthogonal with respect to a square matrix of measures on the real line. …2: Bibliography D
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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory.
Comm. Pure Appl. Math. 52 (11), pp. 1335–1425.
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A survey on orthogonal matrix polynomials satisfying second order differential equations.
J. Comput. Appl. Math. 178 (1-2), pp. 169–190.
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3: 18.38 Mathematical Applications
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Random Matrix Theory
…4: Bibliography B
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Semiclassical asymptotics of orthogonal polynomials, Riemann-Hilbert problem, and universality in the matrix model.
Ann. of Math. (2) 150 (1), pp. 185–266.
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5: 35.4 Partitions and Zonal Polynomials
6: 18.2 General Orthogonal Polynomials
7: 35.1 Special Notation
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively and , and the special functions of matrix argument: Bessel (of the first kind) and (of the second kind) ; confluent hypergeometric (of the first kind) or and (of the second kind) ; Gaussian hypergeometric or ; generalized hypergeometric or .
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►Related notations for the Bessel functions are (Faraut and Korányi (1994, pp. 320–329)), (Terras (1988, pp. 49–64)), and (Faraut and Korányi (1994, pp. 357–358)).
complex variables. | |
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zero matrix. | |
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orthogonal matrix. | |
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8: 28.34 Methods of Computation
9: 3.5 Quadrature
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►For further extensions, applications, and computation of orthogonal polynomials and Gauss-type formulas, see Gautschi (1994, 1996, 2004).
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►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3.
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►Below we give for the classical orthogonal polynomials the recurrence coefficients and in (3.5.30).
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►Complex orthogonal polynomials
of degree , in that satisfy the orthogonality condition
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10: Bibliography K
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Orthogonal Polynomials on -spheres: Gegenbauer, Jacobi and Heun.
In Topics in Polynomials of One and Several Variables and their
Applications,
pp. 299–322.
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Orthogonal polynomials and computer algebra.
In Recent developments in complex analysis and computer algebra
(Newark, DE, 1997), R. P. Gilbert, J. Kajiwara, and Y. S. Xu (Eds.),
Int. Soc. Anal. Appl. Comput., Vol. 4, Dordrecht, pp. 205–234.
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Two-variable Analogues of the Classical Orthogonal Polynomials.
In Theory and Application of Special Functions, R. A. Askey (Ed.),
pp. 435–495.
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Lowering and Raising Operators for Some Special Orthogonal Polynomials.
In Jack, Hall-Littlewood and Macdonald Polynomials,
Contemp. Math., Vol. 417, pp. 227–238.
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Strong asymptotics of polynomials orthogonal with respect to Freud weights.
Internat. Math. Res. Notices 1999 (6), pp. 299–333.
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