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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P. Deift, T. Kriecherbauer, K. T.-R. McLaughlin, S. Venakides, and X. Zhou (1999b) 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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External Links:: ISSN 0010-3640, Document, MathReview (D. S. Lubinsky), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

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A. J. Durán and F. A. Grünbaum (2005) 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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External Links:: ISSN 0377-0427, Document, MathReview, ZentralBlatt
Referenced by:: §18.36(iv).
Encodings:: BibTeX

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3: 18.38 Mathematical Applications

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Random Matrix Theory

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4: Bibliography B

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P. Bleher and A. Its (1999) 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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External Links:: ISSN 0003-486X, Document, MathReview (Thomas Kriecherbauer), ZentralBlatt
Referenced by:: §18.32.
Encodings:: BibTeX

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5: 35.4 Partitions and Zonal Polynomials

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35.4.3

Z_{\kappa}\left(\mathbf{T}\right)=Z_{\kappa}\left(\mathbf{I}\right)\,\left|% \mathbf{T}\right|^{k_{m}}\*\int\limits_{\mathbf{O}(m)}\prod_{j=1}^{m-1}|(% \mathbf{H}\mathbf{T}\mathbf{H}^{-1})_{j}|^{k_{j}-k_{j+1}}\mathrm{d}{\mathbf{H}},

\mathbf{T}\in\boldsymbol{\mathcal{S}}

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35.4.5

Z_{\kappa}\left(\mathbf{H}\mathbf{T}\mathbf{H}^{-1}\right)=Z_{\kappa}\left(% \mathbf{T}\right),

\mathbf{H}\in\mathbf{O}(m)

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Symbols:: $\in$ : element of, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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35.4.7

\int_{\mathbf{O}(m)}Z_{\kappa}\left(\mathbf{S}\mathbf{H}\mathbf{T}\mathbf{H}^{% -1}\right)\mathrm{d}{\mathbf{H}}=\frac{Z_{\kappa}\left(\mathbf{S}\right)Z_{% \kappa}\left(\mathbf{T}\right)}{Z_{\kappa}\left(\mathbf{I}\right)}.

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Symbols:: $\int$ : integral, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{S}$ : real symmetric $m\times m$ matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $\mathrm{d}{\mathbf{H}}$ : normalized Haar measure on $\mathbf{O}(m)$ , $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Referenced by:: §35.9
Permalink:: http://dlmf.nist.gov/35.4.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

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6: 18.2 General Orthogonal Polynomials

§18.2 General Orthogonal Polynomials

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7: 35.1 Special Notation

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$a, b$	complex variables.
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$\boldsymbol{{0}}$	zero matrix.
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$\mathbf{H}$	orthogonal matrix.
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively

\Gamma_{m}\left(a\right)

and

\mathrm{B}_{m}\left(a,b\right)

, and the special functions of matrix argument: Bessel (of the first kind)

A_{\nu}\left(\mathbf{T}\right)

and (of the second kind)

B_{\nu}\left(\mathbf{T}\right)

; confluent hypergeometric (of the first kind)

{{}_{1}F_{1}}\left(a;b;\mathbf{T}\right)

\displaystyle{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)

and (of the second kind)

\Psi\left(a;b;\mathbf{T}\right)

; Gaussian hypergeometric

{{}_{2}F_{1}}\left(a_{1},a_{2};b;\mathbf{T}\right)

\displaystyle{{}_{2}F_{1}}\left({a_{1},a_{2}\atop b};\mathbf{T}\right)

; generalized hypergeometric

{{}_{p}F_{q}}\left(a_{1},\dots,a_{p};b_{1},\dots,b_{q};\mathbf{T}\right)

\displaystyle{{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};% \mathbf{T}\right)

. … ►Related notations for the Bessel functions are

\mathcal{J}_{\nu+\frac{1}{2}(m+1)}(\mathbf{T})=A_{\nu}\left(\mathbf{T}\right)/% A_{\nu}\left(\boldsymbol{{0}}\right)

(Faraut and Korányi (1994, pp. 320–329)),

K_{m}(0,\dots,0,\nu\mathpunct{|}\mathbf{S},\mathbf{T})=\left|\mathbf{T}\right|% ^{\nu}B_{\nu}\left(\mathbf{S}\mathbf{T}\right)

(Terras (1988, pp. 49–64)), and

\mathcal{K}_{\nu}(\mathbf{T})=\left|\mathbf{T}\right|^{\nu}B_{\nu}\left(% \mathbf{S}\mathbf{T}\right)

(Faraut and Korányi (1994, pp. 357–358)).

8: 37.17 Hermite Polynomials on ${\mathbb{R}}^{d}$

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§37.17(i) Product Hermite Polynomials

►Obviously, an orthogonal basis of

\mathcal{V}_{n}({\mathbb{R}}^{d})

consisting of product Hermite polynomials is given by … ►

§37.17(ii) Spherical Hermite Polynomials

… ►For a positive definite symmetric

d\times d

matrix

\mathbf{A}

define the inner product …The matrix

\mathbf{A}

can be factorized as

\mathbf{A}=\mathbf{B}^{\mathrm{T}}\mathbf{B}

for some invertible

d\times d

matrix

\mathbf{B}

. …

9: 28.34 Methods of Computation

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(d)

Solution of the matrix eigenvalue problem for each of the five infinite matrices that correspond to the linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4). See Zhang and Jin (1996, pp. 479–482) and §3.2(iv).

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(f)

Asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See Volkmer (2008). This method also applies to eigenvalues of the Whittaker–Hill equation (§28.31(i)) and eigenvalues of Lamé functions (§29.3(i)).

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10: 3.5 Quadrature

… ►For further extensions, applications, and computation of orthogonal polynomials and Gauss-type formulas, see Gautschi (1994, 1996, 2004). … ►For the classical orthogonal polynomials related to the following Gauss rules, see §18.3. … ►Below we give for the classical orthogonal polynomials the recurrence coefficients

\alpha_{n}

and

\beta_{n}

in (3.5.30). … ►Complex orthogonal polynomials

p_{n}(1/\zeta)

of degree

n=0,1,2,\dots

, in

1/\zeta

that satisfy the orthogonality condition … …