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1: 35.1 Special Notation

$a, b$	complex variables.
…
$\boldsymbol{{0}}$	zero matrix.
$\mathbf{I}$	identity matrix.
…

►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)).

2: 35.5 Bessel Functions of Matrix Argument

§35.5 Bessel Functions of Matrix Argument

►

§35.5(i) Definitions

… ►

§35.5(ii) Properties

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§35.5(iii) Asymptotic Approximations

►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).

3: 35.6 Confluent Hypergeometric Functions of Matrix Argument

§35.6 Confluent Hypergeometric Functions of Matrix Argument

►

§35.6(i) Definitions

… ►

Laguerre Form

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§35.6(ii) Properties

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§35.6(iii) Relations to Bessel Functions of Matrix Argument

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4: 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.8 Generalized Hypergeometric Functions of Matrix Argument

►

§35.8(i) Definition

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Convergence Properties

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Confluence

… ►

Invariance

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5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

►

§35.7(i) Definition

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Jacobi Form

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Confluent Form

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Integral Representation

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►In multivariate statistical analysis based on the multivariate normal distribution, the probability density functions of many random matrices are expressible in terms of generalized hypergeometric functions of matrix argument

{{}_{p}F_{q}}

, with

p\leq 2

and

q\leq 1

. See James (1964), Muirhead (1982), Takemura (1984), Farrell (1985), and Chikuse (2003) for extensive treatments. … ►These references all use results related to the integral formulas (35.4.7) and (35.5.8). … ►In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. …

7: 35.10 Methods of Computation

§35.10 Methods of Computation

… ►See Yan (1992) for the

{{}_{1}F_{1}}

and

{{}_{2}F_{1}}

functions of matrix argument in the case

m=2

, and Bingham et al. (1992) for Monte Carlo simulation on

\mathbf{O}(m)

applied to a generalization of the integral (35.5.8). …

8: 35.2 Laplace Transform

§35.2 Laplace Transform

►

Definition

►For any complex symmetric matrix

\mathbf{Z}

, … ►

Inversion Formula

… ►

Convolution Theorem

…

9: 35 Functions of Matrix Argument

Chapter 35 Functions of Matrix Argument

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10: 18.36 Miscellaneous Polynomials

… ►

§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. Classes of such polynomials have been found that generalize the classical OP’s in the sense that they satisfy second-order matrix differential equations with coefficients independent of the degree. …