35 Functions of Matrix Argument Properties 35.6 Confluent Hypergeometric Functions of Matrix Argument 35.8 Generalized Hypergeometric Functions of Matrix Argument

§35.7 Gaussian Hypergeometric Function of Matrix Argument

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Keywords:: Gaussian hypergeometric function, of matrix argument
Permalink:: http://dlmf.nist.gov/35.7
See also:: Annotations for Ch.35

§35.7(i) Definition

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Keywords:: Gaussian hypergeometric function, definition, of matrix argument
Notes:: See Herz (1955), Muirhead (1982, pp. 264–281, 290, 472), Faraut and Korányi (1994, pp. 337–340).
Permalink:: http://dlmf.nist.gov/35.7.i
See also:: Annotations for §35.7 and Ch.35

35.7.1		${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{\|\kappa\|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},$
$-c+\frac{1}{2}(j+1)\notin\mathbb{N}$ , $1\leq j\leq m$ ; $\\|\mathbf{T}\\|<1$ .
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\\|\mathbf{T}\\|$ : spectral norm of $\mathbf{T}$ , $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\kappa$ : vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.7.E1 Encodings: TeX, pMML, png See also: Annotations for §35.7(i), §35.7 and Ch.35

Jacobi Form

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Keywords:: Gaussian hypergeometric function, Jacobi form, of matrix argument
See also:: Annotations for §35.7(i), §35.7 and Ch.35

35.7.2		$P^{(\gamma,\delta)}_{\nu}\left(\mathbf{T}\right)=\frac{\Gamma_{m}\left(\gamma+% \nu+\frac{1}{2}(m+1)\right)}{\Gamma_{m}\left(\gamma+\frac{1}{2}(m+1)\right)}\*% {{}_{2}F_{1}}\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1% }{2}(m+1)};\mathbf{T}\right),$
$\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ ; $\gamma,\delta,\nu\in\mathbb{C}$ ; $\Re\left(\gamma\right)>-1$ .
ⓘ Defines: $P^{(\NVar{\gamma},\NVar{\delta})}_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Jacobi function of matrix argument Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\mathbb{C}$ : complex plane, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $<$ : less-than for matrices, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros Permalink: http://dlmf.nist.gov/35.7.E2 Encodings: TeX, pMML, png See also: Annotations for §35.7(i), §35.7(i), §35.7 and Ch.35

§35.7(ii) Basic Properties

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Keywords:: Gaussian hypergeometric function, basic properties, of matrix argument
Notes:: See Herz (1955), Muirhead (1982, pp. 264–281, 290, 472). For (35.7.8) see Zheng (1997). See also Koornwinder and Sprinkhuizen-Kuyper (1978), Macdonald (1990), Ding et al. (1996).
Permalink:: http://dlmf.nist.gov/35.7.ii
See also:: Annotations for §35.7 and Ch.35

Case $m=2$

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Keywords:: Gaussian hypergeometric function, case $m=2$ , of matrix argument
See also:: Annotations for §35.7(ii), §35.7 and Ch.35

35.7.3		${{}_{2}F_{1}}\left({a,b\atop c};\begin{bmatrix}t_{1}&0\\ 0&t_{2}\end{bmatrix}\right)=\sum_{k=0}^{\infty}\frac{{\left(a\right)_{k}}{% \left(c-a\right)_{k}}{\left(b\right)_{k}}{\left(c-b\right)_{k}}}{k!\,{\left(c% \right)_{2k}}{\left(c-\tfrac{1}{2}\right)_{k}}}\*(t_{1}t_{2})^{k}{{}_{2}F_{1}}% \left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer Referenced by: Erratum (V1.0.25) for Equation (35.7.3) Permalink: http://dlmf.nist.gov/35.7.E3 Encodings: TeX, pMML, png Notational change (effective with 1.0.25): Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${{}_{2}F_{1}}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF. See also: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

Confluent Form

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Keywords:: Gaussian hypergeometric function, confluent form, of matrix argument
See also:: Annotations for §35.7(ii), §35.7 and Ch.35

35.7.4		$\lim_{c\to\infty}{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}-c\mathbf{T}^{-1}% \right)=\left\|\mathbf{T}\right\|^{b}\Psi\left(b;b-a+\tfrac{1}{2}(m+1);\mathbf{T% }\right).$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind), $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $b$ : complex variable, $c$ : complex variable and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.7.E4 Encodings: TeX, pMML, png See also: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

Integral Representation

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Keywords:: Gaussian hypergeometric function, integral representation, of matrix argument
See also:: Annotations for §35.7(ii), §35.7 and Ch.35

35.7.5		${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathrm{B}_{m}\left% (a,c-a\right)}\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left\|% \mathbf{X}\right\|^{a-\frac{1}{2}(m+1)}\*{\left\|\mathbf{I}-\mathbf{X}\right\|}^{% c-a-\frac{1}{2}(m+1)}{\left\|\mathbf{I}-\mathbf{T}\mathbf{X}\right\|}^{-b}\,% \mathrm{d}{\mathbf{X}},$
$\Re\left(a\right),\Re\left(c-a\right)>\frac{1}{2}(m-1)$ , $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ .
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\int$ : integral, $\mathrm{B}_{\NVar{m}}\left(\NVar{a},\NVar{b}\right)$ : multivariate beta function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{X}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $\,\mathrm{d}$ : differential of matrix, $b$ : complex variable, $<$ : less-than for matrices, $c$ : complex variable, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros Referenced by: §35.7(iv) Permalink: http://dlmf.nist.gov/35.7.E5 Encodings: TeX, pMML, png See also: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

Transformations of Parameters

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Keywords:: Gaussian hypergeometric function, of matrix argument, transformations of parameters
See also:: Annotations for §35.7(ii), §35.7 and Ch.35

35.7.6		${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\left\|\mathbf{I}-\mathbf{T}% \right\|^{c-a-b}{{}_{2}F_{1}}\left({c-a,c-b\atop c};\mathbf{T}\right)=\left\|% \mathbf{I}-\mathbf{T}\right\|^{-a}{{}_{2}F_{1}}\left({a,c-b\atop c};-\mathbf{T}% (\mathbf{I}-\mathbf{T})^{-1}\right)=\left\|\mathbf{I}-\mathbf{T}\right\|^{-b}{{}% _{2}F_{1}}\left({c-a,b\atop c};-\mathbf{T}(\mathbf{I}-\mathbf{T})^{-1}\right).$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\left\|\NVar{\mathbf{X}}\right\|$ : determinant of $\NVar{\mathbf{X}}$ , $b$ : complex variable and $c$ : complex variable Permalink: http://dlmf.nist.gov/35.7.E6 Encodings: TeX, pMML, png See also: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

Gauss Formula

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Keywords:: Gauss formula, Gaussian hypergeometric function, of matrix argument
See also:: Annotations for §35.7(ii), §35.7 and Ch.35

35.7.7		${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}\right)=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(c-a-b\right)}{\Gamma_{m}\left(c-a\right)\Gamma_{m}\left% (c-b\right)}},$
$\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$ .
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $b$ : complex variable, $c$ : complex variable and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.7.E7 Encodings: TeX, pMML, png See also: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

Reflection Formula

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Keywords:: Gaussian hypergeometric function, of matrix argument, reflection formula
See also:: Annotations for §35.7(ii), §35.7 and Ch.35

35.7.8		${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(c-a-b\right)}{\Gamma_{m}\left(c-a\right)\Gamma_{m}\left% (c-b\right)}}\*{{}_{2}F_{1}}\left({a,b\atop a+b-c+\frac{1}{2}(m+1)};\mathbf{I}% -\mathbf{T}\right),$
$\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ ; ${\frac{1}{2}}(j+1)-a\in\mathbb{N}$ for some $j=1,\ldots,m$ ; ${\frac{1}{2}}(j+1)-c\notin\mathbb{N}$ and $c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}$ for all $j=1,\ldots,m$ .
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $\in$ : element of, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, $\Re$ : real part, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $c$ : complex variable, $j$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros Referenced by: §35.7(ii), Erratum (V1.0.26) for Equation (35.7.8) Permalink: http://dlmf.nist.gov/35.7.E8 Encodings: TeX, pMML, png Clarification (effective with 1.0.26): Originally the constraint was written as $\Re\left(c\right),\Re\left(c-a-b\right)>\frac{1}{2}(m-1)$ . The constraint has been replaced with $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ ; ${\frac{1}{2}}(j+1)-a\in\mathbb{N}$ for some $j=1,\ldots,m$ ; ${\frac{1}{2}}(j+1)-c\notin\mathbb{N}$ and $c-a-b-{\frac{1}{2}}(m-j)\notin\mathbb{N}$ for all $j=1,\ldots,m$ ; for details see https://arxiv.org/abs/2002.05248. See also: Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

§35.7(iii) Partial Differential Equations

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Keywords:: Gaussian hypergeometric function, functions of matrix argument, of matrix argument, orthogonal invariance, partial differential equations
Notes:: See Muirhead (1982, pp. 264–281, 290, 472), Faraut and Korányi (1994, pp. 337–340). See also Gross and Richards (1991).
Permalink:: http://dlmf.nist.gov/35.7.iii
See also:: Annotations for §35.7 and Ch.35

Let $f:{\boldsymbol{\Omega}}\to\mathbb{C}$ (a) be orthogonally invariant, so that $f(\mathbf{T})$ is a symmetric function of $t_{1},\dots,t_{m}$ , the eigenvalues of the matrix argument $\mathbf{T}\in{\boldsymbol{\Omega}}$ ; (b) be analytic in $t_{1},\dots,t_{m}$ in a neighborhood of $\mathbf{T}=\boldsymbol{{0}}$ ; (c) satisfy $f(\boldsymbol{{0}})=1$ . Subject to the conditions (a)–(c), the function $f(\mathbf{T})={{}_{2}F_{1}}\left(a,b;c;\mathbf{T}\right)$ is the unique solution of each partial differential equation

35.7.9		$t_{j}(1-t_{j})\frac{{\partial}^{2}F}{{\partial t_{j}}^{2}}-\frac{1}{2}\sum_{% \begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{m}\frac{t_{k}(1-t_{k})}{t_{j}-t_{k}}\frac{\partial F}{% \partial t_{k}}+\left({c-\tfrac{1}{2}(m-1)-\left(a+b-\tfrac{1}{2}(m-3)\right)t% _{j}}+\frac{1}{2}\sum_{\begin{subarray}{c}k=1\\ k\neq j\end{subarray}}^{m}\frac{t_{j}(1-t_{j})}{t_{j}-t_{k}}\right)\frac{% \partial F}{\partial t_{j}}=abF,$
ⓘ Symbols: $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $a$ : complex variable, $t_{j}$ : eigenvalues of $\mathbf{T}$ , $b$ : complex variable, $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer and $m$ : positive integer Referenced by: §35.7(iii) Permalink: http://dlmf.nist.gov/35.7.E9 Encodings: TeX, pMML, png See also: Annotations for §35.7(iii), §35.7 and Ch.35

for $j=1,\dots,m$ .

Systems of partial differential equations for the ${{}_{0}F_{1}}$ (defined in §35.8) and ${{}_{1}F_{1}}$ functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9).

§35.7(iv) Asymptotic Approximations

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Keywords:: Gaussian hypergeometric function, asymptotic approximations, of matrix argument
Referenced by:: §35.10, §35.9
Permalink:: http://dlmf.nist.gov/35.7.iv
See also:: Annotations for §35.7 and Ch.35

Butler and Wood (2002) applies Laplace’s method (§2.3(iii)) to (35.7.5) to derive uniform asymptotic approximations for the functions

35.7.10		${{}_{2}F_{1}}\left({\alpha a,\alpha b\atop\alpha c};\mathbf{T}\right)$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $c$ : complex variable and $\alpha\to\infty$ : parameter Permalink: http://dlmf.nist.gov/35.7.E10 Encodings: TeX, pMML, png See also: Annotations for §35.7(iv), §35.7 and Ch.35

and

35.7.11		${{}_{2}F_{1}}\left({a,b\atop c};\mathbf{I}-\alpha^{-1}\mathbf{T}\right)$
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $c$ : complex variable and $\alpha\to\infty$ : parameter Permalink: http://dlmf.nist.gov/35.7.E11 Encodings: TeX, pMML, png See also: Annotations for §35.7(iv), §35.7 and Ch.35

as $\alpha\to\infty$ . These approximations are in terms of elementary functions.

For other asymptotic approximations for Gaussian hypergeometric functions of matrix argument, see Herz (1955), Muirhead (1982, pp. 264–281, 290, 472, 563), and Butler and Wood (2002).

35.6 Confluent Hypergeometric Functions of Matrix Argument 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.7.1		${{{}_{2}F_{1}}\left({a,b\atop c};\mathbf{T}\right)=\sum_{k=0}^{\infty}\frac{1}% {k!}\sum_{\|\kappa\|=k}\frac{{\left[a\right]_{\kappa}}{\left[b\right]_{\kappa}}}% {{\left[c\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right)},$
$-c+\frac{1}{2}(j+1)\notin\mathbb{N}$ , $1\leq j\leq m$ ; $\\|\mathbf{T}\\|<1$ .
ⓘ Symbols: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{\mathbf{T}}\right)$ : generalized hypergeometric function of matrix argument, $!$ : factorial (as in $n!$ ), $\notin$ : not an element of, $\mathbb{N}$ : set of all positive integers, ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : partitional shifted factorial, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\\|\mathbf{T}\\|$ : spectral norm of $\mathbf{T}$ , $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer, $m$ : positive integer and $\kappa$ : vector of nonnegative integers Permalink: http://dlmf.nist.gov/35.7.E1 Encodings: TeX, pMML, png See also: Annotations for §35.7(i), §35.7 and Ch.35

§35.7 Gaussian Hypergeometric Function of Matrix Argument

Contents

§35.7(i) Definition

Jacobi Form

§35.7(ii) Basic Properties

Case $m=2$

Confluent Form

Integral Representation

Transformations of Parameters

Gauss Formula

Reflection Formula

§35.7(iii) Partial Differential Equations

§35.7(iv) Asymptotic Approximations

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§35.7 Gaussian Hypergeometric Function of Matrix Argument

Contents

§35.7(i) Definition

Jacobi Form

§35.7(ii) Basic Properties

Case m=2

Confluent Form

Integral Representation

Transformations of Parameters

Gauss Formula

Reflection Formula

§35.7(iii) Partial Differential Equations

§35.7(iv) Asymptotic Approximations

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Case $m=2$