§35.7 Gaussian Hypergeometric Function of Matrix Argument

Keywords:: Gaussian hypergeometric function
Permalink:: http://dlmf.nist.gov/35.7

§35.7(i) Definition
§35.7(ii) Basic Properties
§35.7(iii) Partial Differential Equations
§35.7(iv) Asymptotic Approximations

§35.7(i) Definition

Notes:: See Herz (1955), Muirhead (1982, pp. 264–281, 290, 472), Faraut and Korányi (1994, pp. 337–340).
Keywords:: Gaussian hypergeometric function
Permalink:: http://dlmf.nist.gov/35.7.i

35.7.1 ${\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};\mathbf{T}\right)=\sum _{{k=0}}^{\infty}\frac{1}{k!}\sum _{{|\kappa|=k}}\frac{\left[a\right]_{{\kappa}}\left[b\right]_{{\kappa}}}{\left[c\right]_{{\kappa}}}\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)},$ $-c+\frac{1}{2}(j+1)\notin\NatNumber$ , $1\leq j\leq m$ ; $||\mathbf{T}||<1$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $!$ : $n!$ : factorial, $\notin$ : not an element of, $\NatNumber$ : set of all positive integers, $\left[a\right]_{{\kappa}}$ : partitional shifted factorial, $\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right)$ : zonal polynomial, $a$ : complex variable, $b$ : complex variable, $c$ : complex variable, $j$ : nonnegative integer, $k$ : nonnegative integer and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.7.E1
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¶ Jacobi Form

Keywords:: Gaussian hypergeometric function

35.7.2 $\mathop{P^{{(\gamma,\delta)}}_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)=\frac{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(\gamma+\nu+\frac{1}{2}(m+1)\right)}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(\gamma+\frac{1}{2}(m+1)\right)}\*\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({-\nu,\gamma+\delta+\nu+\frac{1}{2}(m+1)\atop\gamma+\frac{1}{2}(m+1)};\mathbf{T}\right),$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ ; $\gamma,\delta,\nu\in\Complex$ ; $\realpart{(\gamma)}>-1$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{P^{{(\alpha,\beta)}}_{{n}}\/}\nolimits\!\left(x\right)$ : Jacobi polynomial, $\Complex$ : complex plane (excluding infinity), $\in$ : element of, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\realpart{}$ : real part and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.7.E2
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§35.7(ii) Basic Properties

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).
Keywords:: Gaussian hypergeometric function
Permalink:: http://dlmf.nist.gov/35.7.ii

¶ Case $m=2$

Keywords:: Gaussian hypergeometric function

35.7.3 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};\left(\begin{matrix}t_{1}&0\\ 0&t_{2}\end{matrix}\right)\right)=\sum _{{k=0}}^{\infty}\frac{\left(a\right)_{{k}}\left(c-a\right)_{{k}}\left(b\right)_{{k}}\left(c-b\right)_{{k}}}{k!\,\left(c\right)_{{2k}}\left(c-\tfrac{1}{2}\right)_{{k}}}\*(t_{1}t_{2})^{k}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a+k,b+k\atop c+2k};t_{1}+t_{2}-t_{1}t_{2}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $!$ : $n!$ : factorial, $a$ : complex variable, $t_{j}$ : eigenvalues, $b$ : complex variable, $c$ : complex variable and $k$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/35.7.E3
Encodings:: TeX, pMML, png

¶ Confluent Form

Keywords:: Gaussian hypergeometric function

35.7.4 $\lim _{{c\to\infty}}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};\mathbf{I}-c\mathbf{T}^{{-1}}\right)=|\mathbf{T}|^{b}\mathop{\Psi\/}\nolimits\!\left(b;b-a+\tfrac{1}{2}(m+1);\mathbf{T}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\Psi\/}\nolimits\!\left(a;b;\mathbf{T}\right)$ : confluent hypergeometric function of matrix argument (second kind), $a$ : complex variable, $b$ : complex variable, $c$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.7.E4
Encodings:: TeX, pMML, png

¶ Integral Representation

Keywords:: Gaussian hypergeometric function

35.7.5 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};\mathbf{T}\right)=\frac{1}{\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,c-a\right)}\int\limits _{{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}}|\mathbf{X}|^{{a-\frac{1}{2}(m+1)}}\*{|\mathbf{I}-\mathbf{X}|}^{{c-a-\frac{1}{2}(m+1)}}{|\mathbf{I}-\mathbf{T}\mathbf{X}|}^{{-b}}d\mathbf{X},$ $\realpart{(a)},\realpart{(c-a)}>\frac{1}{2}(m-1)$ , $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(a,b\right)$ : multivariate beta function, $\realpart{}$ : real part, $a$ : complex variable, $b$ : complex variable, $c$ : complex variable and $m$ : positive integer
Referenced by:: §35.7(iv)
Permalink:: http://dlmf.nist.gov/35.7.E5
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¶ Transformations of Parameters

Keywords:: Gaussian hypergeometric function

35.7.6 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};\mathbf{T}\right)=|\mathbf{I}-\mathbf{T}|^{{c-a-b}}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({c-a,c-b\atop c};\mathbf{T}\right)=|\mathbf{I}-\mathbf{T}|^{{-a}}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,c-b\atop c};-\mathbf{T}(\mathbf{I}-\mathbf{T})^{{-1}}\right)=|\mathbf{I}-\mathbf{T}|^{{-b}}\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({c-a,b\atop c};-\mathbf{T}(\mathbf{I}-\mathbf{T})^{{-1}}\right).$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a$ : complex variable, $b$ : complex variable and $c$ : complex variable
Permalink:: http://dlmf.nist.gov/35.7.E6
Encodings:: TeX, pMML, png

¶ Gauss Formula

Keywords:: Gaussian hypergeometric function

35.7.7 ${\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};\mathbf{I}\right)=\frac{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c-a\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c-b\right)}},$ $\realpart{(c)},\realpart{(c-a-b)}>\frac{1}{2}(m-1)$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\realpart{}$ : real part, $a$ : complex variable, $b$ : complex variable, $c$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.7.E7
Encodings:: TeX, pMML, png

¶ Reflection Formula

Keywords:: Gaussian hypergeometric function

35.7.8 ${\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};\mathbf{T}\right)=\frac{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c-a-b\right)}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c-a\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c-b\right)}}\*\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop a+b-c+\frac{1}{2}(m+1)};\mathbf{I}-\mathbf{T}\right),$ $\realpart{(c)},\realpart{(c-a-b)}>\frac{1}{2}(m-1)$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\realpart{}$ : real part, $a$ : complex variable, $b$ : complex variable, $c$ : complex variable and $m$ : positive integer
Referenced by:: §35.7(ii)
Permalink:: http://dlmf.nist.gov/35.7.E8
Encodings:: TeX, pMML, png

§35.7(iii) 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).
Keywords:: Gaussian hypergeometric function, functions of matrix argument, partial differential equations
Permalink:: http://dlmf.nist.gov/35.7.iii

Let $f:{\boldsymbol{\Omega}}\to\Complex$ (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 $f(\mathbf{T})=\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left(a,b;c;\mathbf{T}\right)$ is the unique solution of each partial differential equation

35.7.9 $t_{j}(1-t_{j})\frac{{\partial}^{2}F}{{\partial t_{j}}^{2}}-\frac{1}{2}\sum _{{\substack{k=1\\ k\neq j}}}^{m}\frac{t_{k}(1-t_{k})}{t_{j}-t_{k}}\frac{\partial F}{\partial t_{k}}+\left({c-\tfrac{1}{2}(m-1)-\left(a+b-\tfrac{1}{2}(m-3)\right)t_{j}}+\frac{1}{2}\sum _{{\substack{k=1\\ k\neq j}}}^{m}\frac{t_{j}(1-t_{j})}{t_{j}-t_{k}}\right)\frac{\partial F}{\partial t_{j}}=abF,$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $a$ : complex variable, $t_{j}$ : eigenvalues, $b$ : 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

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

Systems of partial differential equations for the $\mathop{{{}_{{0}}F_{{1}}}\/}\nolimits$ (defined in §35.8) and $\mathop{{{}_{{1}}F_{{1}}}\/}\nolimits$ 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

Keywords:: Gaussian hypergeometric function
Referenced by:: §35.10, §35.9
Permalink:: http://dlmf.nist.gov/35.7.iv

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 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({\alpha a,\alpha b\atop\alpha c};\mathbf{T}\right)$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a$ : complex variable, $b$ : complex variable, $c$ : complex variable and $\alpha\to\infty$ : parameter
Permalink:: http://dlmf.nist.gov/35.7.E10
Encodings:: TeX, pMML, png

and

35.7.11 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop c};\mathbf{I}-\alpha^{{-1}}\mathbf{T}\right)$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a$ : complex variable, $b$ : complex variable, $c$ : complex variable and $\alpha\to\infty$ : parameter
Permalink:: http://dlmf.nist.gov/35.7.E11
Encodings:: TeX, pMML, png

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).