35 Functions of Matrix Argument Properties 35.7 Gaussian Hypergeometric Function of Matrix Argument 35.9 Applications

§35.8 Generalized Hypergeometric Functions of Matrix Argument

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Keywords:: generalized hypergeometric functions, of matrix argument
Referenced by:: §35.7(iii)
Permalink:: http://dlmf.nist.gov/35.8
See also:: Annotations for Ch.35

§35.8(i) Definition
§35.8(ii) Relations to Other Functions
§35.8(iii) ${{}_{3}F_{2}}$ Case
§35.8(iv) General Properties
§35.8(v) Mellin–Barnes Integrals

§35.8(i) Definition

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Keywords:: definition, expansion in zonal polynomials, generalized hypergeometric functions, of matrix argument
Permalink:: http://dlmf.nist.gov/35.8.i
See also:: Annotations for §35.8 and Ch.35

Let $p$ and $q$ be nonnegative integers; $a_{1},\dots,a_{p}\in\mathbb{C}$ ; $b_{1},\dots,b_{q}\in\mathbb{C}$ ; $-b_{j}+\tfrac{1}{2}(k+1)\notin\mathbb{N}$ , $1\leq j\leq q$ , $1\leq k\leq m$ . The generalized hypergeometric function ${{}_{p}F_{q}}$ with matrix argument $\mathbf{T}\in\boldsymbol{\mathcal{S}}$ , numerator parameters $a_{1},\dots,a_{p}$ , and denominator parameters $b_{1},\dots,b_{q}$ is

35.8.1		${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right% )=\sum_{k=0}^{\infty}\frac{1}{k!}\sum_{\|\kappa\|=k}\frac{{\left[a_{1}\right]_{% \kappa}}\cdots{\left[a_{p}\right]_{\kappa}}}{{\left[b_{1}\right]_{\kappa}}% \cdots{\left[b_{q}\right]_{\kappa}}}Z_{\kappa}\left(\mathbf{T}\right).$
ⓘ Defines: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ 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 Symbols: $!$ : factorial (as in $n!$ ), ${\left[\NVar{a}\right]_{\NVar{\kappa}}}$ : 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, $k$ : nonnegative integer, $p$ : nonnegative integer, $q$ : nonnegative integer and $\kappa$ : vector of nonnegative integers Referenced by: §35.10, §35.10, §35.8(i), §35.8(i), §35.8(i), §35.8(i) Permalink: http://dlmf.nist.gov/35.8.E1 Encodings: TeX, pMML, png See also: Annotations for §35.8(i), §35.8 and Ch.35

Convergence Properties

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Keywords:: convergence properties, generalized hypergeometric functions, of matrix argument
Notes:: See Gross and Richards (1987), Faraut and Korányi (1994, pp. 318–340). See also Herz (1955), James (1964), Gross and Richards (1991), Muirhead (1982, pp. 259–262).
See also:: Annotations for §35.8(i), §35.8 and Ch.35

If $-a_{j}+\tfrac{1}{2}(k+1)\in\mathbb{N}$ for some $j, k$ satisfying $1\leq j\leq p$ , $1\leq k\leq m$ , then the series expansion (35.8.1) terminates.

If $p\leq q$ , then (35.8.1) converges for all $\mathbf{T}$ .

If $p=q+1$ , then (35.8.1) converges absolutely for $\|\mathbf{T}\|<1$ and diverges for $\|\mathbf{T}\|>1$ .

If $p>q+1$ , then (35.8.1) diverges unless it terminates.

§35.8(ii) Relations to Other Functions

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Keywords:: generalized hypergeometric functions, of matrix argument, relations to other functions
Notes:: See Herz (1955), Muirhead (1982, pp. 259–262). See also James (1964), Gross and Richards (1987, 1991).
Permalink:: http://dlmf.nist.gov/35.8.ii
See also:: Annotations for §35.8 and Ch.35

35.8.2		${{}_{0}F_{0}}\left({-\atop-};\mathbf{T}\right)=\mathrm{etr}\left(\mathbf{T}% \right),$
$\mathbf{T}\in\boldsymbol{\mathcal{S}}$ .
ⓘ 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, $\mathrm{etr}\left(\NVar{\mathbf{X}}\right)$ : exponential of trace, $\boldsymbol{\mathcal{S}}$ : space of real symmetric $m\times m$ matrices and $\mathbf{T}$ : real symmetric $m\times m$ matrix Permalink: http://dlmf.nist.gov/35.8.E2 Encodings: TeX, pMML, png See also: Annotations for §35.8(ii), §35.8 and Ch.35

35.8.3		${{}_{2}F_{1}}\left({a,b\atop b};\mathbf{T}\right)={{}_{1}F_{0}}\left({a\atop-}% ;\mathbf{T}\right)=\left\|\mathbf{I}-\mathbf{T}\right\|^{-a},$
$\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ .
ⓘ 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, $a$ : 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, $<$ : less-than for matrices and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros Permalink: http://dlmf.nist.gov/35.8.E3 Encodings: TeX, pMML, png See also: Annotations for §35.8(ii), §35.8 and Ch.35

35.8.4		$A_{\nu}\left(\mathbf{T}\right)=\dfrac{1}{\Gamma_{m}\left(\nu+\frac{1}{2}(m+1)% \right)}{{}_{0}F_{1}}\left({-\atop\nu+\frac{1}{2}(m+1)};-\mathbf{T}\right),$
$\mathbf{T}\in\boldsymbol{\mathcal{S}}$ .
ⓘ Symbols: $A_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (first kind), ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ 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, $\boldsymbol{\mathcal{S}}$ : space of real symmetric $m\times m$ matrices, $\mathbf{T}$ : real symmetric $m\times m$ matrix and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.8.E4 Encodings: TeX, pMML, png See also: Annotations for §35.8(ii), §35.8 and Ch.35

§35.8(iii) ${{}_{3}F_{2}}$ Case

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Keywords:: generalized hypergeometric functions, of matrix argument, ${{}_{3}F_{2}}$ case
Notes:: See Gross and Richards (1991), Macdonald (1990).
Permalink:: http://dlmf.nist.gov/35.8.iii
See also:: Annotations for §35.8 and Ch.35

Kummer Transformation

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Keywords:: Kummer transformation, Kummer’s transformations, for ${{}_{3}F_{2}}$ hypergeometric functions of matrix argument, generalized hypergeometric functions, of matrix argument
See also:: Annotations for §35.8(iii), §35.8 and Ch.35

Let $c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}$ . Then

35.8.5		${{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{2}\right)\Gamma_{m}\left(c\right)}{\Gamma_{m}\left(b_% {2}-a_{3}\right)\Gamma_{m}\left(c+a_{3}\right)}\*{{}_{3}F_{2}}\left({b_{1}-a_{% 1},b_{1}-a_{2},a_{3}\atop b_{1},c+a_{3}};\mathbf{I}\right),$
$\Re\left(b_{2}\right),\Re\left(c\right)>\frac{1}{2}(m-1)$ .
ⓘ 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/35.8.E5 Encodings: TeX, pMML, png See also: Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

Pfaff–Saalschütz Formula

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Keywords:: ${{}_{3}F_{2}}$ functions of matrix argument, Pfaff–Saalschütz formula, generalized hypergeometric functions, of matrix argument
See also:: Annotations for §35.8(iii), §35.8 and Ch.35

Let $a_{1}+a_{2}+a_{3}+\frac{1}{2}(m+1)=b_{1}+b_{2}$ ; one of the $a_{j}$ be a negative integer; $\Re\left(b_{1}-a_{1}\right)$ , $\Re\left(b_{1}-a_{2}\right)$ , $\Re\left(b_{1}-a_{3}\right)$ , $\Re\left(b_{1}-a_{1}-a_{2}-a_{3}\right)>\frac{1}{2}(m-1)$ . Then

35.8.6		${{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{1}-a_{1}\right)\Gamma_{m}\left(b_{1}-a_{2}\right)}{% \Gamma_{m}\left(b_{1}\right)\Gamma_{m}\left(b_{1}-a_{1}-a_{2}\right)}\*\frac{% \Gamma_{m}\left(b_{1}-a_{3}\right)\Gamma_{m}\left(b_{1}-a_{1}-a_{2}-a_{3}% \right)}{\Gamma_{m}\left(b_{1}-a_{1}-a_{3}\right)\Gamma_{m}\left(b_{1}-a_{2}-a% _{3}\right)}.$
ⓘ 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, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $a$ : complex variable, $\mathbf{I}$ : $m\times m$ identity matrix, $b$ : complex variable and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.8.E6 Encodings: TeX, pMML, png See also: Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

Thomae Transformation

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Keywords:: ${{}_{3}F_{2}}$ functions of matrix argument, Thomae transformation, generalized hypergeometric functions, of matrix argument
See also:: Annotations for §35.8(iii), §35.8 and Ch.35

Again, let $c=b_{1}+b_{2}-a_{1}-a_{2}-a_{3}$ . Then

35.8.7		${{}_{3}F_{2}}\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=% \frac{\Gamma_{m}\left(b_{1}\right)\Gamma_{m}\left(b_{2}\right)\Gamma\left(c% \right)}{\Gamma_{m}\left(a_{1}\right)\Gamma_{m}\left(c+a_{2}\right)\Gamma\left% (c+a_{3}\right)}\*{{}_{3}F_{2}}\left({b_{1}-a_{1},b_{2}-a_{2},c\atop c+a_{2},c% +a_{3}};\mathbf{I}\right),$
$\Re\left(b_{1}\right)$ , $\Re\left(b_{2}\right)$ , $\Re\left(c\right)>\frac{1}{2}(m-1)$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{\mathbf{T}}\right)$ 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/35.8.E7 Encodings: TeX, pMML, png See also: Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

§35.8(iv) General Properties

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Keywords:: general properties, generalized hypergeometric functions, of matrix argument
Notes:: See Herz (1955), James (1964), Muirhead (1982, pp. 259–262), Gross and Richards (1987, 1991), Ding et al. (1996), Faraut and Korányi (1994, pp. 318–340).
Permalink:: http://dlmf.nist.gov/35.8.iv
See also:: Annotations for §35.8 and Ch.35

Value at $\mathbf{T}=\boldsymbol{{0}}$

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Keywords:: generalized hypergeometric functions, of matrix argument, value at $\mathbf{T}=\boldsymbol{{0}}$
See also:: Annotations for §35.8(iv), §35.8 and Ch.35

35.8.8		${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\boldsymbol{{0}}% \right)=1.$
ⓘ 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, $a$ : complex variable, $b$ : complex variable, $p$ : nonnegative integer, $q$ : nonnegative integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros Permalink: http://dlmf.nist.gov/35.8.E8 Encodings: TeX, pMML, png See also: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

Confluence

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Keywords:: confluence, generalized hypergeometric functions, of matrix argument
See also:: Annotations for §35.8(iv), §35.8 and Ch.35

35.8.9		$\lim_{\gamma\to\infty}{{}_{p+1}F_{q}}\left({a_{1},\dots,a_{p},\gamma\atop b_{1% },\dots,b_{q}};\gamma^{-1}\mathbf{T}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_% {p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right),$
ⓘ 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, $p$ : nonnegative integer and $q$ : nonnegative integer Referenced by: §35.7(iii) Permalink: http://dlmf.nist.gov/35.8.E9 Encodings: TeX, pMML, png See also: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

35.8.10		$\lim_{\gamma\to\infty}{{}_{p}F_{q+1}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots% ,b_{q},\gamma};\gamma\mathbf{T}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}% \atop b_{1},\dots,b_{q}};\mathbf{T}\right).$
ⓘ 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, $p$ : nonnegative integer and $q$ : nonnegative integer Referenced by: §35.7(iii) Permalink: http://dlmf.nist.gov/35.8.E10 Encodings: TeX, pMML, png See also: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

Invariance

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Keywords:: generalized hypergeometric functions, invariance, of matrix argument
See also:: Annotations for §35.8(iv), §35.8 and Ch.35

35.8.11		${{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{H}% \mathbf{T}\mathbf{H}^{-1}\right)={{}_{p}F_{q}}\left({a_{1},\dots,a_{p}\atop b_% {1},\dots,b_{q}};\mathbf{T}\right),$
$\mathbf{H}\in\mathbf{O}(m)$ .
ⓘ 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, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $b$ : complex variable, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $p$ : nonnegative integer, $q$ : nonnegative integer and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.8.E11 Encodings: TeX, pMML, png See also: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

Laplace Transform

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Keywords:: Laplace transform, generalized hypergeometric functions, of matrix argument
See also:: Annotations for §35.8(iv), §35.8 and Ch.35

35.8.12		${\int_{\boldsymbol{\Omega}}\mathrm{etr}\left(-\mathbf{T}\mathbf{X}\right)\left% \|\mathbf{X}\right\|^{\gamma-\frac{1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{1},\dots,% a_{p}\atop b_{1},\dots,b_{q}};-\mathbf{X}\right)\mathrm{d}{\mathbf{X}}}=\Gamma% _{m}\left(\gamma\right)\left\|\mathbf{T}\right\|^{-\gamma}{{}_{p+1}F_{q}}\left({% a_{1},\dots,a_{p},\gamma\atop b_{1},\dots,b_{q}};-\mathbf{T}^{-1}\right),$
$\Re\left(\gamma\right)>\frac{1}{2}(m-1)$ .
ⓘ 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, $\mathrm{etr}\left(\NVar{\mathbf{X}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $\Re$ : real part, $a$ : complex variable, $\mathbf{T}$ : 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, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable, $p$ : nonnegative integer, $q$ : nonnegative integer and $m$ : positive integer Permalink: http://dlmf.nist.gov/35.8.E12 Encodings: TeX, pMML, png See also: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

Euler Integral

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Keywords:: Euler integral, generalized hypergeometric functions, of matrix argument
See also:: Annotations for §35.8(iv), §35.8 and Ch.35

35.8.13		$\int\limits_{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}\left\|\mathbf{X}\right\|^{a% _{1}-\frac{1}{2}(m+1)}{\left\|\mathbf{I}-\mathbf{X}\right\|}^{b_{1}-a_{1}-\frac{% 1}{2}(m+1)}\*{{}_{p}F_{q}}\left({a_{2},\dots,a_{p+1}\atop b_{2},\dots,b_{q+1}}% ;\mathbf{T}\mathbf{X}\right)\mathrm{d}{\mathbf{X}}=\frac{1}{\mathrm{B}_{m}% \left(b_{1}-a_{1},a_{1}\right)}{{}_{p+1}F_{q+1}}\left({a_{1},\dots,a_{p+1}% \atop b_{1},\dots,b_{q+1}};\mathbf{T}\right),$
$\Re\left(b_{1}-a_{1}\right),\Re\left(a_{1}\right)>\frac{1}{2}(m-1)$ .
ⓘ 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, $\int$ : 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, $p$ : nonnegative integer, $q$ : nonnegative integer, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros Permalink: http://dlmf.nist.gov/35.8.E13 Encodings: TeX, pMML, png See also: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

§35.8(v) Mellin–Barnes Integrals

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Keywords:: Mellin–Barnes integrals, generalized hypergeometric functions, of matrix argument
Permalink:: http://dlmf.nist.gov/35.8.v
See also:: Annotations for §35.8 and Ch.35

Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions ${{}_{p}F_{q}}$ and ${{}_{p+1}F_{p}}$ of matrix argument. A similar result for the ${{}_{0}F_{1}}$ function of matrix argument is given in Faraut and Korányi (1994, p. 346). These multidimensional integrals reduce to the classical Mellin–Barnes integrals (§5.19(ii)) in the special case $m=1$ .

See also Faraut and Korányi (1994, pp. 318–340).

© 2010–2020 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.1.0; Release date 2020-12-15. A printed companion is available. 35.7 Gaussian Hypergeometric Function of Matrix Argument 35.9 Applications