§35.8 Generalized Hypergeometric Functions of Matrix Argument

Keywords:: generalized hypergeometric functions
Referenced by:: §35.7(iii)
Permalink:: http://dlmf.nist.gov/35.8

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

§35.8(i) Definition

Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.8.i

Let $p$ and $q$ be nonnegative integers; $a_{1},\dots,a_{p}\in\Complex$ ; $b_{1},\dots,b_{q}\in\Complex$ ; $-b_{j}+\tfrac{1}{2}(k+1)\notin\NatNumber$ , $1\leq j\leq q$ , $1\leq k\leq m$ . The generalized hypergeometric function $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits$ 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 $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right)=\sum _{{k=0}}^{\infty}\frac{1}{k!}\sum _{{|\kappa|=k}}\frac{\left[a_{1}\right]_{{\kappa}}\cdots\left[a_{p}\right]_{{\kappa}}}{\left[b_{1}\right]_{{\kappa}}\cdots\left[b_{q}\right]_{{\kappa}}}\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}\right).$

¶ Convergence Properties

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).
Keywords:: generalized hypergeometric functions

If $-a_{j}+\tfrac{1}{2}(k+1)\in\NatNumber$ 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

Notes:: See Herz (1955), Muirhead (1982, pp. 259–262). See also James (1964), Gross and Richards (1987, 1991).
Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.8.ii

35.8.2 $\mathop{{{}_{{0}}F_{{0}}}\/}\nolimits\!\left({-\atop-};\mathbf{T}\right)=\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{T}\right),$ $\mathbf{T}\in\boldsymbol{\mathcal{S}}$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\in$ : element of, $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace and $\boldsymbol{\mathcal{S}}$ : space of real symmetric matrices
Permalink:: http://dlmf.nist.gov/35.8.E2
Encodings:: TeX, pMML, png

35.8.3 $\mathop{{{}_{{2}}F_{{1}}}\/}\nolimits\!\left({a,b\atop b};\mathbf{T}\right)=\mathop{{{}_{{1}}F_{{0}}}\/}\nolimits\!\left({a\atop-};\mathbf{T}\right)=|\mathbf{I}-\mathbf{T}|^{{-a}},$ $\boldsymbol{{0}}<\mathbf{T}<\mathbf{I}$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $a$ : complex variable and $b$ : complex variable
Permalink:: http://dlmf.nist.gov/35.8.E3
Encodings:: TeX, pMML, png

35.8.4 $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)=\dfrac{1}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(\nu+\frac{1}{2}(m+1)\right)}\mathop{{{}_{{0}}F_{{1}}}\/}\nolimits\!\left({-\atop\nu+\frac{1}{2}(m+1)};-\mathbf{T}\right),$ $\mathbf{T}\in\boldsymbol{\mathcal{S}}$ .

Symbols:: $\mathop{A_{{\nu}}\/}\nolimits\!\left(\mathbf{T}\right)$ : Bessel function of matrix argument (first kind), $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\in$ : element of, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $m$ : positive integer and $\boldsymbol{\mathcal{S}}$ : space of real symmetric matrices
Permalink:: http://dlmf.nist.gov/35.8.E4
Encodings:: TeX, pMML, png

§35.8(iii) $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits$ Case

Notes:: See Gross and Richards (1991), Macdonald (1990).
Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.8.iii

¶ Kummer Transformation

Keywords:: Kummer’s transformations, generalized hypergeometric functions

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

35.8.5 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=\frac{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{2}\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c\right)}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{2}-a_{3}\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c+a_{3}\right)}\*\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({b_{1}-a_{1},b_{1}-a_{2},a_{3}\atop b_{1},c+a_{3}};\mathbf{I}\right),$ $\realpart{(b_{2})},\realpart{(c)}>\frac{1}{2}(m-1)$ .

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/35.8.E5
Encodings:: TeX, pMML, png

¶ Pfaff–Saalschutz Formula

Keywords:: Pfaff–Saalschutz formula, generalized hypergeometric functions

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; $\realpart{(b_{1}-a_{1})}$ , $\realpart{(b_{1}-a_{2})}$ , $\realpart{(b_{1}-a_{3})}$ , $\realpart{(b_{1}-a_{1}-a_{2}-a_{3})}>\frac{1}{2}(m-1)$ . Then

35.8.6 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=\frac{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{1}-a_{1}\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{1}-a_{2}\right)}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{1}\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{1}-a_{1}-a_{2}\right)}\*\frac{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{1}-a_{3}\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{1}-a_{1}-a_{2}-a_{3}\right)}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{1}-a_{1}-a_{3}\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{1}-a_{2}-a_{3}\right)}.$

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $a$ : complex variable, $b$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.8.E6
Encodings:: TeX, pMML, png

¶ Thomae Transformation

Keywords:: Thomae transformation, generalized hypergeometric functions

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

35.8.7 $\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({a_{1},a_{2},a_{3}\atop b_{1},b_{2}};\mathbf{I}\right)=\frac{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{1}\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(b_{2}\right)\mathop{\Gamma\/}\nolimits\!\left(c\right)}{\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a_{1}\right)\mathop{\Gamma _{{m}}\/}\nolimits\!\left(c+a_{2}\right)\mathop{\Gamma\/}\nolimits\!\left(c+a_{3}\right)}\*\mathop{{{}_{{3}}F_{{2}}}\/}\nolimits\!\left({b_{1}-a_{1},b_{2}-a_{2},c\atop c+a_{2},c+a_{3}};\mathbf{I}\right),$ $\realpart{(b_{1})}$ , $\realpart{(b_{2})}$ , $\realpart{(c)}>\frac{1}{2}(m-1)$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : 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/35.8.E7
Encodings:: TeX, pMML, png

§35.8(iv) General Properties

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).
Keywords:: generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.8.iv

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

Keywords:: generalized hypergeometric functions

35.8.8 $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\boldsymbol{{0}}\right)=1.$

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, $p$ : nonnegative integer and $q$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/35.8.E8
Encodings:: TeX, pMML, png

¶ Confluence

Keywords:: generalized hypergeometric functions

35.8.9 $\lim _{{\gamma\to\infty}}\mathop{{{}_{{p+1}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p},\gamma\atop b_{1},\dots,b_{q}};\gamma^{{-1}}\mathbf{T}\right)=\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right),$

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, $p$ : nonnegative integer and $q$ : nonnegative integer
Referenced by:: §35.7(iii)
Permalink:: http://dlmf.nist.gov/35.8.E9
Encodings:: TeX, pMML, png

35.8.10 $\lim _{{\gamma\to\infty}}\mathop{{{}_{{p}}F_{{q+1}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q},\gamma};\gamma\mathbf{T}\right)=\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right).$

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, $p$ : nonnegative integer and $q$ : nonnegative integer
Referenced by:: §35.7(iii)
Permalink:: http://dlmf.nist.gov/35.8.E10
Encodings:: TeX, pMML, png

¶ Invariance

Keywords:: generalized hypergeometric functions

35.8.11 $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{H}\mathbf{T}\mathbf{H}^{{-1}}\right)=\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};\mathbf{T}\right),$ $\mathbf{H}\in\mathbf{O}(m)$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $\in$ : element of, $a$ : complex variable, $\mathbf{O}(m)$ : space of orthogonal matrices, $b$ : complex variable, $m$ : positive integer, $p$ : nonnegative integer and $q$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/35.8.E11
Encodings:: TeX, pMML, png

¶ Laplace Transform

Keywords:: Laplace transform, generalized hypergeometric functions

35.8.12 ${\int _{{\boldsymbol{\Omega}}}\mathop{\mathrm{etr}\/}\nolimits\!\left(-\mathbf{T}\mathbf{X}\right)|\mathbf{X}|^{{\gamma-\frac{1}{2}(m+1)}}\*\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};-\mathbf{X}\right)d\mathbf{X}}=\mathop{\Gamma _{{m}}\/}\nolimits\!\left(\gamma\right)|\mathbf{T}|^{{-\gamma}}\mathop{{{}_{{p+1}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p},\gamma\atop b_{1},\dots,b_{q}};-\mathbf{T}^{{-1}}\right),$ $\realpart{(\gamma)}>\frac{1}{2}(m-1)$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $dx$ : differential of $x$ , $\mathop{\mathrm{etr}\/}\nolimits\!\left(\mathbf{X}\right)$ : exponential of trace, $\int$ : integral, $\mathop{\Gamma _{{m}}\/}\nolimits\!\left(a\right)$ : multivariate gamma function, $\realpart{}$ : real part, $a$ : complex variable, ${\boldsymbol{\Omega}}$ : space of matrices, $b$ : complex variable, $m$ : positive integer, $p$ : nonnegative integer and $q$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/35.8.E12
Encodings:: TeX, pMML, png

¶ Euler Integral

Keywords:: generalized hypergeometric functions

35.8.13 $\int\limits _{{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}}|\mathbf{X}|^{{a_{1}-\frac{1}{2}(m+1)}}{|\mathbf{I}-\mathbf{X}|}^{{b_{1}-a_{1}-\frac{1}{2}(m+1)}}\*\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{2},\dots,a_{{p+1}}\atop b_{2},\dots,b_{{q+1}}};\mathbf{T}\mathbf{X}\right)d\mathbf{X}=\frac{1}{\mathop{\mathrm{B}_{{m}}\/}\nolimits\!\left(b_{1}-a_{1},a_{1}\right)}\mathop{{{}_{{p+1}}F_{{q+1}}}\/}\nolimits\!\left({a_{1},\dots,a_{{p+1}}\atop b_{1},\dots,b_{{q+1}}};\mathbf{T}\right),$ $\realpart{(b_{1}-a_{1})},\realpart{(a_{1})}>\frac{1}{2}(m-1)$ .

Symbols:: $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits\!\left({a_{1},\dots,a_{p}\atop b_{1},\dots,b_{q}};z\right)$ : generalized hypergeometric function, $dx$ : 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, $m$ : positive integer, $p$ : nonnegative integer and $q$ : nonnegative integer
Permalink:: http://dlmf.nist.gov/35.8.E13
Encodings:: TeX, pMML, png

§35.8(v) Mellin–Barnes Integrals

Keywords:: Mellin–Barnes integrals, generalized hypergeometric functions
Permalink:: http://dlmf.nist.gov/35.8.v

Multidimensional Mellin–Barnes integrals are established in Ding et al. (1996) for the functions $\mathop{{{}_{{p}}F_{{q}}}\/}\nolimits$ and $\mathop{{{}_{{p+1}}F_{{p}}}\/}\nolimits$ of matrix argument. A similar result for the $\mathop{{{}_{{0}}F_{{1}}}\/}\nolimits$ 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).