multivariate hypergeometric function

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11: 19.20 Special Cases

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19.20.3 $$R_F(x,a,y)=R_{-\frac{1}{4}}(\frac{3}{4},\frac{1}{2};a^2,xy),$$ $a=\frac{1}{2}(x+y)$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Referenced by:

§19.20(i)

Permalink:

http://dlmf.nist.gov/19.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(i), §19.20 and Ch.19

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19.20.23 $$R_D(x,y,a)=R_{-\frac{3}{4}}(\frac{5}{4},\frac{1}{2};a^2,xy),$$ $a=\frac{1}{2}x+\frac{1}{2}y$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and $R_D(x,y,z)$: elliptic integral symmetric in only two variables

Referenced by:

§19.20(iv)

Permalink:

http://dlmf.nist.gov/19.20.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(iv), §19.20 and Ch.19

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19.20.25 $$R_{-c}(𝐛;𝐳)=\prod _{j=1}^n{z}_j^{-b_j},$$

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and $n$: nonnegative integer

Referenced by:

§19.18(ii), §19.21(ii)

Permalink:

http://dlmf.nist.gov/19.20.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(v), §19.20 and Ch.19

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19.20.26 $$R_{-a}(𝐛;𝐳)=\prod _{j=1}^n{z}_j^{-b_j}{R}_{-a^{\prime }}(𝐛;{𝒛}^{\mathbf{-}\mathrm{𝟏}}),$$ $a+a^{\prime }=c$, ${𝒛}^{\mathbf{-}\mathrm{𝟏}}=(z_1^{-1},\mathrm{\dots },z_n^{-1})$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/19.20.E26

Encodings:

pMML, png, TeX

See also:

Annotations for §19.20(v), §19.20 and Ch.19

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12: 19.25 Relations to Other Functions

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19.25.3 $$\mathrm{\Pi }({\alpha }^2,k)=\frac{1}{2}\pi R_{-\frac{1}{2}}(\frac{1}{2},-\frac{1}{2},1;k_{}^{\prime }{}_{}{}^2,1,1-{\alpha }^2),$$

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Pi }({\alpha }^2,k)$: Legendre’s complete elliptic integral of the third kind, $k$: real or complex modulus, $k^{\prime }$: complementary modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.8 $$E(\varphi ,k)=R_{-\frac{1}{2}}(\frac{1}{2},-\frac{1}{2},\frac{3}{2};c-1,c-k^2,c),$$

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.15 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=R_{-\frac{1}{2}}(\frac{1}{2},\frac{1}{2},-\frac{1}{2},1;c-1,c-k^2,c,c-{\alpha }^2).$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.25(i)

Permalink:

http://dlmf.nist.gov/19.25.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(i), §19.25 and Ch.19

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19.25.42 $${}_2{}^{}F_1^{}(a,b;c;z)=R_{-a}(b,c-b;1-z,1),$$

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Referenced by:

§19.20(i), §19.20(iv)

Permalink:

http://dlmf.nist.gov/19.25.E42

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vii), §19.25 and Ch.19

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19.25.43 $$R_{-a}(b_1,b_2;z_1,z_2)=z_2^{-a}{}_2{}^{}F_1^{}(a,b_1;b_1+b_2;1-(z_1/z_2)).$$

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function and ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function

Referenced by:

§19.19

Permalink:

http://dlmf.nist.gov/19.25.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §19.25(vii), §19.25 and Ch.19

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13: 19.28 Integrals of Elliptic Integrals

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19.28.4 $${\int }_0^1{t}^{\sigma -1}{(1-t)}^{c-1}{R}_{-a}(b_1,b_2;t,1)dt=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(\sigma \right)\mathrm{\Gamma }\left(\sigma +b_2-a\right)}{\mathrm{\Gamma }\left(\sigma +c-a\right)\mathrm{\Gamma }\left(\sigma +b_2\right)},$$ $c=b_1+b_2>0$, $\mathrm{\Re }\sigma >\max (0,a-b_2)$.

ⓘ

Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sigma$: parameter

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

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14: 19.24 Inequalities

… ►The condition $y\le z$ for (19.24.1) and (19.24.2) serves only to identify $y$ as the smaller of the two nonzero variables of a symmetric function; it does not restrict validity. ►

19.24.1 $$\ln 4\le \sqrt{z}{R}_F(0,y,z)+\ln \sqrt{y/z}\le \frac{1}{2}\pi ,$$ ,

ⓘ

Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter and $\ln z$: principal branch of logarithm function

Referenced by:

§19.24(i)

Permalink:

http://dlmf.nist.gov/19.24.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.24(i), §19.24 and Ch.19

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19.24.15 $$R_C(x,\frac{1}{2}(y+z))\le R_F(x,y,z)\le R_C(x,\sqrt{yz}),$$ $x\ge 0$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $R_F(x,y,z)$: symmetric elliptic integral of first kind

Permalink:

http://dlmf.nist.gov/19.24.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §19.24(ii), §19.24(ii), §19.24 and Ch.19

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15: 35.9 Applications

§35.9 Applications

►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 $p\le 2$ and $q\le 1$. … ►For other statistical applications of ${}_p{}^{}F_q^{}$ functions of matrix argument see Perlman and Olkin (1980), Groeneboom and Truax (2000), Bhaumik and Sarkar (2002), Richards (2004) (monotonicity of power functions of multivariate statistical test criteria), Bingham et al. (1992) (Procrustes analysis), and Phillips (1986) (exact distributions of statistical test criteria). …

16: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.4 $$A_{\nu }\left(𝐓\right)=\frac{1}{{\mathrm{\Gamma }}_m\left(\nu +\frac{1}{2}(m+1)\right)}{}_0{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-}{\nu +\frac{1}{2}(m+1)};-𝐓),$$ $𝐓\in 𝓢$.

ⓘ

Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(ii), §35.8 and Ch.35

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35.8.5 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a_1,a_2,a_3}{b_1,b_2};𝐈)=\frac{{\mathrm{\Gamma }}_m\left(b_2\right){\mathrm{\Gamma }}_m\left(c\right)}{{\mathrm{\Gamma }}_m\left(b_2-a_3\right){\mathrm{\Gamma }}_m\left(c+a_3\right)}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{b_1-a_1,b_1-a_2,a_3}{b_1,c+a_3};𝐈),$$ $\mathrm{\Re }\left(b_2\right),\mathrm{\Re }\left(c\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

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35.8.6 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a_1,a_2,a_3}{b_1,b_2};𝐈)=\frac{{\mathrm{\Gamma }}_m\left(b_1-a_1\right){\mathrm{\Gamma }}_m\left(b_1-a_2\right)}{{\mathrm{\Gamma }}_m\left(b_1\right){\mathrm{\Gamma }}_m\left(b_1-a_1-a_2\right)}\frac{{\mathrm{\Gamma }}_m\left(b_1-a_3\right){\mathrm{\Gamma }}_m\left(b_1-a_1-a_2-a_3\right)}{{\mathrm{\Gamma }}_m\left(b_1-a_1-a_3\right){\mathrm{\Gamma }}_m\left(b_1-a_2-a_3\right)}.$$

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

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35.8.7 $${}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{a_1,a_2,a_3}{b_1,b_2};𝐈)=\frac{{\mathrm{\Gamma }}_m\left(b_1\right){\mathrm{\Gamma }}_m\left(b_2\right)\mathrm{\Gamma }\left(c\right)}{{\mathrm{\Gamma }}_m\left(a_1\right){\mathrm{\Gamma }}_m\left(c+a_2\right)\mathrm{\Gamma }\left(c+a_3\right)}{}_3{}^{}F_2^{}(\genfrac{}{}{0pt}{}{b_1-a_1,b_2-a_2,c}{c+a_2,c+a_3};𝐈),$$ $\mathrm{\Re }\left(b_1\right)$, $\mathrm{\Re }\left(b_2\right)$, $\mathrm{\Re }\left(c\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.8.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(iii), §35.8(iii), §35.8 and Ch.35

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35.8.13 $\mathrm{\Re }\left(b_1-a_1\right),\mathrm{\Re }\left(a_1\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $p$: nonnegative integer, $q$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.8.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

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17: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.3 $$L_{\nu }^{(\gamma )}\left(𝐓\right)=\frac{{\mathrm{\Gamma }}_m\left(\gamma +\nu +\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(\gamma +\frac{1}{2}(m+1)\right)}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\nu }{\gamma +\frac{1}{2}(m+1)};𝐓),$$ $\mathrm{\Re }\left(\gamma \right),\mathrm{\Re }\left(\gamma +\nu \right)>-1$.

ⓘ

Defines:

$L_{\nu }^{(\gamma )}\left(𝐓\right)$: Laguerre function of matrix argument

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐓$: real symmetric $m\times m$ matrix and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(i), §35.6(i), §35.6 and Ch.35

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35.6.4 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b-a\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

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35.6.6 $\mathrm{\Re }\left(b_1\right),\mathrm{\Re }\left(b_2\right)>\frac{1}{2}(m-1)$.

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.6.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

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35.6.8 $${\int }_{𝛀}{\left|𝐓\right|}^{c-\frac{1}{2}(m+1)}\mathrm{\Psi }(a;b;𝐓)d𝐓=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(a-c\right){\mathrm{\Gamma }}_m\left(c-b+\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(a\right){\mathrm{\Gamma }}_m\left(a-b+\frac{1}{2}(m+1)\right)},$$ $\mathrm{\Re }\left(a\right)>\mathrm{\Re }\left(c\right)+\frac{1}{2}(m-1)>m-1$, $\mathrm{\Re }\left(c-b\right)>-1$.

ⓘ

Symbols:

$\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind), $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(ii), §35.6 and Ch.35

… ►

35.6.10 $$\underset{a\to \mathrm{\infty }}{lim}{\mathrm{\Gamma }}_m\left(a\right)\mathrm{\Psi }(a+\nu ;\nu +\frac{1}{2}(m+1);a^{-1}𝐓)=B_{\nu }\left(𝐓\right).$$

ⓘ

Symbols:

$B_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (second kind), $\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind), ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(iii), §35.6 and Ch.35

…

18: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.2 $$P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)=\frac{{\mathrm{\Gamma }}_m\left(\gamma +\nu +\frac{1}{2}(m+1)\right)}{{\mathrm{\Gamma }}_m\left(\gamma +\frac{1}{2}(m+1)\right)}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{-\nu ,\gamma +\delta +\nu +\frac{1}{2}(m+1)}{\gamma +\frac{1}{2}(m+1)};𝐓),$$ ; $\gamma ,\delta ,\nu \in ℂ$; $\mathrm{\Re }\left(\gamma \right)>-1$.

ⓘ

Defines:

$P_{\nu }^{(\gamma ,\delta )}\left(𝐓\right)$: Jacobi function of matrix argument

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $ℂ$: complex plane, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, : less-than for matrices, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Permalink:

http://dlmf.nist.gov/35.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(i), §35.7(i), §35.7 and Ch.35

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35.7.5 $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(c-a\right)>\frac{1}{2}(m-1)$, .

ⓘ

Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $b$: complex variable, : less-than for matrices, $c$: complex variable, $m$: positive integer and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.7(iv)

Permalink:

http://dlmf.nist.gov/35.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.7 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈)=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(c-a-b\right)}{{\mathrm{\Gamma }}_m\left(c-a\right){\mathrm{\Gamma }}_m\left(c-b\right)},$$ $\mathrm{\Re }\left(c\right),\mathrm{\Re }\left(c-a-b\right)>\frac{1}{2}(m-1)$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.7.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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35.7.8 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\frac{{\mathrm{\Gamma }}_m\left(c\right){\mathrm{\Gamma }}_m\left(c-a-b\right)}{{\mathrm{\Gamma }}_m\left(c-a\right){\mathrm{\Gamma }}_m\left(c-b\right)}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{a+b-c+\frac{1}{2}(m+1)};𝐈-𝐓),$$ ; $\frac{1}{2}(j+1)-a\in \mathbb{N}$ for some $j=1,\mathrm{\dots },m$; $\frac{1}{2}(j+1)-c\notin \mathbb{N}$ and $c-a-b-\frac{1}{2}(m-j)\notin \mathbb{N}$ for all $j=1,\mathrm{\dots },m$.

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $\in$: element of, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable, $j$: nonnegative integer, $m$: positive integer and $𝟎$: $m\times m$ 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:

pMML, png, TeX

Clarification (effective with 1.0.26):

Originally the constraint was written as $\mathrm{\Re }\left(c\right),\mathrm{\Re }\left(c-a-b\right)>\frac{1}{2}(m-1)$. The constraint has been replaced with ; $\frac{1}{2}(j+1)-a\in \mathbb{N}$ for some $j=1,\mathrm{\dots },m$; $\frac{1}{2}(j+1)-c\notin \mathbb{N}$ and $c-a-b-\frac{1}{2}(m-j)\notin \mathbb{N}$ for all $j=1,\mathrm{\dots },m$; for details see https://arxiv.org/abs/2002.05248.

See also:

Annotations for §35.7(ii), §35.7(ii), §35.7 and Ch.35

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19: Donald St. P. Richards

… ►Richards has published numerous papers on special functions of matrix argument, harmonic analysis, multivariate statistical analysis, probability inequalities, and applied probability. He is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T. … ►

35 Functions of Matrix Argument

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20: Bibliography F

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Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§15.12(iii), §15.12(iii).

Encodings:

BibTeX

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Multivariate Calculation. Use of the Continuous Groups. Springer Series in Statistics, Springer-Verlag, New York.

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External Links:

ISBN 0-387-96049-X, MathReview (Yasuko Chikuse), ZentralBlatt

Referenced by:

§35.9.

Encodings:

BibTeX

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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II. J. Math. Anal. Appl. 7 (3), pp. 440–451.

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External Links:

Document, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

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Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15 (76), pp. 390–395.

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External Links:

FullText, MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.10.

Encodings:

BibTeX

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Computing the hypergeometric function. J. Comput. Phys. 137 (1), pp. 79–100.

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Notes:

Double-precision Fortran

External Links:

ISSN 0021-9991, Document, Code, MathReview (J. P. Singhal), ZentralBlatt

Referenced by:

§15.19(i), 1st item, 2nd item.

Encodings:

BibTeX

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