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21: 16.6 Transformations of Variable

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16.6.1

{{}_{3}F_{2}}\left({a,b,c\atop a-b+1,a-c+1};z\right)=(1-z)^{-a}{{}_{3}F_{2}}% \left({a-b-c+1,\frac{1}{2}a,\frac{1}{2}(a+1)\atop a-b+1,a-c+1};\frac{-4z}{(1-z% )^{2}}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

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16.6.2

{{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).

ⓘ

Symbols:: ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{a_{1},\dots,a_{p}};\NVar{b_{1},\dots,b_% {q}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{a_{1},\dots,a_{p}}\atop\NVar{b_{1},% \dots,b_{q}}};\NVar{z}\right)$ : alternatively ${{}_{\NVar{p}}F_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}};\NVar{z}\right)$ or ${{}_{\NVar{p}}F_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{\mathbf{b}}};% \NVar{z}\right)$
generalized hypergeometric function, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Referenced by:: §16.6
Permalink:: http://dlmf.nist.gov/16.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.6, §16.6 and Ch.16

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22: 16.21 Differential Equation

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16.21.1

\left((-1)^{p-m-n}z(\vartheta-a_{1}+1)\cdots(\vartheta-a_{p}+1)-(\vartheta-b_{% 1})\cdots(\vartheta-b_{q})\right)w=0,

ⓘ

Symbols:: $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters, $b,b_{1},\ldots,b_{q}$ : real or complex parameters, $w$ : solution and $\vartheta$ : differential operator
Referenced by:: §16.21, §16.21
Permalink:: http://dlmf.nist.gov/16.21.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §16.21 and Ch.16

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16.21.2

{G^{1,p}_{p,q}}\left(z{\mathrm{e}}^{(p-m-n-1)\pi\mathrm{i}};{a_{1},\dots,a_{p}% \atop b_{j},b_{1},\dots,b_{j-1},b_{j+1},\dots,b_{q}}\right),

j=1,\dots,q

.

ⓘ

Symbols:: ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};{\NVar{a_{1},\dots,a% _{p}}\atop\NVar{b_{1},\dots,b_{q}}}\right)$ or ${G^{\NVar{m},\NVar{n}}_{\NVar{p},\NVar{q}}}\left(\NVar{z};\NVar{\mathbf{a}};% \NVar{\mathbf{b}}\right)$ : Meijer $G$ -function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $p$ : nonnegative integer, $q$ : nonnegative integer, $z$ : complex variable, $a,a_{1},\ldots,a_{p}$ : real or complex parameters and $b,b_{1},\ldots,b_{q}$ : real or complex parameters
Permalink:: http://dlmf.nist.gov/16.21.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §16.21 and Ch.16

…

23: 15.16 Products

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15.16.1

F\left({a,b\atop c-\frac{1}{2}};z\right)F\left({c-a,c-b\atop c+\frac{1}{2}};z% \right)=\sum_{s=0}^{\infty}\frac{{\left(c\right)_{s}}}{{\left(c+\frac{1}{2}% \right)_{s}}}A_{s}z^{s},

|z|<1

,

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $A_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/15.16.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

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15.16.2

(1-z)^{a+b-c}F\left(2a,2b;2c-1;z\right)=\sum_{s=0}^{\infty}A_{s}z^{s},

|z|<1

.

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $A_{s}$ : coefficients
Permalink:: http://dlmf.nist.gov/15.16.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

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15.16.3

F\left({a,b\atop c};z\right)F\left({a,b\atop c};\zeta\right)=\sum_{s=0}^{% \infty}\frac{{\left(a\right)_{s}}{\left(b\right)_{s}}{\left(c-a\right)_{s}}{% \left(c-b\right)_{s}}}{{\left(c\right)_{s}}{\left(c\right)_{2s}}s!}\left(z% \zeta\right)^{s}F\left({a+s,b+s\atop c+2s};z+\zeta-z\zeta\right),

|z|<1

,

|\zeta|<1

,

|z+\zeta-z\zeta|<1

.

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/15.16.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

►

15.16.4

F\left({a,b\atop c};z\right)F\left({-a,-b\atop-c};z\right)+\frac{ab(a-c)(b-c)}% {c^{2}(1-c^{2})}z^{2}F\left({1+a,1+b\atop 2+c};z\right)F\left({1-a,1-b\atop 2-% c};z\right)=1.

ⓘ

Symbols:: $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Referenced by:: §15.16
Permalink:: http://dlmf.nist.gov/15.16.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16 and Ch.15

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15.16.5

F\left({\frac{1}{2}+\lambda,-\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F% \left({\frac{1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)+F\left({% \frac{1}{2}+\lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({-\frac% {1}{2}-\lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)-F\left({\frac{1}{2}+% \lambda,\frac{1}{2}-\nu\atop 1+\lambda+\mu};z\right)F\left({\frac{1}{2}-% \lambda,\frac{1}{2}+\nu\atop 1+\nu+\mu};1-z\right)=\frac{\Gamma\left(1+\lambda% +\mu\right)\Gamma\left(1+\nu+\mu\right)}{\Gamma\left(\lambda+\mu+\nu+\frac{3}{% 2}\right)\Gamma\left(\frac{1}{2}+\nu\right)},

|\operatorname{ph}z|<\pi

,

|\operatorname{ph}\left(1-z\right)|<\pi

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/15.16.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §15.16, §15.16 and Ch.15

…

24: 8.11 Asymptotic Approximations and Expansions

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8.11.2

\Gamma\left(a,z\right)=z^{a-1}e^{-z}\left(\sum_{k=0}^{n-1}\frac{u_{k}}{z^{k}}+% R_{n}(a,z)\right),

n=1,2,\dots

.

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer, $n$ : nonnegative integer, $u_{k}$ and $R_{n}(a,z)$ : remainder
A&S Ref:: 6.5.32
Permalink:: http://dlmf.nist.gov/8.11.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §8.11(i), §8.11 and Ch.8

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8.11.3

R_{n}(a,z)=O\left(z^{-n}\right),

|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta

,

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : parameter, $n$ : nonnegative integer, $\delta$ : small positive constant and $R_{n}(a,z)$ : remainder
Permalink:: http://dlmf.nist.gov/8.11.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §8.11(i), §8.11 and Ch.8

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8.11.4

\gamma\left(a,z\right)=z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{z^{k}}{{\left(a% \right)_{k+1}}},

a\neq 0,-1,-2,\dots

.

ⓘ

Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $z$ : complex variable, $a$ : parameter and $k$ : nonnegative integer
A&S Ref:: 6.5.33 (corrected)
Referenced by:: §8.11(ii)
Permalink:: http://dlmf.nist.gov/8.11.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §8.11(ii), §8.11 and Ch.8

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8.11.6

\gamma\left(a,z\right)\sim-z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{(-a)^{k}b_{k}(% \lambda)}{(z-a)^{2k+1}},

0<\lambda<1

,

|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta

.

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\operatorname{ph}$ : phase, $x$ : real variable, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer, $\delta$ : small positive constant, $\lambda$ : parameter and $b_{k}(\lambda)$ : coefficients
Notes:: See Paris (2002b)
Referenced by:: §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.E6
Encodings:: pMML, png, TeX
Addition (effective with 1.0.14):: Originally this equation was stated for real variables $a$ and $x(=\lambda a)$ , $0<\lambda<1$ . It has been extended to allow for complexvariables $a$ and $z(=\lambda a)$ , where $0<\lambda<1$ and $|\operatorname{ph}a|\leq\frac{\pi}{2}-\delta$ .
See also:: Annotations for §8.11(iii), §8.11 and Ch.8

►

8.11.7

\Gamma\left(a,z\right)\sim z^{a}e^{-z}\sum_{k=0}^{\infty}\frac{(-a)^{k}b_{k}(% \lambda)}{(z-a)^{2k+1}},

\lambda>1

,

|\operatorname{ph}a|\leq\frac{3\pi}{2}-\delta

.

ⓘ

Symbols:: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\operatorname{ph}$ : phase, $x$ : real variable, $z$ : complex variable, $a$ : parameter, $k$ : nonnegative integer, $\delta$ : small positive constant, $\lambda$ : parameter and $b_{k}(\lambda)$ : coefficients
Referenced by:: §8.11(iii), §8.11(iii), §8.11(iii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v)
Permalink:: http://dlmf.nist.gov/8.11.E7
Encodings:: pMML, png, TeX
Addition (effective with 1.0.14):: Originally this equation was stated for real variables $a$ and $x(=\lambda a)$ , $\lambda>1$ . It has been extended to allow for complexvariables $a$ and $z(=\lambda a)$ , where $\lambda>1$ and $|\operatorname{ph}a|\leq\frac{3\pi}{2}-\delta$ .
See also:: Annotations for §8.11(iii), §8.11 and Ch.8

…

25: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.2

\Psi\left(a;b;\mathbf{T}\right)=\frac{1}{\Gamma_{m}\left(a\right)}\int_{% \boldsymbol{\Omega}}\operatorname{etr}\left(-\mathbf{T}\mathbf{X}\right)\left|% \mathbf{X}\right|^{a-\frac{1}{2}(m+1)}\*{\left|\mathbf{I}+\mathbf{X}\right|}^{% b-a-\frac{1}{2}(m+1)}\,\mathrm{d}{\mathbf{X}},

\Re\left(a\right)>\frac{1}{2}(m-1)

,

\mathbf{T}\in{\boldsymbol{\Omega}}

.

ⓘ

Defines:: $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind)
Symbols:: $\in$ : element of, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $\int$ : integral, $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma 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, ${\boldsymbol{\Omega}}$ : space of positive-definite real symmetric $m\times m$ matrices, $b$ : complex variable and $m$ : positive integer
Permalink:: http://dlmf.nist.gov/35.6.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(i), §35.6 and Ch.35

… ►

35.6.7

{{}_{1}F_{1}}\left({a\atop b};\mathbf{T}\right)=\operatorname{etr}\left(% \mathbf{T}\right){{}_{1}F_{1}}\left({b-a\atop b};-\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, $\operatorname{etr}\left(\NVar{\mathbf{A}}\right)$ : exponential of trace, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix and $b$ : complex variable
Permalink:: http://dlmf.nist.gov/35.6.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(ii), §35.6 and Ch.35

►

35.6.8

\int_{\boldsymbol{\Omega}}\left|\mathbf{T}\right|^{c-\frac{1}{2}(m+1)}\Psi% \left(a;b;\mathbf{T}\right)\,\mathrm{d}{\mathbf{T}}=\frac{\Gamma_{m}\left(c% \right)\Gamma_{m}\left(a-c\right)\Gamma_{m}\left(c-b+\frac{1}{2}(m+1)\right)}{% \Gamma_{m}\left(a\right)\Gamma_{m}\left(a-b+\frac{1}{2}(m+1)\right)},

\Re\left(a\right)>\Re\left(c\right)+\frac{1}{2}(m-1)>m-1

,

\Re\left(c-b\right)>-1

.

ⓘ

Symbols:: $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind), $\int$ : 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, $\left|\NVar{\mathbf{X}}\right|$ : 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, $c$ : 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

… ►

35.6.9

\lim_{a\to\infty}{{}_{1}F_{1}}\left({a\atop\nu+\frac{1}{2}(m+1)};-a^{-1}% \mathbf{T}\right)=\frac{A_{\nu}\left(\mathbf{T}\right)}{A_{\nu}\left(% \boldsymbol{{0}}\right)}.

ⓘ

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, $a$ : complex variable, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $m$ : positive integer and $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros
Permalink:: http://dlmf.nist.gov/35.6.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §35.6(iii), §35.6 and Ch.35

►

35.6.10

\lim_{a\to\infty}\Gamma_{m}\left(a\right)\Psi\left(a+\nu;\nu+\tfrac{1}{2}(m+1)% ;a^{-1}\mathbf{T}\right)=B_{\nu}\left(\mathbf{T}\right).

ⓘ

Symbols:: $B_{\NVar{\nu}}\left(\NVar{\mathbf{T}}\right)$ : Bessel function of matrix argument (second kind), $\Psi\left(\NVar{a};\NVar{b};\NVar{\mathbf{T}}\right)$ : confluent hypergeometric function of matrix argument (second kind), $\Gamma_{\NVar{m}}\left(\NVar{a}\right)$ : multivariate gamma function, $a$ : complex variable, $\mathbf{T}$ : 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

…

26: 15.6 Integral Representations

… ►

15.6.1

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-zt)^{a}}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re b>0

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E1
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.2

\mathbf{F}\left(a,b;c;z\right)=\frac{\Gamma\left(1+b-c\right)}{2\pi\mathrm{i}% \Gamma\left(b\right)}\int_{0}^{(1+)}\frac{t^{b-1}(t-1)^{c-b-1}}{(1-zt)^{a}}\,% \mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

c-b\neq 1,2,3,\dots

,

\Re b>0

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.6
Permalink:: http://dlmf.nist.gov/15.6.E2
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.2_5

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(b\right)\Gamma\left(c-b% \right)}\int_{0}^{\infty}\frac{t^{b-1}\left(t+1\right)^{a-c}}{\left(t-zt+1% \right)^{a}}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re b>0

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Referenced by:: §15.6, Erratum (V1.1.5) for Additions
Permalink:: http://dlmf.nist.gov/15.6.E2_5
Encodings:: pMML, png, TeX
Addition (effective with 1.1.5):: This equation was added.
See also:: Annotations for §15.6 and Ch.15

… ►

15.6.8

\mathbf{F}\left(a,b;c;z\right)=\frac{1}{\Gamma\left(c-d\right)}\int_{0}^{1}% \mathbf{F}\left(a,b;d;zt\right)t^{d-1}(1-t)^{c-d-1}\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\Re c>\Re d>0

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Proof sketch:: Follows from (15.6.9) with $\lambda=a+b$ .
Referenced by:: (15.6.9), §15.6, §15.6, §15.6, §18.17(iv), Erratum (V1.0.14) for Equation (15.6.8), Erratum (V1.0.14) for Equation (15.6.8)
Permalink:: http://dlmf.nist.gov/15.6.E8
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
See also:: Annotations for §15.6 and Ch.15

►

15.6.9

\mathbf{F}\left(a,b;c;z\right)=\int_{0}^{1}\frac{t^{d-1}(1-t)^{c-d-1}}{(1-zt)^% {a+b-\lambda}}\mathbf{F}\left({\lambda-a,\lambda-b\atop d};zt\right)\mathbf{F}% \left({a+b-\lambda,\lambda-d\atop c-d};\frac{(1-t)z}{1-zt}\right)\,\mathrm{d}t,

|\operatorname{ph}\left(1-z\right)|<\pi

;

\lambda\in\mathbb{C}

,

\Re c>\Re d>0

.

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $\mathbf{F}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $\mathbf{F}\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}{\mathbf{F}}_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Olver’s hypergeometric function, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter and $c$ : real or complex parameter
Keywords:: Mellin transform
Notes:: With $\lambda=a+b$ , this equation specializes to (15.6.8).
Referenced by:: (15.6.8), §15.6, §15.6, Erratum (V1.0.18) for Equation (15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)
Permalink:: http://dlmf.nist.gov/15.6.E9
Encodings:: pMML, png, TeX
Addition (effective with 1.0.23):: The Olver hypergeometric function $\mathbf{F}\left(a,b;c;z\right)$ , previously omitted from the left-hand side to make the formula more concise, has been added. The constraint $|\operatorname{ph}\left(1-z\right)|<\pi$ , originally mentioned in the text, has been directly added to the formula.
Addition (effective with 1.0.18):: The constraint $\lambda\in\mathbb{C}$ was added.
See also:: Annotations for §15.6 and Ch.15

…

27: 8.6 Integral Representations

… ►

8.6.3

\gamma\left(a,z\right)=z^{a}\int_{0}^{\infty}\exp\left(-at-ze^{-t}\right)\,% \mathrm{d}t,

\Re a>0

.

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(i)
Permalink:: http://dlmf.nist.gov/8.6.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §8.6(i), §8.6 and Ch.8

… ►

8.6.7

\Gamma\left(a,z\right)=z^{a}\int_{0}^{\infty}\exp\left(at-ze^{t}\right)\,% \mathrm{d}t,

\Re z>0

.

ⓘ

Symbols:: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\exp\NVar{z}$ : exponential function, $\mathrm{e}$ : base of natural logarithm, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $a$ : parameter
Referenced by:: §8.6(i)
Permalink:: http://dlmf.nist.gov/8.6.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §8.6(i), §8.6 and Ch.8

… ►

8.6.10

\gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{% \Gamma\left(s\right)}{a-s}z^{a-s}\,\mathrm{d}s,

|\operatorname{ph}z|<\tfrac{1}{2}\pi

,

a\neq 0,-1,-2,\dots

,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : parameter and $c$ : real constant
Keywords:: Mellin transform
Referenced by:: §8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))
Permalink:: http://dlmf.nist.gov/8.6.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

►

8.6.11

\Gamma\left(a,z\right)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left% (s+a\right)\frac{z^{-s}}{s}\,\mathrm{d}s,

|\operatorname{ph}z|<\tfrac{1}{2}\pi

,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\Gamma\left(\NVar{a},\NVar{z}\right)$ : incomplete gamma function, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable, $a$ : parameter and $c$ : real constant
Keywords:: Mellin transform
Referenced by:: §8.19(i), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))
Permalink:: http://dlmf.nist.gov/8.6.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

►

8.6.12

\Gamma\left(a,z\right)=-\frac{z^{a-1}e^{-z}}{\Gamma\left(1-a\right)}\*\frac{1}% {2\pi i}\int_{c-i\infty}^{c+i\infty}\Gamma\left(s+1-a\right)\frac{\pi z^{-s}}{% \sin\left(\pi s\right)}\,\mathrm{d}s,

|\operatorname{ph}z|<\tfrac{3}{2}\pi

,

a\neq 1,2,3,\dots

.

…

28: 15.12 Asymptotic Approximations

… ►

(d)

$\Re z>\tfrac{1}{2}$ and $\alpha_{-}-\tfrac{1}{2}\pi+\delta\leq\operatorname{ph}c\leq\alpha_{+}+\tfrac{1% }{2}\pi-\delta$ , where

15.12.1

\alpha_{\pm}=\operatorname{arctan}\left(\frac{\operatorname{ph}z-\operatorname% {ph}\left(1-z\right)\mp\pi}{\ln|1-z^{-1}|}\right),

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, $z$ : complex variable and $\alpha_{\pm}$
Permalink:: http://dlmf.nist.gov/15.12.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(ii), §15.12 and Ch.15

with $z$ restricted so that $\pm\alpha_{\pm}\in[0,\tfrac{1}{2}\pi)$ .

… ►

15.12.2

F\left(a,b;c;z\right)=\sum_{s=0}^{m-1}\frac{{\left(a\right)_{s}}{\left(b\right% )_{s}}}{{\left(c\right)_{s}}s!}z^{s}+O\left(c^{-m}\right),

|c|\to\infty

.

ⓘ

Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $F\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ or $F\left({\NVar{a},\NVar{b}\atop\NVar{c}};\NVar{z}\right)$ : $={{}_{2}F_{1}}\left(\NVar{a},\NVar{b};\NVar{c};\NVar{z}\right)$ Gauss’ hypergeometric function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $!$ : factorial (as in $n!$ ), $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $m$ : integer
Referenced by:: §15.19(iv)
Permalink:: http://dlmf.nist.gov/15.12.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(ii), §15.12 and Ch.15

… ►

15.12.3

F\left({a,b\atop c+\lambda};z\right)\sim\frac{\Gamma\left(c+\lambda\right)}{% \Gamma\left(c-b+\lambda\right)}\sum_{s=0}^{\infty}q_{s}(z){\left(b\right)_{s}}% \lambda^{-s-b},

… ►

15.12.4

\left(\frac{{\mathrm{e}}^{t}-1}{t}\right)^{b-1}{\mathrm{e}}^{t(1-c)}\left(1-z+% z{\mathrm{e}}^{-t}\right)^{-a}=\sum_{s=0}^{\infty}q_{s}(z)t^{s}.

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $s$ : nonnegative integer, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter and $q_{s}(z)$ : coefficients
Permalink:: http://dlmf.nist.gov/15.12.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

… ►

15.12.13

G_{0}(\pm\beta)=\left(2+{\mathrm{e}}^{\pm\zeta}\right)^{c-b-(\ifrac{1}{2})}% \left(1+{\mathrm{e}}^{\pm\zeta}\right)^{a-c+(\ifrac{1}{2})}\left(z-1-{\mathrm{% e}}^{\pm\zeta}\right)^{-a+(\ifrac{1}{2})}\sqrt{\frac{\beta}{{\mathrm{e}}^{% \zeta}-{\mathrm{e}}^{-\zeta}}}.

ⓘ

Symbols:: $\mathrm{e}$ : base of natural logarithm, $z$ : complex variable, $a$ : real or complex parameter, $b$ : real or complex parameter, $c$ : real or complex parameter, $\zeta$ : change of variable, $\beta$ and $G_{0}(\pm\zeta)$ : function
Permalink:: http://dlmf.nist.gov/15.12.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §15.12(iii), §15.12 and Ch.15

…

29: 20.12 Mathematical Applications

… ►This ability to uniformize multiply-connected spaces (manifolds), or multi-sheeted functions of a complex variable (Riemann (1899), Rauch and Lebowitz (1973), Siegel (1988)) has led to applications in string theory (Green et al. (1988a, b), Krichever and Novikov (1989)), and also in statistical mechanics (Baxter (1982)). …

30: 35.8 Generalized Hypergeometric Functions of Matrix Argument

… ►

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:: pMML, png, TeX
See also:: Annotations for §35.8(ii), §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:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §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:: pMML, png, TeX
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:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §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:: pMML, png, TeX
See also:: Annotations for §35.8(iv), §35.8(iv), §35.8 and Ch.35

…

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