integer argument

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21: 19.2 Definitions

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19.2.4 $$F(\varphi ,k)={\int }_0^{\varphi }\frac{d\theta }{\sqrt{1-k^2{\sin }^2\theta }}={\int }_0^{\sin \varphi }\frac{dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}},$$

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Defines:

$F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.5, §19.6(ii)

Permalink:

http://dlmf.nist.gov/19.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.5 $$E(\varphi ,k)={\int }_0^{\varphi }\sqrt{1-k^2{\sin }^2\theta }d\theta ={\int }_0^{\sin \varphi }\frac{\sqrt{1-k^2{t}^2}}{\sqrt{1-t^2}}dt.$$

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Defines:

$E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.6(iii), §22.16(ii)

Permalink:

http://dlmf.nist.gov/19.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.6 $$D(\varphi ,k)={\int }_0^{\varphi }\frac{{\sin }^2\theta d\theta }{\sqrt{1-k^2{\sin }^2\theta }}={\int }_0^{\sin \varphi }\frac{t^{2}dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}}=(F(\varphi ,k)-E(\varphi ,k))/k^2.$$

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Defines:

$D(\varphi ,k)$: incomplete elliptic integral of Legendre’s type

Symbols:

$dx$: differential of $x$ , $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $E(\varphi ,k)$: Legendre’s incomplete elliptic integral of the second kind, $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument and $k$: real or complex modulus

Referenced by:

§19.25(i), §19.36(i)

Permalink:

http://dlmf.nist.gov/19.2.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

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19.2.7 $$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)={\int }_0^{\varphi }\frac{d\theta }{\sqrt{1-k^2{\sin }^2\theta }(1-{\alpha }^2{\sin }^2\theta )}={\int }_0^{\sin \varphi }\frac{dt}{\sqrt{1-t^2}\sqrt{1-k^2{t}^2}(1-{\alpha }^2{t}^2)}.$$

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Defines:

$\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind

Symbols:

$dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $\varphi$: real or complex argument, $k$: real or complex modulus and ${\alpha }^2$: real or complex parameter

Referenced by:

§19.5

Permalink:

http://dlmf.nist.gov/19.2.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.2(ii), §19.2 and Ch.19

… ►If $m$ is an integer, then …

22: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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

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

ⓘ

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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35.8.8 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝟎)=1.$$

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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, $a$: complex variable, $b$: complex variable, $p$: nonnegative integer, $q$: nonnegative integer and $𝟎$: $m\times m$ 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

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35.8.9 $$\underset{\gamma \to \mathrm{\infty }}{lim}{}_{p+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p,\gamma }{b_1,\mathrm{\dots },b_q};{\gamma }^{-1}𝐓)={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓),$$

ⓘ

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, $a$: complex variable, $𝐓$: 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

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35.8.10 $$\underset{\gamma \to \mathrm{\infty }}{lim}{}_p{}^{}F_{q+1}^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q,\gamma };\gamma 𝐓)={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓).$$

ⓘ

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, $a$: complex variable, $𝐓$: 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

…

23: 6.1 Special Notation

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$x$	real variable.
…
$n$	nonnegative integer.
…

►Unless otherwise noted, primes indicate derivatives with respect to the argument. …

24: 11.1 Special Notation

… ► ►►►►

$x$	real variable.
…
$n$	integer order.
$k$	nonnegative integer.
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►Unless indicated otherwise, primes denote derivatives with respect to the argument. …

25: Errata

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Originally the function $\chi$ was presented with argument given by a positive integer $n$. It has now been clarified to be valid for argument given by a positive real number $x$.

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26: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.1 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }}{{\left[b\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right).$$

ⓘ

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, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $k$: nonnegative integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.6.E1

Encodings:

pMML, png, TeX

See also:

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

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

ⓘ

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

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35.6.9 $$\underset{a\to \mathrm{\infty }}{lim}{}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{\nu +\frac{1}{2}(m+1)};-a^{-1}𝐓)=\frac{A_{\nu }\left(𝐓\right)}{A_{\nu }\left(𝟎\right)}.$$

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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, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $m$: positive integer and $𝟎$: $m\times m$ 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

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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

…

27: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►The Coulomb functions given in this chapter are most commonly evaluated for real values of $\rho$, $r$, $\eta$, $ϵ$ and nonnegative integer values of $\mathrm{\ell }$, but they may be continued analytically to complex arguments and order $\mathrm{\ell }$ as indicated in §33.13. …

28: 7.1 Special Notation

… ► ►►►

$x$	real variable.
…
$n$	nonnegative integer.
…

►Unless otherwise noted, primes indicate derivatives with respect to the argument. …

29: 33.8 Continued Fractions

… ►With arguments $\eta ,\rho$ suppressed, ►

33.8.1 $$\frac{F_{\mathrm{\ell }}^{\prime }}{F_{\mathrm{\ell }}}=S_{\mathrm{\ell }+1}-\frac{R_{\mathrm{\ell }+1}^2}{T_{\mathrm{\ell }+1}-}\frac{R_{\mathrm{\ell }+2}^2}{T_{\mathrm{\ell }+2}-}\mathrm{\cdots }.$$

ⓘ

Symbols:

$F_{\mathrm{\ell }}(\eta ,\rho )$: regular Coulomb radial function, $\mathrm{\ell }$: nonnegative integer, $R_{\mathrm{\ell }}$: factor, $S_{\mathrm{\ell }}$: factor and $T_{\mathrm{\ell }}$: factor

Referenced by:

§33.8, §33.8

Permalink:

http://dlmf.nist.gov/33.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

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33.8.2 $$\frac{H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }}{H_{\mathrm{\ell }}^{\pm }}=c\pm \frac{\mathrm{i}}{\rho }\frac{ab}{2(\rho -\eta \pm \mathrm{i})+}\frac{(a+1)(b+1)}{2(\rho -\eta \pm 2\mathrm{i})+}\mathrm{\cdots },$$

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Symbols:

$\mathrm{i}$: imaginary unit, $H_{\mathrm{\ell }}^{\pm }(\eta ,\rho )$: irregular Coulomb radial functions, $\mathrm{\ell }$: nonnegative integer, $\rho$: nonnegative real variable, $\eta$: real parameter, $a$: coefficient, $b$: coefficient and $c$: coefficient

Referenced by:

§33.8

Permalink:

http://dlmf.nist.gov/33.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §33.8 and Ch.33

…

30: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.1 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }{\left[b\right]}_{\kappa }}{{\left[c\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $-c+\frac{1}{2}(j+1)\notin \mathbb{N}$, $1\le j\le m$; .

ⓘ

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, $!$: factorial (as in $n!$), $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $\Vert 𝐓\Vert$: spectral norm of $𝐓$ , $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.7.E1

Encodings:

pMML, png, TeX

See also:

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

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35.7.3

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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, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $a$: complex variable, $t_j$: eigenvalues of $𝐓$ , $b$: complex variable, $c$: complex variable and $k$: nonnegative integer

Referenced by:

Erratum (V1.0.25) for Equation (35.7.3)

Permalink:

http://dlmf.nist.gov/35.7.E3

Encodings:

pMML, png, TeX

Notational change (effective with 1.0.25):

Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${}_2{}^{}F_1^{}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

See also:

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

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35.7.4 $$\underset{c\to \mathrm{\infty }}{lim}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈-c{𝐓}^{-1})={\left|𝐓\right|}^b\mathrm{\Psi }(b;b-a+\frac{1}{2}(m+1);𝐓).$$

ⓘ

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, $\mathrm{\Psi }(a;b;𝐓)$: confluent hypergeometric function of matrix argument (second kind), $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $b$: complex variable, $c$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.7.E4

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

… ►

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 $j=1,\mathrm{\dots },m$.

ⓘ

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

…