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21: 35.3 Multivariate Gamma and Beta Functions

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35.3.2 $${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{s_m-\frac{1}{2}(m+1)}\prod _{j=1}^{m-1}{|{(𝐗)}_j|}^{s_j-s_{j+1}}d𝐗,$$ $s_j\in ℂ$, $\mathrm{\Re }\left(s_j\right)>\frac{1}{2}(j-1)$, $j=1,\mathrm{\dots },m$.

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

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35.3.4 $${\mathrm{\Gamma }}_m\left(a\right)={\pi }^{m(m-1)/4}\prod _{j=1}^m\mathrm{\Gamma }\left(a-\frac{1}{2}(j-1)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

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35.3.5 $${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\pi }^{m(m-1)/4}\prod _{j=1}^m\mathrm{\Gamma }\left(s_j-\frac{1}{2}(j-1)\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

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22: 25.11 Hurwitz Zeta Function

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25.11.37 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}\zeta (nk,a)=-n\ln \mathrm{\Gamma }\left(a\right)+\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(a-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$, $\mathrm{\Re }a\ge 1$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer and $a$: real or complex parameter

Keywords:

infinite series, series representation

Source:

Adamchik and Srivastava (1998, (2.12), p. 136); with the choice $\mathrm{ph}\left(-1\right)=\pi$, which is reasonable since the gamma function is single valued

Permalink:

http://dlmf.nist.gov/25.11.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xi), §25.11 and Ch.25

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23: 5.20 Physical Applications

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5.20.3 $${\psi }_n(\beta )={\int }_{{ℝ}^n}{\mathrm{e}}^{-\beta W}dx={(2\pi )}^{n/2}{\beta }^{-(n/2)-(\beta n(n-1)/4)}{\left(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right)\right)}^{-n}\prod _{j=1}^n\mathrm{\Gamma }\left(1+\frac{1}{2}j\beta \right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $ℝ$: real line, $j$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $W$: potential energy, ${\psi }_n(\beta )$: partition function and $\beta =1/(kT)$

Permalink:

http://dlmf.nist.gov/5.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

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24: 5.5 Functional Relations

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5.5.6 $$\mathrm{\Gamma }\left(nz\right)={(2\pi )}^{(1-n)/2}{n}^{nz-(1/2)}\prod _{k=0}^{n-1}\mathrm{\Gamma }\left(z+\frac{k}{n}\right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.20

Referenced by:

§15.4(iii), §5.5(iii)

Permalink:

http://dlmf.nist.gov/5.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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5.5.7 $$\prod _{k=1}^{n-1}\mathrm{\Gamma }\left(\frac{k}{n}\right)={(2\pi )}^{(n-1)/2}{n}^{-1/2}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.5(iii), §5.5(iii), §5.5 and Ch.5

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25: 35.4 Partitions and Zonal Polynomials

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35.4.1 $${\left[a\right]}_{\kappa }=\frac{{\mathrm{\Gamma }}_m\left(a+\kappa \right)}{{\mathrm{\Gamma }}_m\left(a\right)}=\prod _{j=1}^m{\left(a-\frac{1}{2}(j-1)\right)}_{k_j},$$

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Defines:

${\left[a\right]}_{\kappa }$: partitional shifted factorial

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $a$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

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26: 10.65 Power Series

… ►Also, with $\psi \left(x\right)={\mathrm{\Gamma }}^{\prime }\left(x\right)/\mathrm{\Gamma }\left(x\right)$, … ►

§10.65(iii) Cross-Products and Sums of Squares

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10.65.6 $${\mathrm{ber}}_{\nu }^{2}x+{\mathrm{bei}}_{\nu }^{2}x={(\frac{1}{2}x)}^{2\nu }\sum _{k=0}^{\mathrm{\infty }}\frac{1}{\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\nu +2k+1\right)}\frac{{(\frac{1}{4}{x}^2)}^{2k}}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.28

Permalink:

http://dlmf.nist.gov/10.65.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(iii), §10.65 and Ch.10

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10.65.7 $${\mathrm{ber}}_{\nu }x{\mathrm{bei}}_{\nu }^{\prime }x-{\mathrm{ber}}_{\nu }^{\prime }x{\mathrm{bei}}_{\nu }x={(\frac{1}{2}x)}^{2\nu +1}\sum _{k=0}^{\mathrm{\infty }}\frac{1}{\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\nu +2k+2\right)}\frac{{(\frac{1}{4}{x}^2)}^{2k}}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.29

Permalink:

http://dlmf.nist.gov/10.65.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(iii), §10.65 and Ch.10

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10.65.8 $${\mathrm{ber}}_{\nu }x{\mathrm{ber}}_{\nu }^{\prime }x+{\mathrm{bei}}_{\nu }x{\mathrm{bei}}_{\nu }^{\prime }x=\frac{1}{2}{(\frac{1}{2}x)}^{2\nu -1}\sum _{k=0}^{\mathrm{\infty }}\frac{1}{\mathrm{\Gamma }\left(\nu +k+1\right)\mathrm{\Gamma }\left(\nu +2k\right)}\frac{{(\frac{1}{4}{x}^2)}^{2k}}{k!},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathrm{bei}}_{\nu }\left(x\right)$: Kelvin function, ${\mathrm{ber}}_{\nu }\left(x\right)$: Kelvin function, $!$: factorial (as in $n!$), $k$: nonnegative integer, $x$: real variable and $\nu$: complex parameter

A&S Ref:

9.9.30

Permalink:

http://dlmf.nist.gov/10.65.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.65(iii), §10.65 and Ch.10

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27: Errata

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18.12.2 $${}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\alpha +1};\frac{(x-1)z}{2}){}_0{}^{}𝐅_1^{}(\genfrac{}{}{0pt}{}{-}{\beta +1};\frac{(x+1)z}{2})={\left(\frac{1}{2}(1-x)z\right)}^{-\frac{1}{2}\alpha }{J}_{\alpha }\left(\sqrt{2(1-x)z}\right){\left(\frac{1}{2}(1+x)z\right)}^{-\frac{1}{2}\beta }{I}_{\beta }\left(\sqrt{2(1+x)z}\right)=\sum _{n=0}^{\mathrm{\infty }}\frac{P_n^{(\alpha ,\beta )}\left(x\right)}{\mathrm{\Gamma }\left(n+\alpha +1\right)\mathrm{\Gamma }\left(n+\beta +1\right)}{z}^n$$

This equation was updated to include on the left-hand side, its definition in terms of a product of two ${}_0{}^{}𝐅_1^{}$ functions.

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27.14.2 $$f\left(x\right)=\prod _{m=1}^{\mathrm{\infty }}(1-x^m)={(x;x)}_{\mathrm{\infty }},$$

The representation in terms of ${(x;x)}_{\mathrm{\infty }}$ was added to this equation.

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28: 25.8 Sums

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25.8.4 $$\sum _{k=1}^{\mathrm{\infty }}\frac{{(-1)}^k}{k}(\zeta \left(nk\right)-1)=\ln \left(\prod _{j=0}^{n-1}\mathrm{\Gamma }\left(2-{\mathrm{e}}^{(2j+1)\pi \mathrm{i}/n}\right)\right),$$ $n=2,3,4,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $n$: nonnegative integer

Keywords:

infinite series

Source:

Adamchik and Srivastava (1998, (2.7), p. 135)

Permalink:

http://dlmf.nist.gov/25.8.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §25.8 and Ch.25

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29: 30.10 Series and Integrals

… ►Integrals and integral equations for ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$ are given in Arscott (1964b, §8.6), Erdélyi et al. (1955, §16.13), Flammer (1957, Chapter 5), and Meixner (1951). For product formulas and convolutions see Connett et al. (1993). …For expansions in products of spherical Bessel functions, see Flammer (1957, Chapter 6).

30: 14.2 Differential Equations

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14.2.5 $${𝖯}_{\nu +1}^{\mu }\left(x\right){𝖰}_{\nu }^{\mu }\left(x\right)-{𝖯}_{\nu }^{\mu }\left(x\right){𝖰}_{\nu +1}^{\mu }\left(x\right)=\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, : Ferrers function of the first kind, : Ferrers function of the second kind, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.2(iv)

Permalink:

http://dlmf.nist.gov/14.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14

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14.2.11 $$P_{\nu +1}^{\mu }\left(x\right)Q_{\nu }^{\mu }\left(x\right)-P_{\nu }^{\mu }\left(x\right)Q_{\nu +1}^{\mu }\left(x\right)={\mathrm{e}}^{\mu \pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(\nu +\mu +1\right)}{\mathrm{\Gamma }\left(\nu -\mu +2\right)}.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $P_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the first kind, $Q_{\nu }^{\mu }\left(z\right)$: associated Legendre function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $x$: real variable, $\mu$: general order and $\nu$: general degree

Referenced by:

§14.21(iii)

Permalink:

http://dlmf.nist.gov/14.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §14.2(iv), §14.2 and Ch.14