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1: 27.4 Euler Products and Dirichlet Series

§27.4 Euler Products and Dirichlet Series

►The fundamental theorem of arithmetic is linked to analysis through the concept of the Euler product. …In this case the infinite product on the right (extended over all primes $p$) is also absolutely convergent and is called the Euler product of the series. If $f(n)$ is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes … ►Euler products are used to find series that generate many functions of multiplicative number theory. …

2: 27.6 Divisor Sums

… ►Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. …

3: 5.8 Infinite Products

… ►

5.8.2 $$\frac{1}{\mathrm{\Gamma }\left(z\right)}=z{\mathrm{e}}^{\gamma z}\prod _{k=1}^{\mathrm{\infty }}\left(1+\frac{z}{k}\right){\mathrm{e}}^{-z/k},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\mathrm{e}$: base of natural logarithm, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.1.3

Permalink:

http://dlmf.nist.gov/5.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

►

5.8.3 $${\left|\frac{\mathrm{\Gamma }\left(x\right)}{\mathrm{\Gamma }\left(x+\mathrm{i}y\right)}\right|}^2=\prod _{k=0}^{\mathrm{\infty }}\left(1+\frac{y^2}{{(x+k)}^2}\right),$$ $x\ne 0,-1,\mathrm{\dots }$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{i}$: imaginary unit, $k$: nonnegative integer, $x$: real variable and $y$: real variable

A&S Ref:

6.1.25 (where the formula for the reciprocal is given.)

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

… ►

5.8.5 $$\prod _{k=0}^{\mathrm{\infty }}\frac{(a_1+k)(a_2+k)\mathrm{\cdots }(a_m+k)}{(b_1+k)(b_2+k)\mathrm{\cdots }(b_m+k)}=\frac{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_m\right)}{\mathrm{\Gamma }\left(a_1\right)\mathrm{\Gamma }\left(a_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(a_m\right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $m$: nonnegative integer, $k$: nonnegative integer, $a_k$: coefficient and $b_k$: coefficient

Referenced by:

§5.8

Permalink:

http://dlmf.nist.gov/5.8.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.8 and Ch.5

…

4: 16.18 Special Cases

… ►

16.18.1 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\left(\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)/\prod _{k=1}^p\mathrm{\Gamma }\left(a_k\right)\right)G_{p,q+1}^{1,p}(-z;\genfrac{}{}{0pt}{}{1-a_1,\mathrm{\dots },1-a_p}{0,1-b_1,\mathrm{\dots },1-b_q})=\left(\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)/\prod _{k=1}^p\mathrm{\Gamma }\left(a_k\right)\right)G_{q+1,p}^{p,1}(-\frac{1}{z};\genfrac{}{}{0pt}{}{1,b_1,\mathrm{\dots },b_q}{a_1,\mathrm{\dots },a_p}).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $G_{p,q}^{m,n}(z;\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q})$ or $G_{p,q}^{m,n}(z;𝐚;𝐛)$: Meijer $G$-function, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Permalink:

http://dlmf.nist.gov/16.18.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.18 and Ch.16

…

5: 27.3 Multiplicative Properties

… ►

27.3.3 $$\varphi \left(n\right)=n\prod _{p|n}(1-p^{-1}),$$

ⓘ

Symbols:

$\varphi \left(n\right)$: Euler’s totient, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

A&S Ref:

24.3.2 I.C

Permalink:

http://dlmf.nist.gov/27.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §27.3 and Ch.27

…

6: 18.25 Wilson Class: Definitions

… ►

18.25.4 $$w(y^2)=\frac{1}{2y}{\left|\frac{{\prod }_j\mathrm{\Gamma }\left(a_j+iy\right)}{\mathrm{\Gamma }\left(2iy\right)}\right|}^2,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $y$: real variable and $w(x)$: weight function

Permalink:

http://dlmf.nist.gov/18.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

►

18.25.5

ⓘ

Symbols:

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

Referenced by:

§18.25(ii)

Permalink:

http://dlmf.nist.gov/18.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

… ►

18.25.7 $$w(y^2)=\frac{1}{2y}{\left|\frac{{\prod }_j\mathrm{\Gamma }\left(a_j+iy\right)}{\mathrm{\Gamma }\left(2iy\right)}\right|}^2,$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $y$: real variable and $w(x)$: weight function

Permalink:

http://dlmf.nist.gov/18.25.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

►

18.25.8

ⓘ

Symbols:

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

Referenced by:

§18.25(ii)

Permalink:

http://dlmf.nist.gov/18.25.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §18.25(ii), §18.25(ii), §18.25 and Ch.18

…

7: 37.3 Triangular Region with Weight Function

… ►

37.3.2 ,

ⓘ

Defines:

: inner product on triangular region (locally), : real parameter ($>-1$) (locally), : real parameter ($>-1$) (locally) and : real parameter ($>-1$) (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{△}$: triangular region, $x$: real variable, $y$: real variable, : weight function in two-variables on $\mathrm{△}$ , $f$: two-variable function on $\mathrm{△}$ and $g$: two-variable function on $\mathrm{△}$

Referenced by:

(37.3.18), (37.3.19), (37.3.20), §37.3(i)

Permalink:

http://dlmf.nist.gov/37.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(i), §37.3 and Ch.37

… ►The OPs of degree $n$ with respect to the inner product (37.3.2) form the space ${𝒱}_n={𝒱}_n^{\alpha ,\beta ,\gamma }$. … ►

37.3.4 ,

ⓘ

Defines:

: positive constant (locally)

Symbols:

: two-variable Jacobi polynomial on the triangle, ${\delta }_{j,k}$: Kronecker delta, $n$: nonnegative integer, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, : inner product on triangular region, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proved:

Dunkl and Xu (2014, Proposition 2.4.1)(proved)

Referenced by:

(37.3.32)

Permalink:

http://dlmf.nist.gov/37.3.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(ii), §37.3 and Ch.37

… ►

37.3.13

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), : co-monic basis element of two-variable Jacobi polynomial on the triangle, $!$: factorial (as in $n!$), : monic basis element of two-variable Jacobi polynomial on the triangle, $n$: nonnegative integer, $j$: nonnegative integer, $k$: nonnegative integer, $m$: nonnegative integer, $d$: positive integer, usually $\ge 2$ , : inner product on triangular region, : real parameter ($>-1$), : real parameter ($>-1$) and : real parameter ($>-1$)

Proveds:

Engelis (1964); Fackerell and Littler (1974, (11)); Dunkl and Xu (2014, Proposition 5.3.3); for $d=2$; Appell and Kampé de Fériet (1926, §XXXVI); only for (proved)

Source:

Erdélyi et al. (1953b, 12.4(17)); only for

Permalink:

http://dlmf.nist.gov/37.3.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §37.3(iii), §37.3 and Ch.37

…

8: 27.14 Unrestricted Partitions

… ►Euler introduced the reciprocal of the infinite product ►

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

ⓘ

Defines:

$f\left(x\right)$: Euler’s reciprocal function

Symbols:

${(a;q)}_n$: $q$-Pochhammer symbol (or $q$-shifted factorial), $m$: positive integer and $x$: real number

Referenced by:

§27.14(iv), §27.21, Erratum (V1.0.28) for Equation (27.14.2)

Permalink:

http://dlmf.nist.gov/27.14.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.28):

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

See also:

Annotations for §27.14(ii), §27.14 and Ch.27

…

9: 16.11 Asymptotic Expansions

… ►

16.11.2 $$H_{p,q}(z)=\sum _{m=1}^p\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\mathrm{\Gamma }\left(a_m+k\right)\left(\prod _{\begin{array}{c}\mathrm{\ell }=1\\ \mathrm{\ell }\ne m\end{array}}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}-a_m-k\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}-a_m-k\right)\right)z^{-a_m-k}.$$

ⓘ

Defines:

$H_{p,q}(z)$: formal infinite series (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.11(ii), §16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(i), §16.11 and Ch.16

… ►

16.11.6 $$\left(\prod _{\mathrm{\ell }=1}^{q+1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=H_{q+1,q}(-z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$;

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters and $H_{p,q}(z)$: formal infinite series

Permalink:

http://dlmf.nist.gov/16.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.7 $$\left(\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_q{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_q}{b_1,\mathrm{\dots },b_q};z)\sim H_{q,q}(z{\mathrm{e}}^{\mp \pi \mathrm{i}})+E_{q,q}(z).$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series and $H_{p,q}(z)$: formal infinite series

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.8 $$\left(\prod _{\mathrm{\ell }=1}^{q-1}\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_{q-1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q-1}}{b_1,\mathrm{\dots },b_q};-z)\sim H_{q-1,q}(z)+E_{q-1,q}(z{e}^{-\pi \mathrm{i}})+E_{q-1,q}(z{e}^{\pi \mathrm{i}}),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series, $H_{p,q}(z)$: formal infinite series and $e_{k,m}$

Permalink:

http://dlmf.nist.gov/16.11.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

… ►

16.11.9 $$\left(\prod _{\mathrm{\ell }=1}^p\mathrm{\Gamma }\left(a_{\mathrm{\ell }}\right)/\prod _{\mathrm{\ell }=1}^q\mathrm{\Gamma }\left(b_{\mathrm{\ell }}\right)\right){}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};-z)\sim E_{p,q}(z{e}^{-\pi \mathrm{i}})+E_{p,q}(z{e}^{\pi \mathrm{i}}),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters, $b,b_1,\mathrm{\dots },b_q$: real or complex parameters, $E_{p,q}(z)$: formal infinite series and $e_{k,m}$

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.11.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.11(ii), §16.11(ii), §16.11 and Ch.16

…

10: 5.14 Multidimensional Integrals

… ►

5.14.2 $${\int }_{V_n}{\left(1-\sum _{k=1}^n{t}_k\right)}^{z_{n+1}-1}\prod _{k=1}^n{t}_k^{z_k-1}d{t}_k=\frac{\mathrm{\Gamma }\left(z_1\right)\mathrm{\Gamma }\left(z_2\right)\mathrm{\cdots }\mathrm{\Gamma }\left(z_{n+1}\right)}{\mathrm{\Gamma }\left(z_1+z_2+\mathrm{\cdots }+z_{n+1}\right)}.$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $n$: nonnegative integer, $k$: nonnegative integer, $z$: complex variable and $V_n$: simplex

Referenced by:

§5.19(iii)

Permalink:

http://dlmf.nist.gov/5.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14 and Ch.5

… ►

5.14.4 $${\int }_{{[0,1]}^n}{t}_1{t}_2\mathrm{\cdots }{t}_m{|\mathrm{\Delta }(t_1,\mathrm{\dots },t_n)|}^{2c}\prod _{k=1}^n{t}_k^{a-1}{(1-t_k)}^{b-1}d{t}_k=\frac{1}{{(\mathrm{\Gamma }\left(1+c\right))}^n}\prod _{k=1}^m\frac{a+(n-k)c}{a+b+(2n-k-1)c}\prod _{k=1}^n\frac{\mathrm{\Gamma }\left(a+(n-k)c\right)\mathrm{\Gamma }\left(b+(n-k)c\right)\mathrm{\Gamma }\left(1+kc\right)}{\mathrm{\Gamma }\left(a+b+(2n-k-1)c\right)},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $[a,b]$: closed interval, $dx$: differential of $x$ , $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer, $a$: real or complex variable, $b$: real or complex variable and $\mathrm{\Delta }$: product

Permalink:

http://dlmf.nist.gov/5.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

… ►

5.14.5 $${\int }_{{[0,\mathrm{\infty })}^n}{t}_1{t}_2\mathrm{\cdots }{t}_m{|\mathrm{\Delta }(t_1,\mathrm{\dots },t_n)|}^{2c}\prod _{k=1}^n{t}_k^{a-1}{\mathrm{e}}^{-t_k}d{t}_k=\prod _{k=1}^m(a+(n-k)c)\frac{{\prod }_{k=1}^n\mathrm{\Gamma }\left(a+(n-k)c\right)\mathrm{\Gamma }\left(1+kc\right)}{{(\mathrm{\Gamma }\left(1+c\right))}^n},$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $[a,b)$: half-closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $m$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer, $a$: real or complex variable and $\mathrm{\Delta }$: product

Permalink:

http://dlmf.nist.gov/5.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

… ►

5.14.6 $$\frac{1}{{(2\pi )}^{n/2}}{\int }_{{(-\mathrm{\infty },\mathrm{\infty })}^n}{|\mathrm{\Delta }(t_1,\mathrm{\dots },t_n)|}^{2c}\prod _{k=1}^n\exp \left(-\frac{1}{2}{t}_k^2\right)d{t}_k=\frac{{\prod }_{k=1}^n\mathrm{\Gamma }\left(1+kc\right)}{{(\mathrm{\Gamma }\left(1+c\right))}^n},$$ $\mathrm{\Re }c>-1/n$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $(a,b)$: open interval, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $k$: nonnegative integer and $\mathrm{\Delta }$: product

Referenced by:

§5.20

Permalink:

http://dlmf.nist.gov/5.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5

… ►

5.14.7 $\mathrm{\Re }b>-1/n$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $j$: nonnegative integer, $n$: nonnegative integer, $k$: nonnegative integer and $b$: real or complex variable

Referenced by:

§5.14, §5.20

Permalink:

http://dlmf.nist.gov/5.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §5.14, §5.14 and Ch.5