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11: 5.17 Barnes’ $G$-Function (Double Gamma Function)

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5.17.3 $$G\left(z+1\right)={(2\pi )}^{z/2}\exp \left(-\frac{1}{2}z(z+1)-\frac{1}{2}\gamma z^2\right)\prod _{k=1}^{\mathrm{\infty }}\left({\left(1+\frac{z}{k}\right)}^k\exp \left(-z+\frac{z^2}{2k}\right)\right).$$

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $\gamma$: Euler’s constant, $\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $k$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/5.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

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12: 16.5 Integral Representations and Integrals

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16.5.1 $$\left(\prod _{k=1}^p\mathrm{\Gamma }\left(a_k\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)\right){}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\frac{1}{2\pi \mathrm{i}}{\int }_L\left(\prod _{k=1}^p\mathrm{\Gamma }\left(a_k+s\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k+s\right)\right)\mathrm{\Gamma }\left(-s\right){(-z)}^{s}ds,$$

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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, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $p$: 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:

§13.6(vi), §16.11(i), §16.5, §16.5

Permalink:

http://dlmf.nist.gov/16.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.5 and Ch.16

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13: 16.8 Differential Equations

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16.8.8 $${}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\frac{\mathrm{\Gamma }\left(a_k-a_j\right)}{\mathrm{\Gamma }\left(a_k\right)}/\prod _{k=1}^q\frac{\mathrm{\Gamma }\left(b_k-a_j\right)}{\mathrm{\Gamma }\left(b_k\right)}\right){\tilde{w}}_j(z),$$ $|\mathrm{ph}\left(-z\right)|\le \pi$.

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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 $w$: solution

Referenced by:

§16.11(ii)

Permalink:

http://dlmf.nist.gov/16.8.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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16.8.9 $$\left(\prod _{k=1}^{q+1}\mathrm{\Gamma }\left(a_k\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_{q+1}}{b_1,\mathrm{\dots },b_q};z)=\sum _{j=1}^{q+1}{\left(z_0-z\right)}^{-a_j}\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a_j+n\right)}{n!}\left(\prod _{\begin{array}{c}k=1\\ k\ne j\end{array}}^{q+1}\mathrm{\Gamma }\left(a_k-a_j-n\right)/\prod _{k=1}^q\mathrm{\Gamma }\left(b_k-a_j-n\right)\right){}_{q+1}{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1-a_j-n,\mathrm{\dots },a_{q+1}-a_j-n}{b_1-a_j-n,\mathrm{\dots },b_q-a_j-n};z_0){\left(z-z_0\right)}^{-n}.$$

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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, $!$: factorial (as in $n!$), $q$: 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:

§16.8(ii)

Permalink:

http://dlmf.nist.gov/16.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §16.8(ii), §16.8 and Ch.16

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14: 25.2 Definition and Expansions

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25.2.12 $$\zeta \left(s\right)=\frac{{(2\pi )}^s{\mathrm{e}}^{-s-(\gamma s/2)}}{2(s-1)\mathrm{\Gamma }\left(\frac{1}{2}s+1\right)}\prod _{\rho }\left(1-\frac{s}{\rho }\right){\mathrm{e}}^{s/\rho },$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $s$: complex variable and $\rho$: zeros

Keywords:

infinite product, primes

Source:

Titchmarsh (1986b, (2.12.6), (2.12.8), p. 30–31)

A&S Ref:

23.2.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/25.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.2(iv), §25.2 and Ch.25

►product over zeros $\rho$ of $\zeta$ with $\mathrm{\Re }\rho >0$ (see §25.10(i)); $\gamma$ is Euler’s constant (§5.2(ii)).

15: 31.8 Solutions via Quadratures

… ►The variables $\lambda$ and $\nu$ are two coordinates of the associated hyperelliptic (spectral) curve $\mathrm{\Gamma }:{\nu }^2={\prod }_{j=1}^{2g+1}(\lambda -{\lambda }_j)$. …

16: 20.12 Mathematical Applications

… ►For applications of Jacobi’s triple product (20.5.9) to Ramanujan’s $\tau \left(n\right)$ function and Euler’s pentagonal numbers see Hardy and Wright (1979, pp. 132–160) and McKean and Moll (1999, pp. 143–145). …

17: 5.13 Integrals

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5.13.5 $\mathrm{\Re }\left(a_k\right)>0$, $k=1,2,3,4$.

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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 $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $j$: nonnegative integer, $k$: nonnegative integer and $a$: real or complex variable

Referenced by:

§5.13

Permalink:

http://dlmf.nist.gov/5.13.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §5.13, §5.13 and Ch.5

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18: 19.29 Reduction of General Elliptic Integrals

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19.29.7 $${\int }_y^x\frac{a_{\alpha }+b_{\alpha }t}{a_{\delta }+b_{\delta }t}\frac{dt}{s(t)}=\frac{2}{3}{d}_{\alpha \beta }{d}_{\alpha \gamma }{R}_D(U_{\alpha \beta }^2,U_{\alpha \gamma }^2,U_{\alpha \delta }^2)+\frac{2X_{\alpha }{Y}_{\alpha }}{X_{\delta }{Y}_{\delta }{U}_{\alpha \delta }},$$ $U_{\alpha \delta }\ne 0$.

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $dx$: differential of $x$ , $\int$: integral, $X_{\alpha }$ , $U_n$ , $Y_{\alpha }$ , $d_{\alpha \beta }$ , $s(t)$: product, $\alpha$: parameter, $\beta$: parameter, $\gamma$: parameter and $\delta$: parameter

Referenced by:

§19.29(i), §19.29(i), §19.29(ii), §19.29(ii), §19.29(iii), §19.30(ii), §19.34

Permalink:

http://dlmf.nist.gov/19.29.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(i), §19.29 and Ch.19

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19.29.8 $${\int }_y^x\frac{a_{\alpha }+b_{\alpha }t}{a_5+b_{5}t}\frac{dt}{s(t)}=\frac{2}{3}\frac{d_{\alpha \beta }{d}_{\alpha \gamma }{d}_{\alpha \delta }}{d_{\alpha 5}}{R}_J(U_{12}^2,U_{13}^2,U_{23}^2,U_{\alpha 5}^2)+2R_C(S_{\alpha 5}^2,Q_{\alpha 5}^2),$$ $S_{\alpha 5}^2\in ℂ\setminus (-\mathrm{\infty },0)$,

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\int$: integral, $(a,b)$: open interval, $\setminus$: set subtraction, $U_n$ , $S_n$ , $Q_{\alpha \beta }$ , $d_{\alpha \beta }$ , $s(t)$: product, $\alpha$: parameter, $\beta$: parameter, $\gamma$: parameter and $\delta$: parameter

Referenced by:

§19.29(i), §19.29(ii), §19.29(ii), §19.29(ii)

Permalink:

http://dlmf.nist.gov/19.29.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(i), §19.29 and Ch.19

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19.29.16 $$b_{\beta }{b}_{\gamma }I({𝐞}_{\alpha })=d_{\alpha \beta }{d}_{\alpha \gamma }I(-{𝐞}_{\alpha })+2b_{\alpha }\left(\frac{s(x)}{a_{\alpha }+b_{\alpha }x}-\frac{s(y)}{a_{\alpha }+b_{\alpha }y}\right),$$

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Symbols:

$I(𝐦)$: integral, $s(t)$: product, $d_{\alpha \beta }$ , $\alpha$: parameter, $\beta$: parameter and $\gamma$: parameter

Referenced by:

§19.34

Permalink:

http://dlmf.nist.gov/19.29.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §19.29(ii), §19.29 and Ch.19

►where $\alpha ,\beta ,\gamma$ is any permutation of the numbers $1,2,3$, and …

19: 31.15 Stieltjes Polynomials

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31.15.1 $$\frac{d^{2}w}{{dz}^2}+\left(\sum _{j=1}^N\frac{{\gamma }_j}{z-a_j}\right)\frac{dw}{dz}+\frac{\mathrm{\Phi }(z)}{{\prod }_{j=1}^N(z-a_j)}w=0,$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter, $N+1$: number of singularities and $\mathrm{\Phi }(z)$: polynomial

Referenced by:

§31.15(i)

Permalink:

http://dlmf.nist.gov/31.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(i), §31.15 and Ch.31

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31.15.4 $$G({\zeta }_1,{\zeta }_2,\mathrm{\dots },{\zeta }_n)=\prod _{k=1}^n\prod _{\mathrm{\ell }=1}^N{({\zeta }_k-a_{\mathrm{\ell }})}^{{\gamma }_{\mathrm{\ell }}/2}\prod _{j=k+1}^n({\zeta }_k-{\zeta }_j).$$

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Symbols:

$\gamma$: real or complex parameter, $j$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(ii), §31.15 and Ch.31

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31.15.12

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Defines:

$\rho (z)$: weight function (locally)

Symbols:

$z$: complex variable, $\gamma$: real or complex parameter, $j$: nonnegative integer, $a$: complex parameter and $N+1$: number of singularities

Permalink:

http://dlmf.nist.gov/31.15.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §31.15(iii), §31.15 and Ch.31

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20: 10.21 Zeros

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10.21.15 $$J_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +1\right)}\prod _{k=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{j_{\nu ,k}^2}\right),$$ $\nu \ge 0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $j_{\nu ,m}$: zeros of the Bessel function $J_{\nu }\left(x\right)$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.5.10

Referenced by:

§10.21(iii)

Permalink:

http://dlmf.nist.gov/10.21.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(iii), §10.21 and Ch.10

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10.21.16 $$J_{\nu }^{\prime }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu -1}}{2\mathrm{\Gamma }\left(\nu \right)}\prod _{k=1}^{\mathrm{\infty }}\left(1-\frac{z^2}{j_{\nu ,k}^{\prime }{}_{}{}^2}\right),$$ $\nu >0$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $j_{\nu ,m}^{\prime }$: zeros of the Bessel function derivative $J_{\nu }^{\prime }\left(x\right)$ , $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.5.11

Referenced by:

§10.21(iii)

Permalink:

http://dlmf.nist.gov/10.21.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(iii), §10.21 and Ch.10

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