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1: 5.17 Barnes’ $G$-Function (Double Gamma Function)

§5.17 Barnes’ $G$-Function (Double Gamma Function)

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$G\left(1\right)=1,$

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5.17.2 $$G\left(n\right)=(n-2)!(n-3)!\mathrm{\cdots }1!,$$ $n=2,3,\mathrm{\dots }$.

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Symbols:

$G\left(z\right)$: Barnes’ $G$-function (or double gamma function), $!$: factorial (as in $n!$) and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/5.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §5.17 and Ch.5

… ►When $z\to \mathrm{\infty }$ in , …Here $B_{2k+2}$ is the Bernoulli number (§24.2(i)), and $A$ is Glaisher’s constant, given by …

2: 24.1 Special Notation

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Bernoulli Numbers and Polynomials

►The origin of the notation $B_n$, $B_n\left(x\right)$, is not clear. … ►

Euler Numbers and Polynomials

… ►Its coefficients were first studied in Euler (1755); they were called Euler numbers by Raabe in 1851. The notations $E_n$, $E_n\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …

3: 5.1 Special Notation

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$j,m,n$	nonnegative integers.
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$\delta$	arbitrary small positive constant.
…

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4: 19.11 Addition Theorems

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$\mathrm{\Delta }(\theta )=\sqrt{1-k^2{\sin }^2\theta }.$

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$$\delta ={\alpha }^2(1-{\alpha }^2)({\alpha }^2-k^2).$$

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19.11.6_5 $$R_C(\gamma -\delta ,\gamma )=\frac{-1}{\sqrt{\delta }}\arctan \left(\frac{\sqrt{\delta }\sin \theta \sin \varphi \sin \psi }{{\alpha }^2-1-{\alpha }^2\cos \theta \cos \varphi \cos \psi }\right).$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\cos z$: cosine function, $\arctan z$: arctangent function, $\sin z$: sine function, $\varphi$: real or complex argument, ${\alpha }^2$: real or complex parameter, $\theta$: amplitude, $\psi$: angle, $\gamma$ and $\delta$

Referenced by:

§19.11(i), Erratum (V1.0.28) for Equations (15.2.3_5), (19.11.6_5)

Permalink:

http://dlmf.nist.gov/19.11.E6_5

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.28):

This equation was given a number.

See also:

Annotations for §19.11(i), §19.11 and Ch.19

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$$\delta ={\alpha }^2(1-{\alpha }^2)({\alpha }^2-k^2).$$

… ►If $\varphi =\theta$ in §19.11(i) and $\mathrm{\Delta }(\theta )$ is again defined by (19.11.3), then …

5: 31.2 Differential Equations

… ►This equation has regular singularities at $0,1,a,\mathrm{\infty }$, with corresponding exponents $\{0,1-\gamma \}$, $\{0,1-\delta \}$, $\{0,1-ϵ\}$, $\{\alpha ,\beta \}$, respectively (§2.7(i)). … ►The parameters play different roles: $a$ is the singularity parameter; $\alpha ,\beta ,\gamma ,\delta ,ϵ$ are exponent parameters; $q$ is the accessory parameter. … ►Next, $w(z)={(z-1)}^{1-\delta }{w}_2(z)$ satisfies (31.2.1) if $w_2$ is a solution of (31.2.1) with transformed parameters $q_2=q+a\gamma (1-\delta )$; ${\alpha }_2=\alpha +1-\delta$, ${\beta }_2=\beta +1-\delta$, ${\delta }_2=2-\delta$. … ►For example, if $\tilde{z}=z/a$, then the parameters are $\tilde{a}=1/a$, $\tilde{q}=q/a$; $\tilde{\delta }=ϵ$, $\widetilde{ϵ}=\delta$. …For example, $w(z)={(1-z)}^{-\alpha }\tilde{w}(z/(z-1))$, which arises from $\tilde{z}=z/(z-1)$, satisfies (31.2.1) if $\tilde{w}(\tilde{z})$ is a solution of (31.2.1) with $z$ replaced by $\tilde{z}$ and transformed parameters $\tilde{a}=a/(a-1)$, $\tilde{q}=-(q-a\alpha \gamma )/(a-1)$; $\tilde{\beta }=\alpha +1-\delta$, $\tilde{\delta }=\alpha +1-\beta$. …

6: 14.17 Integrals

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§14.17(ii) Barnes’ Integral

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14.17.6 $${\int }_{-1}^1{𝖯}_l^m\left(x\right){𝖯}_n^m\left(x\right)dx=\frac{(n+m)!}{(n-m)!\left(n+\frac{1}{2}\right)}{\delta }_{l,n},$$

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Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

A&S Ref:

8.14.11 8.14.13

Referenced by:

§14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/14.17.E6

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

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14.17.7 $${\int }_{-1}^1{𝖯}_l^m\left(x\right){𝖯}_n^{-m}\left(x\right)dx=\frac{{(-1)}^m}{l+\frac{1}{2}}{\delta }_{l,n},$$

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Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

Referenced by:

§14.17(iii)

Permalink:

http://dlmf.nist.gov/14.17.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right and ${(-1)}^m$ has been moved to the numerator of the fraction.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

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14.17.8 $${\int }_{-1}^1\frac{{𝖯}_n^l\left(x\right){𝖯}_n^m\left(x\right)}{1-x^2}dx=\frac{(n+m)!}{(n-m)!m}{\delta }_{l,m},$$ $m>0$,

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Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

A&S Ref:

8.14.12 8.14.14

Referenced by:

§14.17(iii)

Permalink:

http://dlmf.nist.gov/14.17.E8

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

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14.17.9 $${\int }_{-1}^1\frac{{𝖯}_n^l\left(x\right){𝖯}_n^{-m}\left(x\right)}{1-x^2}dx=\frac{{(-1)}^l}{l}{\delta }_{l,m},$$ $l>0$.

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Symbols:

: Ferrers function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $m$: nonnegative integer, $n$: nonnegative integer and $l$: nonnegative integer

Referenced by:

§14.17(iii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/14.17.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right and ${(-1)}^l$ has been moved to the numerator of the fraction.

See also:

Annotations for §14.17(iii), §14.17 and Ch.14

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7: 13.27 Mathematical Applications

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13.27.1

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Permalink:

http://dlmf.nist.gov/13.27.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.27 and Ch.13

►where $\alpha$, $\beta$, $\gamma$, $\delta$ are real numbers, and $\gamma >0$. …

8: 1.17 Integral and Series Representations of the Dirac Delta

§1.17 Integral and Series Representations of the Dirac Delta

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§1.17(i) Delta Sequences

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Sine and Cosine Functions

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Coulomb Functions (§33.14(iv))

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Airy Functions (§9.2)

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9: 14.16 Zeros

… ►where $m$, $n\in \mathbb{Z}$ and ${\delta }_{\mu }$, ${\delta }_{\nu }\in (0,1)$. … ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)$ if any of the following sets of conditions hold: … ►

$\mu >0$, $n\ge m$, and ${\delta }_{\nu }>{\delta }_{\mu }$.

… ►The number of zeros of ${𝖯}_{\nu }^{\mu }\left(x\right)$ in the interval $(-1,1)$ is $\max (\lceil \nu -|\mu |\rceil ,0)+1$ if either of the following sets of conditions holds: ►

$\mu >0$, $n>m$, and ${\delta }_{\nu }\le {\delta }_{\mu }$.

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10: 3.9 Acceleration of Convergence

… ►Here $\mathrm{\Delta }$ is the forward difference operator: … ►

§3.9(iii) Aitken’s ${\mathrm{\Delta }}^2$-Process

… ► … ►Shanks’ transformation is a generalization of Aitken’s ${\mathrm{\Delta }}^2$-process. … ►Aitken’s ${\mathrm{\Delta }}^2$-process is the case $k=1$. …