Dixon 3F2(1) sum

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1: 34.2 Definition: $\mathit{3}j$ Symbol

§34.2 Definition: $\mathit{3}j$ Symbol

►The quantities $j_1,j_2,j_3$ in the $\mathit{3}j$ symbol are called angular momenta. …where $r,s,t$ is any permutation of $1,2,3$. The corresponding projective quantum numbers $m_1,m_2,m_3$ are given by … ►When both conditions are satisfied the $\mathit{3}j$ symbol can be expressed as the finite sum …

2: 17.7 Special Cases of Higher ${}_r{}^{}\varphi _s^{}$ Functions

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F. H. Jackson’s Terminating $q$-Analog of Dixon’s Sum

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$q$-Analog of Dixon’s ${}_3{}^{}F_2^{}\left(1\right)$ Sum

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Gasper–Rahman $q$-Analog of Watson’s ${}_3{}^{}F_2^{}$ Sum

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First $q$-Analog of Bailey’s ${}_4{}^{}F_3^{}\left(1\right)$ Sum

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Second $q$-Analog of Bailey’s ${}_4{}^{}F_3^{}\left(1\right)$ Sum

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3: Peter A. Clarkson

… ► Dixon and J. …

4: 16.4 Argument Unity

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Lerch Sum

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Dixon’s Well-Poised Sum

… ►The function ${}_3{}^{}F_2^{}(a,b,c;d,e;1)$ is analytic in the parameters $a,b,c,d,e$ when its series expansion converges and the bottom parameters are not negative integers or zero. … ►Balanced ${}_4{}^{}F_3^{}\left(1\right)$ series have transformation formulas and three-term relations. … ►Transformations for both balanced ${}_4{}^{}F_3^{}\left(1\right)$ and very well-poised ${}_7{}^{}F_6^{}\left(1\right)$ are included in Bailey (1964, pp. 56–63). …

5: Bibliography D

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Unification of one-dimensional Fokker-Planck equations beyond hypergeometrics: Factorizer solution method and eigenvalue schemes. Phys. Rev. E (3) 57 (1), pp. 252–275.

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External Links:

ISSN 1063-651X, Document, MathReview (Victor I. Stepanov), ZentralBlatt

Referenced by:

§31.17(ii).

Encodings:

BibTeX

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Pole dynamics for elliptic solutions of the Korteweg-de Vries equation. Math. Phys. Anal. Geom. 3 (1), pp. 49–74.

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External Links:

ISSN 1385-0172, Document, MathReview (Giulio Soliani), ZentralBlatt

Referenced by:

§23.21(ii).

Encodings:

BibTeX

… ►

A simple sum formula for Clebsch-Gordan coefficients. Lett. Math. Phys. 5 (3), pp. 207–211.

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External Links:

ISSN 0377-9017, Document, MathReview, ZentralBlatt

Referenced by:

§14.17(iv).

Encodings:

BibTeX

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Infinite integrals in the theory of Bessel functions. Quart. J. Math., Oxford Ser. 1 (1), pp. 122–145.

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External Links:

Document, ZentralBlatt

Referenced by:

§10.32(iii).

Encodings:

BibTeX

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From Nonlinearity to Coherence: Universal Features of Nonlinear Behaviour in Many-Body Physics. Oxford University Press, Oxford.

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Referenced by:

Profile Peter A. Clarkson.

Encodings:

BibTeX

…

6: 10.32 Integral Representations

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10.32.2 $$I_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\pi }{\mathrm{e}}^{\pm z\cos \theta }{(\sin \theta )}^{2\nu }d\theta =\frac{{(\frac{1}{2}z)}^{\nu }}{{\pi }^{\frac{1}{2}}\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_{-1}^1{(1-t^2)}^{\nu -\frac{1}{2}}{\mathrm{e}}^{\pm zt}dt,$$ $\mathrm{\Re }\nu >-\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.18

Permalink:

http://dlmf.nist.gov/10.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32 and Ch.10

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10.32.8 $$K_{\nu }\left(z\right)=\frac{{\pi }^{\frac{1}{2}}{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-z\cosh t}{(\sinh t)}^{2\nu }dt=\frac{{\pi }^{\frac{1}{2}}{(\frac{1}{2}z)}^{\nu }}{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)}{\int }_1^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{(t^2-1)}^{\nu -\frac{1}{2}}dt,$$ $\mathrm{\Re }\nu >-\frac{1}{2}$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\sinh z$: hyperbolic sine function, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.23

Referenced by:

§10.74(iii)

Permalink:

http://dlmf.nist.gov/10.32.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32 and Ch.10

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10.32.14 $$K_{\nu }\left(z\right)=\frac{1}{2{\pi }^2\mathrm{i}}{\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\cos \left(\nu \pi \right){\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}-t-\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-t+\nu \right){(2z)}^{t}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.32(ii), §10.74(iii)

Permalink:

http://dlmf.nist.gov/10.32.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(ii), §10.32(ii), §10.32 and Ch.10

►In (10.32.14) the integration contour separates the poles of $\mathrm{\Gamma }\left(t\right)$ from the poles of $\mathrm{\Gamma }\left(\frac{1}{2}-t-\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-t+\nu \right)$. … ►

10.32.19 $$K_{\mu }\left(z\right)K_{\nu }\left(z\right)=\frac{1}{8\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(t+\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t+\frac{1}{2}\mu -\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu -\frac{1}{2}\nu \right)}{\mathrm{\Gamma }\left(2t\right)}{(\frac{1}{2}z)}^{-2t}dt,$$ .

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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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.32(iii)

Permalink:

http://dlmf.nist.gov/10.32.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32(iii), §10.32 and Ch.10

…

7: 34.5 Basic Properties: $\mathit{6}j$ Symbol

… ►Examples are provided by: … ►

§34.5(vi) Sums

… ►Equations (34.5.15) and (34.5.16) are the sum rules. They constitute addition theorems for the $\mathit{6}j$ symbol. … ►For other sums see Ginocchio (1991).

8: 34.3 Basic Properties: $\mathit{3}j$ Symbol

… ►When any one of $j_1,j_2,j_3$ is equal to $0,\frac{1}{2}$, or $1$, the $\mathit{3}j$ symbol has a simple algebraic form. …For these and other results, and also cases in which any one of $j_1,j_2,j_3$ is $\frac{3}{2}$ or $2$, see Edmonds (1974, pp. 125–127). … ►Even permutations of columns of a $\mathit{3}j$ symbol leave it unchanged; odd permutations of columns produce a phase factor ${(-1)}^{j_1+j_2+j_3}$, for example, … ►

§34.3(vi) Sums

►For sums of products of $\mathit{3}j$ symbols, see Varshalovich et al. (1988, pp. 259–262). …

9: 4.17 Special Values and Limits

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Table 4.17.1: Trigonometric functions: values at multiples of $\frac{1}{12}\pi$.

► ►►►

$\theta$	$\sin \theta$	$\cos \theta$	$\tan \theta$	$\csc \theta$	$\sec \theta$	$\cot \theta$
…
$\pi /3$	$\frac{1}{2}\sqrt{3}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2}{3}\sqrt{3}$	$2$	$\frac{1}{3}\sqrt{3}$
…

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4.17.1 $$\underset{z\to 0}{lim}\frac{\sin z}{z}=1,$$

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Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.74

Permalink:

http://dlmf.nist.gov/4.17.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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4.17.2 $$\underset{z\to 0}{lim}\frac{\tan z}{z}=1.$$

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Symbols:

$\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.3.75

Permalink:

http://dlmf.nist.gov/4.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

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4.17.3 $$\underset{z\to 0}{lim}\frac{1-\cos z}{z^2}=\frac{1}{2}.$$

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Symbols:

$\cos z$: cosine function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/4.17.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §4.17 and Ch.4

10: 34.4 Definition: $\mathit{6}j$ Symbol

§34.4 Definition: $\mathit{6}j$ Symbol

►The $\mathit{6}j$ symbol is defined by the following double sum of products of $\mathit{3}j$ symbols: …where the summation is taken over all admissible values of the $m$’s and $m^{\prime }$’s for each of the four $\mathit{3}j$ symbols; compare (34.2.2) and (34.2.3). ►Except in degenerate cases the combination of the triangle inequalities for the four $\mathit{3}j$ symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths $j_1,j_2,j_3,l_1,l_2,l_3$; see Figure 34.4.1. … ►For alternative expressions for the $\mathit{6}j$ symbol, written either as a finite sum or as other terminating generalized hypergeometric series ${}_4{}^{}F_3^{}$ of unit argument, see Varshalovich et al. (1988, §§9.2.1, 9.2.3).