finite sum of 6j symbols

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1—10 of 12 matching pages

1: 34.6 Definition: $\mathit{9j}$ Symbol

… ►The

\mathit{9j}

symbol may be defined either in terms of

\mathit{3j}

symbols or equivalently in terms of

\mathit{6j}

symbols: ►

34.6.1

\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{\mbox{\scriptsize all }m_{rs}}\begin{% pmatrix}j_{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix},

ⓘ

Defines:: $\begin{Bmatrix}\NVar{j_{11}}&\NVar{j_{12}}&\NVar{j_{13}}\\ \NVar{j_{21}}&\NVar{j_{22}}&\NVar{j_{23}}\\ \NVar{j_{31}}&\NVar{j_{32}}&\NVar{j_{33}}\end{Bmatrix}$ : $\mathit{9j}$ symbol
Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol, $j,j_{r}$ : non-negative integers or non-negative integers plus one half., $r$ : nonnegative integer and $s$ : integer
Permalink:: http://dlmf.nist.gov/34.6.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §34.6 and Ch.34

…

2: 34.4 Definition: $\mathit{6j}$ Symbol

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

… ►The

\mathit{6j}

symbol can be expressed as the finite sum … ►For alternative expressions for the

\mathit{6j}

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).

3: 34.5 Basic Properties: $\mathit{6j}$ Symbol

… ►Examples are provided by: …

4: Bibliography R

… ►

J. Raynal (1979) On the definition and properties of generalized

6

j

symbols. J. Math. Phys. 20 (12), pp. 2398–2415.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (S. Saran), ZentralBlatt
Referenced by:: §16.4(iii), §16.4(iii).
Encodings:: BibTeX

… ►

H. Rosengren (1999) Another proof of the triple sum formula for Wigner

9j

-symbols. J. Math. Phys. 40 (12), pp. 6689–6691.

ⓘ

External Links:: ISSN 0022-2488, Document, MathReview (Michael Wüstner), ZentralBlatt
Referenced by:: §34.6.
Encodings:: BibTeX

… ►

G. Rota, D. Kahaner, and A. Odlyzko (1973) On the foundations of combinatorial theory. VIII. Finite operator calculus. J. Math. Anal. Appl. 42, pp. 684–760.

ⓘ

External Links:: ISSN 0022-247X, Document, Link, MathReview (P. Saphar), ZentralBlatt
Referenced by:: §18.2(xii).
Encodings:: BibTeX

… ►

M. Rotenberg, R. Bivins, N. Metropolis, and J. K. Wooten, Jr. (1959) The

3

j

and

6

j

Symbols. The Technology Press, MIT, Cambridge, MA.

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External Links:: ISBN 0-262-18004-9
Referenced by:: (34.1.1), §34.1, §34.1, §34.14, Erratum (V1.0.14) for Section 34.1, Erratum (V1.0.15) for Section 34.1.
Encodings:: BibTeX

… ►

K. Rottbrand (2000) Finite-sum rules for Macdonald’s functions and Hankel’s symbols. Integral Transform. Spec. Funct. 10 (2), pp. 115–124.

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External Links:: ISSN 1065-2469, Document, MathReview, ZentralBlatt
Referenced by:: §10.60(iii).
Encodings:: BibTeX

…

$\mathbb{C}$	complex plane (excluding infinity).
…
empty sums	zero.
…
$n!!$	double factorial: $2\cdot 4\cdot 6\cdots n$ if $n=2,4,6,\dotsc$ ; $1\cdot 3\cdot 5\cdots n$ if $n=1,3,5,\dotsc$ ; 1 if $n=0,-1$ .
…
$<\infty$	is finite, or converges.
…

$(a,b]$ or $[a,b)$	half-closed intervals.
…
$[a_{j,k}]$ or $[a_{jk}]$	matrix with $(j,k)$ th element $a_{j,k}$ or $a_{jk}$ .
…

7: Errata

… ►

Subsection 31.11(iv)

Just below (31.11.17), $P_{j}$ has been replaced with $P_{j}^{6}$ .

… ►

Chapters 10 Bessel Functions, 18 Orthogonal Polynomials, 34 3j, 6j, 9j Symbols

The Legendre polynomial $P_{n}$ was mistakenly identified as the associated Legendre function $P_{n}$ in §§10.54, 10.59, 10.60, 18.18, 18.41, 34.3 (and was thus also affected by the bug reported below). These symbols now link correctly to their definitions. Reported by Roy Hughes on 2022-05-23

… ►

Section 34.1

The relation between Clebsch-Gordan and $\mathit{3j}$ symbols was clarified, and the sign of $m_{3}$ was changed for readability. The reference Condon and Shortley (1935) for the Clebsch-Gordan coefficients was replaced by Edmonds (1974) and Rotenberg et al. (1959) and the references for $\mathit{3j}$ , $\mathit{6j}$ , $\mathit{9j}$ symbols were made more precise in §34.1.

… ►

Section 34.1

The reference for Clebsch-Gordan coefficients, Condon and Shortley (1935), was replaced by Edmonds (1974) and Rotenberg et al. (1959). The references for $\mathit{3j}$ , $\mathit{6j}$ , $\mathit{9j}$ symbols were made more precise.

… ►

Equation (34.7.4)

34.7.4

\begin{pmatrix}j_{13}&j_{23}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}\begin{Bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{Bmatrix}=\sum_{m_{r1},m_{r2},r=1,2,3}\begin{pmatrix}j% _{11}&j_{12}&j_{13}\\ m_{11}&m_{12}&m_{13}\end{pmatrix}\*\begin{pmatrix}j_{21}&j_{22}&j_{23}\\ m_{21}&m_{22}&m_{23}\end{pmatrix}\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{31}&m_{32}&m_{33}\end{pmatrix}\*\begin{pmatrix}j_{11}&j_{21}&j_{31}\\ m_{11}&m_{21}&m_{31}\end{pmatrix}\begin{pmatrix}j_{12}&j_{22}&j_{32}\\ m_{12}&m_{22}&m_{32}\end{pmatrix}

Originally the third $\mathit{3j}$ symbol in the summation was written incorrectly as $\begin{pmatrix}j_{31}&j_{32}&j_{33}\\ m_{13}&m_{23}&m_{33}\end{pmatrix}.$

Reported 2015-01-19 by Yan-Rui Liu.

…

8: Bibliography F

… ►

J. L. Fields and Y. L. Luke (1963b) Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. J. Math. Anal. Appl. 6 (3), pp. 394–403.

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External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.11(iii).
Encodings:: BibTeX

… ►

J. L. Fields (1983) Uniform asymptotic expansions of a class of Meijer

G

-functions for a large parameter. SIAM J. Math. Anal. 14 (6), pp. 1204–1253.

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External Links:: ISSN 0036-1410, Document, MathReview (Roop N. Kesarwani), ZentralBlatt
Referenced by:: §16.22.
Encodings:: BibTeX

… ►

J. P. M. Flude (1998) The Edmonds asymptotic formulas for the

3j

and

6j

symbols. J. Math. Phys. 39 (7), pp. 3906–3915.

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External Links:: ISSN 0022-2488, Document, MathReview (A. J. Coleman), ZentralBlatt
Referenced by:: §34.8.
Encodings:: BibTeX

… ►

P. J. Forrester and N. S. Witte (2002) Application of the

\tau

-function theory of Painlevé equations to random matrices:

\mathrm{P}_{\mathrm{V}}

\mathrm{P}_{\mathrm{III}}

, the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55 (6), pp. 679–727.

ⓘ

External Links:: ISSN 0010-3640, Document, MathReview (Mark Adler), ZentralBlatt
Referenced by:: §32.10(iii), §32.10(v), §32.14, §32.6(iv), §32.6(v).
Encodings:: BibTeX

… ►

T. Fort (1948) Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

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External Links:: MathReview (W. J. Trjitzinsky), ZentralBlatt
Referenced by:: §26.1, §26.1, Notations S, Notations S.
Encodings:: BibTeX

…

9: Bibliography W

… ►

R. S. Ward (1987) The Nahm equations, finite-gap potentials and Lamé functions. J. Phys. A 20 (10), pp. 2679–2683.

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External Links:: ISSN 0305-4470, Document, MathReview (James M. D’Archangelo), ZentralBlatt
Referenced by:: §29.19(ii).
Encodings:: BibTeX

… ►

G. N. Watson (1935b) The surface of an ellipsoid. Quart. J. Math., Oxford Ser. 6, pp. 280–287.

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External Links:: Document, ZentralBlatt
Referenced by:: §19.33(i).
Encodings:: BibTeX

… ►

J. K. G. Watson (1999) Asymptotic approximations for certain

6

j

and

9

j

symbols. J. Phys. A 32 (39), pp. 6901–6902.

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External Links:: ISSN 0305-4470, Document, MathReview, ZentralBlatt
Referenced by:: §34.8, §34.8.
Encodings:: BibTeX

… ►

D. V. Widder (1941) The Laplace Transform. Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, NJ.

ⓘ

External Links:: FullText, MathReview (J. D. Tamarkin), ZentralBlatt
Referenced by:: §1.14(vi), §1.15(viii).
Encodings:: BibTeX

… ►

C. Y. Wu (1982) A series of inequalities for Mills’s ratio. Acta Math. Sinica 25 (6), pp. 660–670.

ⓘ

External Links:: MathReview (J. S. Huang), ZentralBlatt
Referenced by:: §7.8.
Encodings:: BibTeX

…

10: 18.38 Mathematical Applications

… ►

$3j$ and $6j$ Symbols

… ► ►The

6j

symbol (34.4.3), with an alternative expression as a terminating balanced

{{}_{4}F_{3}}

of unit argument, can be expressend in terms of Racah polynomials (18.26.3). The orthogonality relations (34.5.14) for the

6j

symbols can be rewritten in terms of orthogonality relations for Racah polynomials as given by (18.25.9)–(18.25.12). … …

finite sum of 6j symbols

1—10 of 12 matching pages

1: 34.6 Definition: $\mathit{9j}$ Symbol

2: 34.4 Definition: $\mathit{6j}$ Symbol

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

3: 34.5 Basic Properties: $\mathit{6j}$ Symbol

4: Bibliography R

5: Mathematical Introduction

6: 10.22 Integrals

§10.22(ii) Integrals over Finite Intervals

7: Errata

8: Bibliography F

9: Bibliography W

10: 18.38 Mathematical Applications

$3j$ and $6j$ Symbols

finite sum of 6j symbols

1—10 of 12 matching pages

1: 34.6 Definition: 9 ⁢ j Symbol

2: 34.4 Definition: 6 ⁢ j Symbol

§34.4 Definition: 6 ⁢ j Symbol

3: 34.5 Basic Properties: 6 ⁢ j Symbol

4: Bibliography R

5: Mathematical Introduction

6: 10.22 Integrals

§10.22(ii) Integrals over Finite Intervals

7: Errata

8: Bibliography F

9: Bibliography W

10: 18.38 Mathematical Applications

3 ⁢ j and 6 ⁢ j Symbols

1: 34.6 Definition: $\mathit{9j}$ Symbol

2: 34.4 Definition: $\mathit{6j}$ Symbol

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

3: 34.5 Basic Properties: $\mathit{6j}$ Symbol

$3j$ and $6j$ Symbols