finite sum of 6j symbols

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1—10 of 13 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.

ⓘ

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.

ⓘ

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.

ⓘ

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.

ⓘ

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.

ⓘ

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.

ⓘ

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.

ⓘ

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.

ⓘ

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.

ⓘ

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: 25.11 Hurwitz Zeta Function

… ►

25.11.10

\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{{\left(s\right)_{n}}}{n!}\zeta% \left(n+s\right)(1-a)^{n},

s\neq 1

|a-1|<1

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $!$ : factorial (as in $n!$ ), $n$ : nonnegative integer, $a$ : real or complex parameter, $s$ : complex variable and $z$ : complex variable
Keywords:: infinite series, series representation
Proof sketch:: Derivable using Taylor’s theorem (1.10.1) with $z=a$ , $z_{0}=1$ , (25.11.17), (5.2.4), (25.11.2).
Referenced by:: §25.11(iv), §25.11(iv), Erratum (V1.0.9) for Chapters 7, 25
Permalink:: http://dlmf.nist.gov/25.11.E10
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.9):: We have rewritten the original summation $\sum_{n=0}^{\infty}\frac{\Gamma\left(n+s\right)}{n!\Gamma\left(s\right)}\zeta% \left(n+s\right)(1-a)^{n}$ more concisely as $\sum_{n=0}^{\infty}\frac{{\left(s\right)_{n}}}{n!}\zeta\left(n+s\right)(1-a)^{n}$ using the Pochhammer symbol.
See also:: Annotations for §25.11(iv), §25.11 and Ch.25

… ►

25.11.36Removed because it is just (25.15.1) combined with (25.15.3).

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet $L$ -function and $k$ : nonnegative integer
Referenced by:: §25.11(x), Erratum (V1.0.23) for Equation (25.11.36), Erratum (V1.0.23) for Equation (25.11.36)
Permalink:: http://dlmf.nist.gov/25.11.E36
Removal (effective with 1.1.4):: This equation has been removed because it is just (25.15.1) combined with (25.15.3).
Clarification (effective with 1.0.23):: The Dirichlet $L$ -function was inserted on the left-hand side. The upper-index of the finitesum which originally was $k$ , was replaced with $k-1$ since $\chi(k)=0$ .
See also:: Annotations for §25.11(x), §25.11 and Ch.25

… ►

§25.11(xi) Sums

… ►For further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360). … ►

25.11.43

\zeta\left(s,a\right)-\frac{a^{1-s}}{s-1}-\frac{1}{2}a^{-s}\sim\sum_{k=1}^{% \infty}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}.

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\sim$ : Poincaré asymptotic expansion, $!$ : factorial (as in $n!$ ), $k$ : nonnegative integer, $a$ : real or complex parameter and $s$ : complex variable
Keywords:: asymptotic approximation
Source:: Paris (2005b, (1.3), (1.4), p. 298)
Referenced by:: §25.11(xii), Erratum (V1.0.9) for Chapters 7, 25
Permalink:: http://dlmf.nist.gov/25.11.E43
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.9):: We have rewritten the original summation $\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}\frac{\Gamma\left(s+2k-1\right)}{\Gamma% \left(s\right)}a^{1-s-2k}$ more concisely as $\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}$ using the Pochhammer symbol.
See also:: Annotations for §25.11(xii), §25.11 and Ch.25

…

finite sum of 6j symbols

1—10 of 13 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: 25.11 Hurwitz Zeta Function

§25.11(xi) Sums

finite sum of 6j symbols

1—10 of 13 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: 25.11 Hurwitz Zeta Function

§25.11(xi) Sums

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