higher-order 3nj symbols

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21: 34.13 Methods of Computation

§34.13 Methods of Computation

►Methods of computation for $\mathit{3}j$ and $\mathit{6}j$ symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981). ►For $\mathit{9}j$ symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989). …

22: 34.1 Special Notation

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$2j_1,2j_2,2j_3,2l_1,2l_2,2l_3$	nonnegative integers.
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►The main functions treated in this chapter are the Wigner $\mathit{3}j,\mathit{6}j,\mathit{9}j$ symbols, respectively, … ►An often used alternative to the $\mathit{3}j$ symbol is the Clebsch–Gordan coefficient ►

34.1.1

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Symbols:

: $\mathit{3}j$ symbol and $j,j_r$: non-negative integers or non-negative integers plus one half.

Sources:

Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))

Permalink:

http://dlmf.nist.gov/34.1.E1

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3}j$. The sign of $m_3$ was changed for clarity.

See also:

Annotations for §34.1 and Ch.34

… ►For other notations for $\mathit{3}j$, $\mathit{6}j$, $\mathit{9}j$ symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).

23: 34.9 Graphical Method

§34.9 Graphical Method

►The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …For specific examples of the graphical method of representing sums involving the $\mathit{3}j,\mathit{6}j$, and $\mathit{9}j$ symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).

24: 34.8 Approximations for Large Parameters

§34.8 Approximations for Large Parameters

►For large values of the parameters in the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols, different asymptotic forms are obtained depending on which parameters are large. … ►and the symbol $o\left(1\right)$ denotes a quantity that tends to zero as the parameters tend to infinity, as in §2.1(i). … ►Uniform approximations in terms of Airy functions for the $\mathit{3}j$ and $\mathit{6}j$ symbols are given in Schulten and Gordon (1975b). For approximations for the $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.

25: 34.3 Basic Properties: $\mathit{3}j$ Symbol

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

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§34.3(ii) Symmetry

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§34.3(iv) Orthogonality

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§34.3(vi) Sums

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26: 34 3j, 6j, 9j Symbols

Chapter 34 $\mathit{3}j,\mathit{6}j,\mathit{9}j$ Symbols

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27: 34.5 Basic Properties: $\mathit{6}j$ Symbol

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

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§34.5(ii) Symmetry

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§34.5(iv) Orthogonality

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§34.5(vi) Sums

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28: Bibliography R

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Some properties of Bernoulli’s numbers (J. Indian Math. Soc. 3 (1911), 219–234.). In Collected Papers,

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Referenced by:

§24.7(i).

Encodings:

BibTeX

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Notes:

Symbolic and extended-precision general purpose computer algebra system.

External Links:

Home Page

Referenced by:

§ ‣ Software Cross Index.

Encodings:

BibTeX

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An Introduction to Combinatorial Analysis. John Wiley & Sons Inc., New York.

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Notes:

Reprinted by Princeton University Press, Princeton, NJ, 1980 and reissued by Dover Publications Inc., New York, 2002.

External Links:

MathReview (L. Moser), ZentralBlatt

Referenced by:

§1.4(iii), §26.14(ii), §26.18, §26.3(ii), §26.3(iii), Ch.26.

Encodings:

BibTeX

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Calculation of $3n$-$j$ symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.

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External Links:

Document

Referenced by:

§34.13.

Encodings:

BibTeX

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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

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29: 16.24 Physical Applications

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§16.24(iii) $\mathit{3}j$, $\mathit{6}j$, and $\mathit{9}j$ Symbols

►The $\mathit{3}j$ symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. They can be expressed as ${}_3{}^{}F_2^{}$ functions with unit argument. The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner $\mathit{6}j$ symbols. These are balanced ${}_4{}^{}F_3^{}$ functions with unit argument. …

30: Bibliography G

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Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.

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External Links:

ISBN 1-4020-0101-0, MathReview Entry

Referenced by:

Profile Mourad E.H. Ismail.

Encodings:

BibTeX

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A new identity for some six-$j$ symbols. J. Math. Phys. 32 (6), pp. 1430–1432.

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External Links:

ISSN 0022-2488, Document, MathReview

Referenced by:

§34.5(vi).

Encodings:

BibTeX

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Matrices, moments and quadrature with applications. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ.

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External Links:

ISBN 978-0-691-14341-5, MathReview (G. A. Evans)

Referenced by:

§18.2(ix), §18.40(ii).

Encodings:

BibTeX

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Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

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Notes:

In Russian. English translation: Differential Equations 18(3), pp. 317–326.

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).

Encodings:

BibTeX

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Spectral Methods and Their Applications. World Scientific Publishing Co. Inc., River Edge, NJ-Singapore.

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External Links:

ISBN 981-02-3333-7, MathReview (K. Moszyński), ZentralBlatt

Referenced by:

§18.38(i).

Encodings:

BibTeX

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