3j symbols
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11: 34.9 Graphical Method
§34.9 Graphical Method
… ►For specific examples of the graphical method of representing sums involving the , and symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).12: 34.6 Definition: Symbol
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►The
symbol may be defined either in terms of
symbols or equivalently in terms of
symbols:
►
34.6.1
►
34.6.2
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13: 18.38 Mathematical Applications
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►
and Symbols
►The symbol (34.2.6), with an alternative expression as a terminating of unit argument, can be expressed in terms of Hahn polynomials (18.20.5) or, by (18.21.1), dual Hahn polynomials. The orthogonality relations in §34.3(iv) for the symbols can be rewritten in terms of orthogonality relations for Hahn or dual Hahn polynomials as given by §§18.2(i), 18.2(iii) and Table 18.19.1 or by §18.25(iii), respectively. … ►The symbol (34.4.3), with an alternative expression as a terminating balanced of unit argument, can be expressend in terms of Racah polynomials (18.26.3). …14: 34.8 Approximations for Large Parameters
§34.8 Approximations for Large Parameters
►For large values of the parameters in the , , and symbols, different asymptotic forms are obtained depending on which parameters are large. … ►
34.8.1
,
…
►Uniform approximations in terms of Airy functions for the and
symbols are given in Schulten and Gordon (1975b).
For approximations for the , , and
symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.
15: 34.4 Definition: Symbol
§34.4 Definition: Symbol
►The symbol is defined by the following double sum of products of symbols: …where the summation is taken over all admissible values of the ’s and ’s for each of the four symbols; compare (34.2.2) and (34.2.3). ►Except in degenerate cases the combination of the triangle inequalities for the four symbols in (34.4.1) is equivalent to the existence of a tetrahedron (possibly degenerate) with edges of lengths ; see Figure 34.4.1. … ►where is defined as in §16.2. …16: 16.4 Argument Unity
17: 34.7 Basic Properties: Symbol
18: 34.5 Basic Properties: Symbol
19: 10 Bessel Functions
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20: 18 Orthogonal Polynomials
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