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11: 25.13 Periodic Zeta Function
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►The notation is used for the polylogarithm with real:
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25.13.1
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is periodic in with period 1, and equals when is an integer.
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25.13.2
, ,
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25.13.3
if ; if .
12: 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).13: 34.10 Zeros
§34.10 Zeros
►In a symbol, if the three angular momenta do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the symbol is zero. Similarly the symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four symbols in the summation. …Such zeros are called nontrivial zeros. ►For further information, including examples of nontrivial zeros and extensions to symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).14: 34.13 Methods of Computation
§34.13 Methods of Computation
►Methods of computation for and 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 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). …15: 34.7 Basic Properties: Symbol
§34.7 Basic Properties: Symbol
… ►§34.7(ii) Symmetry
… ►§34.7(iv) Orthogonality
… ►§34.7(vi) Sums
… ►It constitutes an addition theorem for the symbol. …16: 16.4 Argument Unity
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►The function is well-poised if
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►See Raynal (1979) for a statement in terms of symbols (Chapter 34).
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►Transformations for both balanced and very well-poised are included in Bailey (1964, pp. 56–63).
A similar theory is available for very well-poised ’s which are 2-balanced.
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17: 15.4 Special Cases
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15.4.31
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15.4.32
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►where the limit interpretation (15.2.6), rather than (15.2.5), has to be taken when in (15.4.33) , and in (15.4.34) .
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18: 34.1 Special Notation
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►The main functions treated in this chapter are the Wigner symbols, respectively,
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►An often used alternative to the symbol is the Clebsch–Gordan coefficient
…For other notations for , , symbols, see Edmonds (1974, pp. 52, 97, 104–105) and Varshalovich et al. (1988, §§8.11, 9.10, 10.10).
nonnegative integers. | |
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19: 15.8 Transformations of Variable
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►Alternatively, if is a negative integer, then we interchange and in .
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►The hypergeometric functions that correspond to Groups 3 and 4 have a nonlinear function of as variable.
The transformation formulas between two hypergeometric functions in Group 2, or two hypergeometric functions in Group 3, are the linear transformations (15.8.1).
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