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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: 34.10 Zeros
…
►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).
13: 34.13 Methods of Computation
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►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).
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14: 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. …15: 16.24 Physical Applications
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►
§16.24(iii) , , and Symbols
… ►They can be expressed as functions with unit argument. …These are balanced functions with unit argument. Lastly, special cases of the symbols are functions with unit argument. …16: 24.16 Generalizations
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►For , Bernoulli and Euler polynomials of order
are defined respectively by
…When they reduce to the Bernoulli and Euler numbers of
order
:
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►For extensions of to complex values of , , and , and also for uniform asymptotic expansions for large and large , see Temme (1995b) and López and Temme (1999b, 2010b).
…
►
is a polynomial in of degree .
…
►Generalized Bernoulli numbers and polynomials belonging to are defined by
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17: 34.1 Special Notation
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►
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►The main functions treated in this chapter are the Wigner symbols, respectively,
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►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. | |
… |
18: 9.4 Maclaurin Series
19: 10 Bessel Functions
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20: 23 Weierstrass Elliptic and Modular
Functions
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