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21: 17.7 Special Cases of Higher Functions
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-Analog of Bailey’s Sum
… ►-Analog of Gauss’s Sum
… ►-Analog of Dixon’s Sum
… ►Gasper–Rahman -Analog of Watson’s Sum
… ►Andrews’ -Analog of the Terminating Version of Watson’s Sum (16.4.6)
…22: 18.17 Integrals
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►For the beta function see §5.12, and for the confluent hypergeometric function see (16.2.1) and Chapter 13.
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►For the confluent hypergeometric function see (16.2.1) and Chapter 13.
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►For the hypergeometric function see §§15.1 and 15.2(i).
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►For the generalized hypergeometric function see (16.2.1).
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►For further integrals, see Apelblat (1983, pp. 189–204), Erdélyi et al. (1954a, pp. 38–39, 94–95, 170–176, 259–261, 324), Erdélyi et al. (1954b, pp. 42–44, 271–294), Gradshteyn and Ryzhik (2000, pp. 788–806), Gröbner and Hofreiter (1950, pp. 23–30), Marichev (1983, pp. 216–247), Oberhettinger (1972, pp. 64–67), Oberhettinger (1974, pp. 83–92), Oberhettinger (1990, pp. 44–47 and 152–154), Oberhettinger and Badii (1973, pp. 103–112), Prudnikov et al. (1986b, pp. 420–617), Prudnikov et al. (1992a, pp. 419–476), and Prudnikov et al. (1992b, pp. 280–308).
23: Tom H. Koornwinder
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►Koornwinder has published numerous papers on special functions, harmonic analysis, Lie groups, quantum groups, computer algebra, and their interrelations, including an interpretation of Askey–Wilson polynomials on quantum SU(2), and a five-parameter extension (the Macdonald–Koornwinder polynomials) of Macdonald’s polynomials for root systems BC.
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24: 34.12 Physical Applications
§34.12 Physical Applications
►The angular momentum coupling coefficients (, , and symbols) are essential in the fields of nuclear, atomic, and molecular physics. …, and symbols are also found in multipole expansions of solutions of the Laplace and Helmholtz equations; see Carlson and Rushbrooke (1950) and Judd (1976).25: 9 Airy and Related Functions
Chapter 9 Airy and Related Functions
…26: 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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27: 34 3j, 6j, 9j Symbols
Chapter 34 Symbols
…28: 3.2 Linear Algebra
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►Assume that can be factored as in (3.2.5), but without partial pivoting.
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►where is the largest of the absolute values of the eigenvalues of the matrix ; see §3.2(iv).
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►is called the characteristic polynomial of and its zeros are the eigenvalues of .
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►has the same eigenvalues as .
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►Many methods are available for computing eigenvalues; see Golub and Van Loan (1996, Chapters 7, 8), Trefethen and Bau (1997, Chapter 5), and Wilkinson (1988, Chapters 8, 9).