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1: 34.13 Methods of Computation
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βΊMethods 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).
βΊFor 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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2: Bibliography H
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Sums with cylindrical functions that reduce to the probability function and to related functions.
Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).
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Lamé polynomials of large order.
SIAM J. Math. Anal. 8 (5), pp. 800–842.
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The higher arithmetic.
American Scientist 97, pp. 364–368.
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Calculation of Coulomb wave functions for high energies.
Phys. Rev. 54 (8), pp. 627–628.
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On congruences involving Bernoulli numbers and irregular primes. II.
Rep. Fac. Sci. Technol. Meijo Univ. 31, pp. 1–8.
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3: 27.2 Functions
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27.2.9
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βΊIt is the special case of the function that counts the number of ways of expressing as the product of factors, with the order of factors taken into account.
…Note that .
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βΊTable 27.2.2 tabulates the Euler totient function , the divisor function (), and the sum of the divisors (), for .
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4: Bibliography
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Tables of for Complex Argument.
Pergamon Press, New York.
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Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”.
Trans. Camb. Phil. Soc. 8, pp. 595–599.
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Frequency response characteristics of the multiport planar elliptic patch.
IEEE Trans. Microwave Theory Tech. 40 (8), pp. 1726–1730.
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Algorithm 588. Fast Hankel transforms using related and lagged convolutions.
ACM Trans. Math. Software 8 (4), pp. 369–370.
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-identities of Auluck, Carlitz, and Rogers.
Duke Math. J. 33 (3), pp. 575–581.
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5: 26.10 Integer Partitions: Other Restrictions
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denotes the number of partitions of into distinct parts.
denotes the number of partitions of into at most distinct parts.
denotes the number of partitions of into parts with difference at least .
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βΊNote that , with strict inequality for .
It is known that for , , with strict inequality for sufficiently large, provided that , or ; see Yee (2004).
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6: 34.1 Special Notation
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βΊThe main functions treated in this chapter are the Wigner symbols, respectively,
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βΊFor 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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7: Bibliography B
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The generating function of Jacobi polynomials.
J. London Math. Soc. 13, pp. 8–12.
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Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces.
J. Phys. A 8 (4), pp. 566–584.
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Coulomb functions for large charges and small velocities.
Phys. Rev. (2) 97 (2), pp. 542–554.
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Numerical aspects of Mathieu eigenvalues.
Rend. Circ. Mat. Palermo (2) 15, pp. 51–97.
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A Dictionary of Inequalities.
Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 97, Longman, Harlow.
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8: 15.7 Continued Fractions
9: 21.9 Integrable Equations
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10: Bibliography K
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Special functions and the Bieberbach conjecture.
Amer. Math. Monthly 95 (8), pp. 689–696.
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Introduction to Elliptic Curves and Modular Forms.
2nd edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York.
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The addition formula for Laguerre polynomials.
SIAM J. Math. Anal. 8 (3), pp. 535–540.
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Bessel Functions and their Applications.
Analytical Methods and Special Functions, Vol. 8, Taylor & Francis Ltd., London-New York.
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The Painlevé-Kowalevski and poly-Painlevé tests for integrability.
Stud. Appl. Math. 86 (2), pp. 87–165.
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