generalized Mehler–Fock transformation
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11: 9.1 Special Notation
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►Other notations that have been used are as follows: and for and (Jeffreys (1928), later changed to and ); , (Fock (1945)); (Szegő (1967, §1.81)); , (Tumarkin (1959)).
nonnegative integer, except in §9.9(iii). | |
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12: 16.24 Physical Applications
§16.24 Physical Applications
►§16.24(i) Random Walks
►Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. … ►§16.24(iii) , , and Symbols
… ►The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner symbols. …13: 8.16 Generalizations
§8.16 Generalizations
►For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). Other generalizations are considered in Guthmann (1991) and Paris (2003).14: 14.1 Special Notation
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►The main functions treated in this chapter are the Legendre functions , , , ; Ferrers functions , (also known as the Legendre functions on the cut); associated Legendre functions , , ; conical functions , , , , (also known as Mehler functions).
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, , | real variables. |
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, | general order and degree, respectively. |
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15: 16.6 Transformations of Variable
§16.6 Transformations of Variable
►Quadratic
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16.6.2
►For Kummer-type transformations of functions see Miller (2003) and Paris (2005a), and for further transformations see Erdélyi et al. (1953a, §4.5), Miller and Paris (2011), Choi and Rathie (2013) and Wang and Rathie (2013).
16: Bibliography F
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Generalized parabolic cylinder functions.
Asymptotic Anal. 5 (6), pp. 517–531.
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Theory and Computation of Spheroidal Harmonics with General Arguments.
Master’s Thesis, The University of Western Australia, Department of Physics.
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Electromagnetic Diffraction and Propagation Problems.
International Series of Monographs on Electromagnetic Waves,
Vol. 1, Pergamon Press, Oxford.
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Diffraction of radio waves around the earth’s surface.
Acad. Sci. USSR. J. Phys. 9, pp. 255–266.
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Generalized Fermi-Dirac integrals—FD, FDG, FDH.
Comput. Phys. Comm. 39 (2), pp. 181–185.
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17: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
►The function is defined by ►
10.46.1
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►For asymptotic expansions of as in various sectors of the complex -plane for fixed real values of and fixed real or complex values of , see Wright (1935) when , and Wright (1940b) when .
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►The Laplace transform of can be expressed in terms of the Mittag-Leffler function:
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18: Bibliography G
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On the generalization of a method for computing Bessel function integrals.
J. Comput. Appl. Math. 6 (2), pp. 167–168.
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Formulas of the Dirichlet-Mehler Type.
In Fractional Calculus and its Applications, B. Ross (Ed.),
Lecture Notes in Math., Vol. 457, pp. 207–215.
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On the computation of generalized Fermi-Dirac and Bose-Einstein integrals.
Comput. Phys. Comm. 74 (2), pp. 233–238.
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Generalized Functions. Vol. 1: Properties and Operations.
Academic Press, New York.
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Generalized Fermi-Dirac functions and derivatives: Properties and evaluation.
Comput. Phys. Comm. 136 (3), pp. 294–309.
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