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generalized Mehler–Fock transformation

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11: 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) 3 j , 6 j , and 9 j Symbols
The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner 6 j symbols. …
12: 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).
13: 14.1 Special Notation
x , y , τ

real variables.

μ , ν

general order and degree, respectively.

The main functions treated in this chapter are the Legendre functions P ν ( x ) , Q ν ( x ) , P ν ( z ) , Q ν ( z ) ; Ferrers functions P ν μ ( x ) , Q ν μ ( x ) (also known as the Legendre functions on the cut); associated Legendre functions P ν μ ( z ) , Q ν μ ( z ) , Q ν μ ( z ) ; conical functions P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) , Q ^ - 1 2 + i τ μ ( x ) , P - 1 2 + i τ μ ( x ) , Q - 1 2 + i τ μ ( x ) (also known as Mehler functions). …
14: 16.6 Transformations of Variable
§16.6 Transformations of Variable
Quadratic
Cubic
16.6.2 F 2 3 ( a , 2 b - a - 1 , 2 - 2 b + a b , a - b + 3 2 ; z 4 ) = ( 1 - z ) - a F 2 3 ( 1 3 a , 1 3 a + 1 3 , 1 3 a + 2 3 b , a - b + 3 2 ; - 27 z 4 ( 1 - z ) 3 ) .
For Kummer-type transformations of F 2 2 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).
15: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
The function ϕ ( ρ , β ; z ) is defined by
10.46.1 ϕ ( ρ , β ; z ) = k = 0 z k k ! Γ ( ρ k + β ) , ρ > - 1 .
For asymptotic expansions of ϕ ( ρ , β ; z ) as z in various sectors of the complex z -plane for fixed real values of ρ and fixed real or complex values of β , see Wright (1935) when ρ > 0 , and Wright (1940b) when - 1 < ρ < 0 . … The Laplace transform of ϕ ( ρ , β ; z ) can be expressed in terms of the Mittag-Leffler function: …
16: Bibliography F
  • M. Faierman (1992) Generalized parabolic cylinder functions. Asymptotic Anal. 5 (6), pp. 517–531.
  • P. Falloon (2001) Theory and Computation of Spheroidal Harmonics with General Arguments. Master’s Thesis, The University of Western Australia, Department of Physics.
  • V. A. Fock (1965) Electromagnetic Diffraction and Propagation Problems. International Series of Monographs on Electromagnetic Waves, Vol. 1, Pergamon Press, Oxford.
  • V. Fock (1945) Diffraction of radio waves around the earth’s surface. Acad. Sci. USSR. J. Phys. 9, pp. 255–266.
  • L. W. Fullerton and G. A. Rinker (1986) Generalized Fermi-Dirac integrals—FD, FDG, FDH. Comput. Phys. Comm. 39 (2), pp. 181–185.
  • 17: 14.12 Integral Representations
    §14.12(i) - 1 < x < 1
    Mehler–Dirichlet Formula
    14.12.1 P ν μ ( cos θ ) = 2 1 / 2 ( sin θ ) μ π 1 / 2 Γ ( 1 2 - μ ) 0 θ cos ( ( ν + 1 2 ) t ) ( cos t - cos θ ) μ + ( 1 / 2 ) d t , 0 < θ < π , μ < 1 2 .
    14.12.2 P ν - μ ( x ) = ( 1 - x 2 ) - μ / 2 Γ ( μ ) x 1 P ν ( t ) ( t - x ) μ - 1 d t , μ > 0 ;
    14.12.5 P ν - μ ( x ) = ( x 2 - 1 ) - μ / 2 Γ ( μ ) 1 x P ν ( t ) ( x - t ) μ - 1 d t , μ > 0 .
    18: Bibliography G
  • B. Gabutti (1980) On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.
  • G. Gasper (1975) Formulas of the Dirichlet-Mehler Type. In Fractional Calculus and its Applications, B. Ross (Ed.), Lecture Notes in Math., Vol. 457, pp. 207–215.
  • W. Gautschi (1993) On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.
  • I. M. Gel’fand and G. E. Shilov (1964) Generalized Functions. Vol. 1: Properties and Operations. Academic Press, New York.
  • Z. Gong, L. Zejda, W. Dappen, and J. M. Aparicio (2001) Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.
  • 19: 17.15 Generalizations
    §17.15 Generalizations
    20: 15.17 Mathematical Applications
    The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … Harmonic analysis can be developed for the Jacobi transform either as a generalization of the Fourier-cosine transform1.14(ii)) or as a specialization of a group Fourier transform. … Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). … By considering, as a group, all analytic transformations of a basis of solutions under analytic continuation around all paths on the Riemann sheet, we obtain the monodromy group. …