# generalized Mehler–Fock transformation

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##### 11: 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) $\mathit{3j}$, $\mathit{6j}$, and $\mathit{9j}$ Symbols
The coefficients of transformations between different coupling schemes of three angular momenta are related to the Wigner $\mathit{6j}$ 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
 $x$, $y$, $\tau$ real variables. … general order and degree, respectively. …
The main functions treated in this chapter are the Legendre functions $\mathsf{P}_{\nu}\left(x\right)$, $\mathsf{Q}_{\nu}\left(x\right)$, $P_{\nu}\left(z\right)$, $Q_{\nu}\left(z\right)$; Ferrers functions $\mathsf{P}^{\mu}_{\nu}\left(x\right)$, $\mathsf{Q}^{\mu}_{\nu}\left(x\right)$ (also known as the Legendre functions on the cut); associated Legendre functions $P^{\mu}_{\nu}\left(z\right)$, $Q^{\mu}_{\nu}\left(z\right)$, $\boldsymbol{Q}^{\mu}_{\nu}\left(z\right)$; conical functions $\mathsf{P}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\mathsf{Q}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $\widehat{\mathsf{Q}}^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $P^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$, $Q^{\mu}_{-\frac{1}{2}+i\tau}\left(x\right)$ (also known as Mehler functions). …
##### 14: 16.6 Transformations of Variable
###### Cubic
16.6.2 ${{}_{3}F_{2}}\left({a,2b-a-1,2-2b+a\atop b,a-b+\frac{3}{2}};\frac{z}{4}\right)% =(1-z)^{-a}{{}_{3}F_{2}}\left({\frac{1}{3}a,\frac{1}{3}a+\frac{1}{3},\frac{1}{% 3}a+\frac{2}{3}\atop b,a-b+\frac{3}{2}};\frac{-27z}{4(1-z)^{3}}\right).$
For Kummer-type transformations of ${{}_{2}F_{2}}$ 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 $\phi\left(\rho,\beta;z\right)$ is defined by
10.46.1 $\phi\left(\rho,\beta;z\right)=\sum_{k=0}^{\infty}\frac{z^{k}}{k!\Gamma\left(% \rho k+\beta\right)},$ $\rho>-1$.
For asymptotic expansions of $\phi\left(\rho,\beta;z\right)$ as $z\to\infty$ in various sectors of the complex $z$-plane for fixed real values of $\rho$ and fixed real or complex values of $\beta$, see Wright (1935) when $\rho>0$, and Wright (1940b) when $-1<\rho<0$. … The Laplace transform of $\phi\left(\rho,\beta;z\right)$ 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
###### Mehler–Dirichlet Formula
14.12.1 $\mathsf{P}^{\mu}_{\nu}\left(\cos\theta\right)=\frac{2^{1/2}(\sin\theta)^{\mu}}% {\pi^{1/2}\Gamma\left(\frac{1}{2}-\mu\right)}\int_{0}^{\theta}\frac{\cos\left(% \left(\nu+\frac{1}{2}\right)t\right)}{(\cos t-\cos\theta)^{\mu+(1/2)}}\mathrm{% d}t,$ $0<\theta<\pi$, $\Re\mu<\tfrac{1}{2}$.
14.12.2 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{\left(1-x^{2}\right)^{-\mu/2}}{% \Gamma\left(\mu\right)}\int_{x}^{1}\mathsf{P}_{\nu}\left(t\right)(t-x)^{\mu-1}% \mathrm{d}t,$ $\Re\mu>0$;
14.12.5 $P^{-\mu}_{\nu}\left(x\right)=\frac{\left(x^{2}-1\right)^{-\mu/2}}{\Gamma\left(% \mu\right)}\int_{1}^{x}P_{\nu}\left(t\right)(x-t)^{\mu-1}\mathrm{d}t,$ $\Re\mu>0$.
##### 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.
##### 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. …