generalized Mehler–Fock transformation
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§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. …
§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).
►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).
general order and degree, respectively.
§16.6 Transformations of Variable►
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).
§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function►The function is defined by ►
10.46.1 .… ►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 . … ►The Laplace transform of can be expressed in terms of the Mittag-Leffler function: …
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… ►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 transform (§1.14(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. …