# Mehler?Fock transformation

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##### 2: 15.14 Integrals
###### §15.14 Integrals
The Mellin transform of the hypergeometric function of negative argument is given by … Fourier transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §§1.14 and 2.14). Laplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). …
##### 3: 12.16 Mathematical Applications
PCFs are also used in integral transforms with respect to the parameter, and inversion formulas exist for kernels containing PCFs. …Integral transforms and sampling expansions are considered in Jerri (1982).
##### 4: 2.5 Mellin Transform Methods
###### §2.5 Mellin Transform Methods
The Mellin transform of $f(t)$ is defined by …The inversion formula is given by …
##### 5: 35.2 Laplace Transform
###### Convolution Theorem
If $g_{j}$ is the Laplace transform of $f_{j}$, $j=1,2$, then $g_{1}g_{2}$ is the Laplace transform of the convolution $f_{1}*f_{2}$, where …
##### 6: 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. …
##### 7: 19.15 Advantages of Symmetry
Symmetry in $x,y,z$ of $R_{F}\left(x,y,z\right)$, $R_{G}\left(x,y,z\right)$, and $R_{J}\left(x,y,z,p\right)$ replaces the five transformations (19.7.2), (19.7.4)–(19.7.7) of Legendre’s integrals; compare (19.25.17). Symmetry unifies the Landen transformations of §19.8(ii) with the Gauss transformations of §19.8(iii), as indicated following (19.22.22) and (19.36.9). (19.21.12) unifies the three transformations in §19.7(iii) that change the parameter of Legendre’s third integral. …
##### 8: 14.31 Other Applications
###### §14.31(ii) Conical Functions
These functions are also used in the Mehler–Fock integral transform14.20(vi)) for problems in potential and heat theory, and in elementary particle physics (Sneddon (1972, Chapter 7) and Braaksma and Meulenbeld (1967)). The conical functions and Mehler–Fock transform generalize to Jacobi functions and the Jacobi transform; see Koornwinder (1984a) and references therein. …
##### 9: 16.20 Integrals and Series
Extensive lists of Laplace transforms and inverse Laplace transforms of the Meijer $G$-function are given in Prudnikov et al. (1992a, §3.40) and Prudnikov et al. (1992b, §3.38). …
##### 10: 16.6 Transformations of Variable
###### §16.6 Transformations of Variable
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).