generalized Mehler?Fock transformation

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3: 16.2 Definition and Analytic Properties
Polynomials
Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …
4: 1.16 Distributions
$\Lambda:\mathcal{D}(I)\rightarrow\mathbb{C}$ is called a distribution, or generalized function, if it is a continuous linear functional on $\mathcal{D}(I)$, that is, it is a linear functional and for every $\phi_{n}\to\phi$ in $\mathcal{D}(I)$, … More generally, if $\alpha(x)$ is an infinitely differentiable function, then …
§1.16(vii) Fourier Transforms of Tempered Distributions
Then its Fourier transform is …
7: 19.2 Definitions
§19.2(iii) Bulirsch’s Integrals
Bulirsch’s integrals are linear combinations of Legendre’s integrals that are chosen to facilitate computational application of Bartky’s transformation (Bartky (1938)). …
8: 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. …
9: 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).