# hypergeometric functions

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##### 1: 15.2 Definitions and Analytical Properties
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###### §15.2(i) Gauss Series
โบThe hypergeometric function $F\left(a,b;c;z\right)$ is defined by the Gauss series … … โบOn the circle of convergence, $|z|=1$, the Gauss series: … โบ
##### 2: 17.1 Special Notation
###### §17.1 Special Notation
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 $k,j,m,n,r,s$ nonnegative integers. …
โบThe main functions treated in this chapter are the basic hypergeometric (or $q$-hypergeometric) function ${{}_{r}\phi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$, the bilateral basic hypergeometric (or bilateral $q$-hypergeometric) function ${{}_{r}\psi_{s}}\left(a_{1},a_{2},\dots,a_{r};b_{1},b_{2},\dots,b_{s};q,z\right)$, and the $q$-analogs of the Appell functions $\Phi^{(1)}\left(a;b,b^{\prime};c;q;x,y\right)$, $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)$, $\Phi^{(3)}\left(a,a^{\prime};b,b^{\prime};c;q;x,y\right)$, and $\Phi^{(4)}\left(a,b;c,c^{\prime};q;x,y\right)$. โบAnother function notation used is the “idem” function: …
##### 3: 16.2 Definition and Analytic Properties
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###### §16.2(i) Generalized Hypergeometric Series
โบโบUnless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … โบ
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##### 7: 19.16 Definitions
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###### §19.16(ii) $R_{-a}\left(\mathbf{b};\mathbf{z}\right)$
โบAll elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function โบ โบ โบ
##### 8: 15.18 Physical Applications
###### §15.18 Physical Applications
โบThe hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). …
##### 9: 15.14 Integrals
###### §15.14 Integrals
โบThe Mellin transform of the hypergeometric function of negative argument is given by … โบLaplace 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). … โบ
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