# hypergeometric R-function

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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บ
###### §15.2(ii) Analytic Properties
โบFormula (15.4.6) reads $F\left(a,b;a;z\right)=(1-z)^{-b}$. …
##### 2: 16.2 Definition and Analytic Properties
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###### Polynomials
โบNote also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via … โบ
##### 3: 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บFine (1988) uses $F(a,b;t:q)$ for a particular specialization of a ${{}_{2}\phi_{1}}$ function.
##### 4: 15.10 Hypergeometric Differential Equation
###### §15.10 Hypergeometric Differential Equation
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15.10.1 $z(1-z)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(c-(a+b+1)z\right)\frac% {\mathrm{d}w}{\mathrm{d}z}-abw=0.$
โบThis is the hypergeometric differential equation. … โบ
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##### 8: 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
##### 9: 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)). …
##### 10: 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