# hypergeometric differential equation

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##### 1: 15.10 Hypergeometric Differential Equation
###### §15.10(i) Fundamental Solutions
This is the hypergeometric differential equation. …
##### 2: 15.11 Riemann’s Differential Equation
15.11.3 $w=P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.$
15.11.4 $w=P\begin{Bmatrix}0&1&\infty&\\ 0&0&a&z\\ 1-c&c-a-b&b&\end{Bmatrix}$
15.11.6 $P\begin{Bmatrix}\alpha&\beta&\gamma&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}\widetilde{\alpha}&\widetilde{% \beta}&\widetilde{\gamma}&\\ a_{1}&b_{1}&c_{1}&t\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}.$
The reduction of a general homogeneous linear differential equation of the second order with at most three regular singularities to the hypergeometric differential equation is given by …
15.11.8 $z^{\lambda}(1-z)^{\mu}P\begin{Bmatrix}0&1&\infty&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}0&1&\infty&\\ a_{1}+\lambda&b_{1}+\mu&c_{1}-\lambda-\mu&z\\ a_{2}+\lambda&b_{2}+\mu&c_{2}-\lambda-\mu&\end{Bmatrix},$
##### 3: 16.8 Differential Equations
###### §16.8(ii) The Generalized HypergeometricDifferentialEquation
When no $b_{j}$ is an integer, and no two $b_{j}$ differ by an integer, a fundamental set of solutions of (16.8.3) is given by … We have the connection formula …
##### 4: 15.17 Mathematical Applications
###### §15.17(i) DifferentialEquations
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … … The three singular points in Riemann’s differential equation (15.11.1) lead to an interesting Riemann sheet structure. …These monodromy groups are finite iff the solutions of Riemann’s differential equation are all algebraic. …
##### 5: 15.19 Methods of Computation
A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods. …
##### 6: 15.5 Derivatives and Contiguous Functions
An equivalent equation to the hypergeometric differential equation (15.10.1) is …
##### 7: 16.13 Appell Functions
The following four functions of two real or complex variables $x$ and $y$ cannot be expressed as a product of two ${{}_{2}F_{1}}$ functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …
##### 8: 19.4 Derivatives and Differential Equations
If $\phi=\pi/2$, then these two equations become hypergeometric differential equations (15.10.1) for $K\left(k\right)$ and $E\left(k\right)$. …
##### 9: 16.23 Mathematical Applications
###### §16.23(i) DifferentialEquations
These equations are frequently solvable in terms of generalized hypergeometric functions, and the monodromy of generalized hypergeometric functions plays an important role in describing properties of the solutions. … …
##### 10: 19.18 Derivatives and Differential Equations
###### §19.18(ii) DifferentialEquations
If $n=2$, then elimination of $\partial_{2}v$ between (19.18.11) and (19.18.12), followed by the substitution $(b_{1},b_{2},z_{1},z_{2})=(b,c-b,1-z,1)$, produces the Gauss hypergeometric equation (15.10.1). …