# hypergeometric equation

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##### 1: 15.10 Hypergeometric Differential Equation
###### §15.10 Hypergeometric Differential Equation
This is the hypergeometric differential equation. …
##### 2: 15.17 Mathematical Applications
The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … …
##### 3: 31.12 Confluent Forms of Heun’s Equation
This is analogous to the derivation of the confluent hypergeometric equation from the hypergeometric equation in §13.2(i). …
##### 4: 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 $z^{\lambda}(1-z)^{\mu}P\begin{Bmatrix}0&1&\infty&\\ a_{1}&b_{1}&c_{1}&z\\ a_{2}&b_{2}&c_{2}&\end{Bmatrix}=P\begin{Bmatrix}0&1&\infty&\\ a_{1}+\lambda&b_{1}+\mu&c_{1}-\lambda-\mu&z\\ a_{2}+\lambda&b_{2}+\mu&c_{2}-\lambda-\mu&\end{Bmatrix},$
##### 5: 16.8 Differential Equations
###### §16.8(ii) The Generalized Hypergeometric Differential Equation
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 …
##### 7: 16.23 Mathematical Applications
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. … …
##### 8: 19.18 Derivatives and Differential Equations
###### §19.18(ii) Differential Equations
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). …
##### 9: Bibliography N
• J. Negro, L. M. Nieto, and O. Rosas-Ortiz (2000) Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.
• ##### 10: Bibliography S
• F. C. Smith (1939a) On the logarithmic solutions of the generalized hypergeometric equation when $p=q+1$ . Bull. Amer. Math. Soc. 45 (8), pp. 629–636.
• F. C. Smith (1939b) Relations among the fundamental solutions of the generalized hypergeometric equation when $p=q+1$. II. Logarithmic cases. Bull. Amer. Math. Soc. 45 (12), pp. 927–935.
• C. Snow (1952) Hypergeometric and Legendre Functions with Applications to Integral Equations of Potential Theory. National Bureau of Standards Applied Mathematics Series, No. 19, U. S. Government Printing Office, Washington, D.C..