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hypergeometric equation

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The hypergeometric equation is the following linear ordinary differential equation:
 $$x (1 - x) y'' + (c - (a + b + 1) x ) y' - aby = 0$$
(Here, $a$, $b$, and $c$ are complex constants.)

The solutions of this equation may be expressed in terms of the hypergeometric function, hence the name.

The hypergeometric equation is a Fuchsian differential equation with singularities at $0$, $1$, and $\infty$.  By a suitable change of variables, any second order Fuchsian differential equation may be converted into a hypergeometric equation.