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.