hypergeometric equation

The hypergeometric equation is the following linear ordinary differential equation:

$$x(1-x)y^{\prime \prime }+(c-(a+b+1)x)y^{\prime }-aby=0$$

(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 $\mathrm{\infty }$. By a suitable change of variables, any second order Fuchsian differential equation may be converted into a hypergeometric equation.