Definition:Hypergeometric Differential Equation

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A hypergeometric differential equation is a second order ODE of the form:

$x \paren {1 - x} \dfrac {\d^2 y} {\d x^2} + \paren {c - \paren {a + b + 1} x} \dfrac {\d y} {\d x} - a b y = 0$

where $a$, $b$ and $c$ are complex numbers.

Also presented as

Some sources present this as:

$x \paren {x - 1} \dfrac {\d^2 y} {\d x^2} + \paren {\paren {a + b + 1} x - c} \dfrac {\d y} {\d x} + a b y = 0$

which reduces to the given form on multiplication of all terms by $-1$.

Also see

  • Results about hypergeometric differential equations can be found here.