Definition:Hypergeometric Differential Equation
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Definition
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.
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 31$: Hypergeometric Functions: Hypergeometric Differential Equation: $31.1$
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hypergeometric differential equation