\begin{definition}[Definition:Legendre's Associated Differential Equation]
'''Legendre's associated differential equation''' is a second order ODE of the form:
:$\ds \paren {1 - x^2} \frac {\d^2 y} {\d x^2} - 2 x \frac {\d y} {\d x} + \paren {n \paren {n + 1} - \frac {m^2} {1 - x^2} } y = 0$
where $m$ and $n$ are complex numbers.
Solutions of this equation are called '''associated Legendre functions'''.
\end{definition}