Editing Associated Legendre polynomials (section)

==Negative {{mvar|m}} and/or negative {{mvar|ℓ}}==

The differential equation is clearly invariant under a change in sign of ''m''.

The functions for negative ''m'' were shown above to be proportional to those of positive ''m'':
<math display="block">P_\ell ^{-m} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} P_\ell ^{m}</math>

(This followed from the Rodrigues' formula definition.  This definition also makes the various recurrence formulas work for positive or negative {{mvar|m}}.)
<math display="block">\text{If}\quad |m| > \ell\,\quad\text{then}\quad P_\ell^{m} = 0.\,</math>

The differential equation is also invariant under a change from {{mvar|ℓ}} to {{math|−''ℓ'' − 1}}, and the functions for negative {{mvar|ℓ}} are defined by

<math display="block">P_{-\ell} ^{m} = P_{\ell-1} ^{m},\ (\ell=1,\,2,\, \dots).</math>