Editing Associated Legendre polynomials (section)

==Generalization via hypergeometric functions==
{{main|Legendre function}}
These functions may actually be defined for general complex parameters and argument:<ref>{{cite journal|first1=H. A.|last1=Mavromatis|first2=R. S. |last2=Alassar|title=A generalized formula for the integral of three Associated Legendre Polynomials|year=1999|journal=Appl. Math. Lett.|volume=12|number=3|pages=101-105|doi=10.1016/S0893-9659(98)00180-3}}</ref>

<math display="block">P_{\lambda}^{\mu}(z) = \frac{1}{\Gamma(1-\mu)} \left[\frac{1+z}{1-z}\right]^{\mu/2} \,_2F_1 (-\lambda, \lambda+1; 1-\mu; \frac{1-z}{2})</math>

where <math>\Gamma</math> is the [[gamma function]] and <math> _2F_1</math> is the [[hypergeometric function]]

<math display="block">\,_2F_1 (\alpha, \beta; \gamma; z) = \frac{\Gamma(\gamma)}{\Gamma(\alpha)\Gamma(\beta)} \sum_{n=0}^\infty\frac{\Gamma(n+\alpha)\Gamma(n+\beta)}{\Gamma(n+\gamma)\ n!}z^n,</math>

They are called the '''Legendre functions''' when defined in this more general way. They satisfy the same differential equation as before:

<math display="block">(1-z^2)\,y'' -2zy' + \left(\lambda[\lambda+1] - \frac{\mu^2}{1-z^2}\right)\,y = 0.\,</math>

Since this is a second order differential equation, it has a second solution, <math>Q_\lambda^{\mu}(z)</math>, defined as:

<math display="block">Q_{\lambda}^{\mu}(z) = \frac{\sqrt{\pi}\ \Gamma(\lambda+\mu+1)}{2^{\lambda+1}\Gamma(\lambda+3/2)}\frac{1}{z^{\lambda+\mu+1}}(1-z^2)^{\mu/2} \,_2F_1 \left(\frac{\lambda+\mu+1}{2}, \frac{\lambda+\mu+2}{2}; \lambda+\frac{3}{2}; \frac{1}{z^2}\right)</math>

<math>P_\lambda^{\mu}(z)</math> and <math>Q_\lambda^{\mu}(z)</math> both obey the various recurrence formulas given previously.