14 Legendre and Related FunctionsComplex Arguments14.28 Sums14.30 Spherical and Spheroidal Harmonics

Solutions of the equation

14.29.1 | $$\left(1-{z}^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\nu \left(\nu +1\right)-\frac{{\mu}_{1}^{2}}{2\left(1-z\right)}-\frac{{\mu}_{2}^{2}}{2\left(1+z\right)}\right)w=0$$ | ||

are called *Generalized Associated Legendre Functions*. As in the case of
(14.21.1), the solutions are hypergeometric functions, and
(14.29.1) reduces to (14.21.1) when
${\mu}_{1}={\mu}_{2}=\mu $. For properties see Virchenko and Fedotova (2001) and
Braaksma and Meulenbeld (1967).

For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).