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

Solutions of the equation

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

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).