§14.29 Generalizations

Keywords:: associated Legendre functions
Permalink:: http://dlmf.nist.gov/14.29

Solutions of the equation

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

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\mu$ : general order and $\nu$ : general degree
Referenced by:: §14.29
Permalink:: http://dlmf.nist.gov/14.29.E1
Encodings:: TeX, pMML, png

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).