14 Legendre and Related Functions Complex Arguments 14.28 Sums 14.30 Spherical and Spheroidal Harmonics

§14.29 Generalizations

ⓘ

Keywords:: associated Legendre functions, generalized
Permalink:: http://dlmf.nist.gov/14.29
See also:: Annotations for Ch.14

Solutions of the equation

14.29.1		$\left(1-z^{2}\right)\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-2z\frac{% \mathrm{d}w}{\mathrm{d}z}+{\left(\nu(\nu+1)-\frac{\mu_{1}^{2}}{2(1-z)}-\frac{% \mu_{2}^{2}}{2(1+z)}\right)w}=0$
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 See also: Annotations for §14.29 and Ch.14

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).

14.28 Sums 14.30 Spherical and Spheroidal Harmonics

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