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14 Legendre and Related FunctionsComplex Arguments

§14.29 Generalizations

Solutions of the equation

14.29.1 (1z2)d2wdz22zdwdz+(ν(ν+1)μ122(1z)μ222(1+z))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 μ1=μ2=μ. 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).