Legendre functions

In mathematics, the Legendre functions P_λ, Q_λ and associated Legendre functions Pμ
λ, Qμ
λ are generalizations of Legendre polynomials to non-integer degree.

Associated Legendre functions are solutions of the general Legendre equation

where the complex numbers λ and μ are called the degree and order of the associated Legendre functions, respectively. The Legendre polynomials are the associated Legendre functions of order μ=0.

This is a second order linear equation with three regular singular points (at 1, −1, and ∞). Like all such equations, it can be converted into a hypergeometric differential equation by a change of variable, and its solutions can be expressed using hypergeometric functions.

These functions may actually be defined for general complex parameters and argument:

where ${\displaystyle \Gamma }$ is the gamma function and ${\displaystyle _{2}F_{1}}$ is the hypergeometric function.

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