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

Differential equation

Associated Legendre functions are solutions of the Legendre equation


where the complex numbers λ and μ are called the degree and order of the associated Legendre functions respectively. 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 the hypergeometric differential equation

Hypergeometric differential equation

In mathematics, the Gaussian or ordinary hypergeometric function 2F1 is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation...

 by a change of variable, and its solutions can be expressed using hypergeometric functions.

Definition

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


where is the gamma function

Gamma function

In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...

 and is the hypergeometric function.

The second order differential equation has a second solution,
, defined as:

Integral representations

The Legendre functions can be written as contour integrals. For example
where the contour circles around the points 1 and z in the positive direction and does not circle around −1.
For real x, we have

Legendre function as characters

The real integral representation of are very useful in the study of harmonic analysis on where is the double coset space of (see Zonal spherical function

Zonal spherical function

In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G...

). Actually the Fourier transform on is given by
where

External links

• Legendre function P on the Wolfram functions site.
• Legendre function Q on the Wolfram functions site.
• Associated Legendre function P on the Wolfram functions site.
• Associated Legendre function Q on the Wolfram functions site.