The Legendre polynomial (for ) is the polynomial in one variable given by the formula
Alternatively they can be defined via a generating function:
The Legendre polynomials satisfy:
- the following differential equation of the second order
- the recursion relations
- the mixed differential recursion relations
The Legendre polynomials form a complete system of orthogonal polynomials on the interval , that is an unnormalised basis of the Hilbert space .
The first few Legendre polynomials are , , , , . Their values at are
One also has the integral formulas
A generalization of Legendre polynomials are the Legendre functions where is not necessarily an integer and which are given by
for and also
These are satisfying the orthogonality relations
and the differential equation
is a special case of a hypergeometric function, namely
Legendre polynomials enter the expressions for the spherical function?s for sphere in 3d:
(If it is not clear from mathML rendering – both fractions are under square root – including both the numerators and denominators).
If and are two points of the unit sphere in spherical coordinates (polar angle, azimuth), and is the angle between the two corresponding rays from the origin then
what for reduces to the spherical law of cosine from spherical trigonometry:
The orthogonality relation for Legendre polynomials gives Laplace’s formula
where are unit vectors and is some spherical function.
The following orthogonality integral relation is over product of unit spheres in :
where the arguments of Legendre polynomials are the inner products of the unit vectors.
Zoran: the last formula is in my own formula notes which I have written as a student many years ago and used hundreds of times, but it now looks to me suspicious; I have no time to check it right now.