The generating function for the Legendre polynomials is a mathematical tool that, when expanded as a power series, produces Legendre polynomials as its coefficients in terms of a variable.

Legendre polynomials often arise in problems involving spherical harmonics. An example (see diagram above) is the multipole expansion . As the two points and the angle between them form a triangle, the Legendre polynomials are related to the cosine rule , which can be rearranged to

where and .

Since , we have and hence . This implies that we can expand the RHS of eq343 as a binomial series:

The coefficients of are the Legendre polynomials. Therefore, the generating function for the Legendre polynomials is