Legendre functions and polynomials

Syntax: | `y = LEGENDRE(n,x) y = PLMU(n,m,x) y = PLMN(n,m,x)`

The most elementary of the Legendre functions, the Legendre polynomials, , can be defined by the generating function:

More explicit representations are:

where is the hypergeometric function.

Unnormalized associated Legendre functions of degree n

Normalized associated Legendre functions

where , is a non-negative integer and satisfy the differential equation:

Note that `PLMU(n,0,x) = LEGENDRE(n,x)`