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)

  Laguerre polynomials
  Prime factors