gen_legendre_P( n , m , z )

The generalized or associated Legendre polynomial of z in SageMath. Defined by

$P_n^m (z) = \frac{ (-1)^m }{ 2^n n! } (1-z^2)^{m/2} \frac{ d^{n+m} }{ dz^{n+m} } ( z^2-1 )^n$

A solution of the differential equation

$( 1-z^2 ) \frac{ d^2 f }{ dz^2 } -2z \frac{ d f }{ dz } + \left[ n(n+1) - \frac{ m^2 }{ 1-z^2 } \right] f = 0$

The second linearly independent solution of this equation is gen_legendre_Q.

Explicit form:

Plot on the real axis:

Special values:

Related functions:   gen_legendre_Q

Function category: orthogonal polynomials sagemath-docs