jacobi_P( n , a , b , z )

The Jacobi polynomial of z in SageMath. Defined by

$P_n^{(a,b)} (z) = \frac{ (-1)^n }{ 2^n n! } (1-z)^{-a} (1+z)^{-b} \frac{ d^n }{ dz^n } (1-z)^{a+n} (1+z)^{b+n}$

A solution of the differential equation

$( 1-z^2 ) \frac{ d^2 f }{ dz^2 } + ( b-a - (a+b+2)z ) \frac{ d f }{ dz } + n(n+a+b+1) f = 0$

Explicit form:

Plot on the real axis: