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:
Series expansion about the origin:
Special values:
Related functions: gegenbauer
Function category: orthogonal polynomials sagemath-docs