ultraspherical( n , a , z )

The ultraspherical or Gegenbauer polynomial of z in SageMath. Defined in terms of jacobi_P by

\[ C_n^{(a)} (z) = \frac{ \Gamma(n+2a) }{ \Gamma(2a) } \frac{ \Gamma(a+\frac{1}{2}) }{ \Gamma(n+a+\frac{1}{2}) } P_n^{(a-\frac{1}{2},a-\frac{1}{2})} (z) \]

A solution of the differential equation

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

Alias of gegenbauer.

Explicit form:

Plot on the real axis:

Special values:

Related functions:   gegenbauer   jacobi_P

Function category: orthogonal polynomials sagemath-docs