gegenbauer( 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 ultraspherical.
Explicit form:
Plot on the real axis:
Special values:
Related functions: ultraspherical jacobi_P
Function category: orthogonal polynomials sagemath-docs