Gegenbauer C
The Gegenbauer polynomial $C_n^{\lambda}$ of degree $n \in \{0,1,2,\ldots\}$ and order $\lambda$ defined by $$C_n^{\lambda}(x)=\displaystyle\sum_{k=0}^{\left\lfloor \frac{n}{2} \right\rfloor} \dfrac{(-1)^k\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2x)^{n-2k},$$ where $\left\lfloor \frac{n}{2} \right\rfloor$ denotes the floor function, $\Gamma$ denotes gamma, and $k!$ denotes the factorial.
Properties
Orthogonality of Gegenbauer C on (-1,1)
(n+2)C_(n+2)^(lambda)(x)=2(lambda+n+1)xC_(n+1)^(lambda)(x)-(2lambda+n)C_n^(lambda)(x)
nC_n^(lambda)(x)=2lambda(xC_(n-1)^(lambda+1)(x)-C_(n-2)^(lambda+1)(x))
(n+2lambda)C_n^(lambda)(x)=2lambda(C_n^(lambda+1)(x)-xC_(n-1)^(lambda+1)(x))
nC_n^(lambda)(x)=(n-1+2lambda)xC_(n-1)^(lambda)(x)-2lambda(1-x^2)C_(n-2)^(lambda-1)(x)
[[d/dx[C_n^(lambda)(x)]=2lambda C_(n+1)^(lambda+1)(x)]]
Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials
Relationship between Chebyshev T and Gegenbauer C
Relationship between Chebyshev U and Gegenbauer C
