Gegenbauer C
The Gegenbauer polynomial $C_n^{\lambda}$ of degree $n \in \{0,1,2,\ldots\}$ and order $\lambda$ defined by $$C_n^{\lambda}(z)=\displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} \dfrac{(-1)^k\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2z)^{n-2k},$$ where $\lfloor \frac{n}{2} \rfloor$ 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)
Theorem: The following formula holds: $$(n+2\lambda)C_n^{\lambda}(x) = 2\lambda(C_n^{\lambda+1}(x)-xC_{n-1}^{\lambda+1}(x))$$.
Proof: █
Theorem: The following formula holds: $$nC_n^{\lambda}(x) = (n-1+2\lambda)xC_{n-1}^{\lambda}(x) - 2\lambda(1-x^2)C_{n-2}^{\lambda-1}(x).$$
Proof: █
Theorem: The following formula holds: $$C_n^{\lambda '}(x) = 2\lambda C_{n+1}^{\lambda+1}(x).$$
Proof: █
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
