Gegenbauer polynomials

Remarks

Equivalently, $C_0^{(\lambda)} = 1$, $C_1^{(\lambda)} = 2\lambda x$ and $(n + 1) C_{n+1}^{(\lambda)} = 2(n + \lambda) x \, C_n^{(\lambda)} - (n + 2\lambda - 1) C_{n-1}^{(\lambda)}$ for $n \ge 1$. Example: $C_2^{(\lambda)}(x) = 2\lambda(\lambda + 1) x^2 - \lambda$. Check: with $u = 2xt - t^2$, the expansion $(1 - u)^{-\lambda} = 1 + \lambda u + \frac{\lambda(\lambda + 1)}{2} u^2 + \cdots$ has $t^2$-coefficient $-\lambda + 2\lambda(\lambda + 1) x^2$. They are orthogonal: $\int_{-1}^{1} C_m^{(\lambda)}(x) C_n^{(\lambda)}(x) (1 - x^2)^{\lambda - 1/2} \, dx = 0$ for $m \ne n$. Special cases: $C_n^{(1/2)}$ is the Legendre polynomial $P_n$, and $C_n^{(1)}$ is the Chebyshev polynomial of the second kind $U_n$. Non-example: $\lambda = 0$ is excluded, because the generating function is then the constant $1$ and gives $C_n^{(0)} = 0$ for all $n \ge 1$.