Gegenbauer C

The Gegenbauer polynomial of degree $n$ and order $\lambda$ is the coefficient of $t^n$ in the expansion of $\dfrac{1}{(1-2xt+t^2)^{\lambda}}$ in the sense that $$\dfrac{1}{(1-2xt+t^2)^{\lambda}} = \sum_{k=0}^{\infty} C_k^{\lambda}(x)t^k.$$

Properties

Theorem: The following formula holds: $$C_n^{\lambda}(x) = \displaystyle\sum_{k=0}^{\lfloor \frac{n}{2} \rfloor} (-1)^k \dfrac{\Gamma(n-k+\lambda)}{\Gamma(\lambda)k!(n-2k)!} (2x)^{n-2k}.$$

Proof: █

Theorem (Orthogonality): The following formula holds: $$\displaystyle\int_{-1}^1 (1-x^2)^{\lambda-\frac{1}{2}} C_n^{\lambda}(x)C_m^{\lambda}(x) dx = 2^{1-2\lambda} \pi \dfrac{\Gamma(n+2\lambda)}{(n+\lambda)(\Gamma(\lambda))^2\Gamma(n+1)}\delta_{mn},$$ where $\delta_{mn}=0$ if $m\neq 0$ and $\delta_{mn}=1$ when $m=n$.

Proof: █