ゲーゲンバウアー多項式（グラフ）

\(\normalsize Gegenbauer\ polynomial\ C_n^\lambda(x)\\ (1)\ (1-x^2)y''-(2\lambda+1)xy'+n(n+2\lambda)y=0\\ \hspace{25px}y=C_n^\lambda(x)\\ (2)\ {\large\frac {1}{(1-2xt+t^2)^\lambda}}={\large\displaystyle \sum_{\small n=0}^{\small\infty}} C_n^\lambda(x)t^n\\ (3)\hspace{5px}{\large\int_{\tiny -1}^{\tiny 1}}(1-x^2)^{\lambda-\frac{1}{2}}C_n^\lambda(x)C_m^\lambda(x)dx\\ \hspace{110px}={\large\frac{\pi\Gamma(n+2\lambda)}{2^{2\lambda-1}(n+\lambda)n!\Gamma^2(\lambda)}}\delta_{mn}\\ (4)\hspace{2px} C_n^\lambda(x)={\normalsize\frac{\Gamma(n+2\lambda)}{n!\Gamma(2\lambda)}}{}_{\small 2}F_{\small 1} ({\small-}n,n+2\lambda;\lambda+\frac{1}{2};\frac{1-x}{2})\\\)