ゲーゲンバウアー多項式
α =1 の場合のゲーゲンバウアー多項式
α =2 の場合のゲーゲンバウアー多項式
α =3 の場合のゲーゲンバウアー多項式
次の母関数により定義される:
1 ( 1 − 2 x t + t 2 ) α = ∑ n = 0 ∞ C n ( α ) ( x ) t n 1 − x t ( 1 − 2 x t + t 2 ) α + 1 = ∑ n = 0 ∞ n + 2 α 2 α C n ( α ) ( x ) t n {\displaystyle {\begin{aligned}{\frac {1}{(1-2xt+t^{2})^{\alpha }}}&=\sum _{n=0}^{\infty }C_{n}^{(\alpha )}(x)t^{n}\\{\frac {1-xt}{(1-2xt+t^{2})^{\alpha +1}}}&=\sum _{n=0}^{\infty }{\frac {n+2\alpha }{2\alpha }}C_{n}^{(\alpha )}(x)t^{n}\end{aligned}}}
次の漸化式を満たす:
C 0 ( α ) ( x ) = 1 C 1 ( α ) ( x ) = 2 α x C n ( α ) ( x ) = 1 n [ 2 x ( n + α − 1 ) C n − 1 ( α ) ( x ) − ( n + 2 α − 2 ) C n − 2 ( α ) ( x ) ] {\displaystyle {\begin{aligned}C_{0}^{(\alpha )}(x)&=1\\C_{1}^{(\alpha )}(x)&=2\alpha x\\C_{n}^{(\alpha )}(x)&={\frac {1}{n}}\left[2x(n+\alpha -1)C_{n-1}^{(\alpha )}(x)-(n+2\alpha -2)C_{n-2}^{(\alpha )}(x)\right]\end{aligned}}}