ゲーゲンバウアー多項式

性質

{\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}}}
{\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}}}
${\displaystyle (1-x^{2})y''-(2\alpha +1)xy'+n(n+2\alpha )y=0}$
${\displaystyle C_{n}^{(\alpha )}(x)={\frac {(-2)^{n}}{n!}}{\frac {\Gamma (n+\alpha )\Gamma (n+2\alpha )}{\Gamma (\alpha )\Gamma (2n+2\alpha )}}(1-x^{2})^{-\alpha +1/2}{\frac {d^{n}}{dx^{n}}}\left[(1-x^{2})^{n+\alpha -1/2}\right]}$
• 次の直交関係を満たす：
${\displaystyle \int _{-1}^{1}C_{n}^{(\alpha )}(x)C_{m}^{(\alpha )}(x)(1-x^{2})^{\alpha -{\frac {1}{2}}}\,dx={\frac {\pi 2^{1-2\alpha }\Gamma (n+2\alpha )}{n!(n+\alpha )[\Gamma (\alpha )]^{2}}}\delta _{nm}}$
• ある角度余弦を引数とする関数値について、次式が成り立つ：
${\displaystyle C_{n}^{(\alpha )}(\cos \theta )=\sum _{r=0}^{\infty }{\frac {\Gamma (\alpha +r)\Gamma (n+\alpha -r)}{r!(n-r)![\Gamma (\alpha )]^{2}}}\cos(2r-n)\theta }$
• ${\displaystyle \alpha =1/2}$  の場合がルジャンドル多項式に、${\displaystyle \alpha =1}$  の場合が第二種チェビシェフ多項式に相当する。

参考文献

• 森口, 繁一、宇田川, 銈久、一松, 信『岩波数学公式 Ⅲ』、1987年、新装版。ISBN 4-00-005509-7
• Milton Abramowitz; Irene A. Stegun, ed (1965-06-01). Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Dover Books on Mathematics. Dover Publications. ISBN 0-486-61272-4