チェビシェフ多項式の別表記
チェビシェフ多項式の別表記
(1)
\[ T_{n}(x)=\frac{1}{2}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}+\left(x-i\sqrt{1-x^{2}}\right)^{n}\right) \]
(2)
\[ U_{n-1}(x)=\frac{1}{2i\sqrt{1-x^{2}}}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}-\left(x-i\sqrt{1-x^{2}}\right)^{n}\right) \]
(1)
\begin{align*} T_{n}(x) & =\cos\left(n\cos^{\circ}x\right)\\ & =\frac{1}{2}\left(e^{in\cos^{\circ}x}+e^{-in\cos^{\circ}x}\right)\\ & =\frac{1}{2}\left(\left(e^{i\cos^{\circ}x}\right)^{n}+\left(e^{-i\cos^{\circ}x}\right)^{n}\right)\\ & =\frac{1}{2}\left(\left(\cos\cos^{\circ}x+i\sin\cos^{\circ}x\right)^{n}+\left(\cos\cos^{\circ}x-i\sin\cos^{\circ}x\right)^{n}\right)\\ & =\frac{1}{2}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}+\left(x-i\sqrt{1-x^{2}}\right)^{n}\right) \end{align*}
(2)
\begin{align*} U_{n-1}(x) & =\frac{\sin(n\cos^{\circ}x)}{\sin\cos^{\circ}x}\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left(e^{in\cos^{\circ}x}-e^{-in\cos^{\circ}x}\right)\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left(\left(\cos\cos^{\circ}x+i\sin\cos^{\circ}x\right)^{n}-\left(\cos\cos^{\circ}x-i\sin\cos^{\circ}x\right)^{n}\right)\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left(\left(x+i\sqrt{1-x^{2}}\right)^{n}-\left(x-i\sqrt{1-x^{2}}\right)^{n}\right) \end{align*}
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- チェビシェフの微分方程式\[ \left(1-x^{2}\right)T_{n}''(x)-xT_{n}'(x)+n^{2}T_{n}(x)=0 \]
- 第3種・第4種チェビシェフ多項式の直交性\[ \int_{-1}^{1}V_{m}(x)V_{n}(x)\sqrt{\frac{1+x}{1-x}}dx=\pi\delta_{mn} \]
- チェビシェフ多項式の直交性\[ \int_{-1}^{1}T_{m}(x)T_{n}(x)\frac{dx}{\sqrt{1-x^{2}}}=\frac{\pi}{2}\left(\delta_{mn}+\delta_{0m}\delta_{0n}\right) \]
- (*)チェビシェフ多項式のロドリゲス公式\[ T_{n}(x)=\frac{(-1)^{n}\sqrt{\pi}\sqrt{1-x^{2}}}{2^{n}\Gamma\left(n+\frac{1}{2}\right)}\frac{d^{n}}{dx^{n}}\left(1-x^{2}\right)^{n-\frac{1}{2}} \]
- 第3種・第4種チェビシェフ多項式の定義\[ V_{n}(x)=\frac{\cos\left(\left(n+\frac{1}{2}\right)\cos^{\circ}x\right)}{\cos\left(\frac{1}{2}\cos^{\circ}x\right)} \]