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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チェビシェフ多項式の別表記

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