チェビシェフ多項式の昇降演算子

チェビシェフ多項式の昇降演算子

(1)

\[ \left(\left(1-x^{2}\right)\frac{d}{dx}\mp nx\right)T_{n}(x)=\mp nT_{n\pm1}(x) \]

(2)

\[ \left[\left(1-x^{2}\right)\left(n\pm1\right)\frac{d}{dx}\mp x\left(n\pm1\right)^{2}\right]U_{n-1}(x)=\mp n(n\pm1)U_{n-1\pm1}(x) \]

(1)

\begin{align*} \left(\left(1-x^{2}\right)\frac{d}{dx}\mp nx\right)T_{n}(x) & =n\left\{ \left(1-x^{2}\right)U_{n-1}(x)\mp xT_{n}(x)\right\} \\ & =n\left\{ \sin^{2}tU_{n-1}(\cos t)\mp\cos tT_{n}(\cos t)\right\} \qquad,\qquad x=\cos t\\ & =n\left\{ \sin t\sin(nt)\mp\cos t\cos(nt)\right\} \\ & =\mp n\left\{ \cos t\cos(nt)\mp\sin t\sin(nt)\right\} \\ & =\mp n\cos\left((n\pm1)t\right)\\ & =\mp nT_{n\pm1}(x) \end{align*}

(2)

\begin{align*} \LHS & =\frac{d}{dx}\left(\left(1-x^{2}\right)\frac{d}{dx}\mp nx\right)T_{n}(x)\\ & =\left(-2x\frac{d}{dx}+\left(1-x^{2}\right)\frac{d^{2}}{dx^{2}}\mp n\mp nx\frac{d}{dx}\right)T_{n}(x)\\ & =\left(-2x+\left(1-x^{2}\right)\frac{d}{dx}\mp nx\right)nU_{n-1}(x)\mp nT_{n}(x)\\ & =\left(-2x+\left(1-x^{2}\right)\frac{d}{dx}\mp nx\right)nU_{n-1}(x)\mp\left(xU_{n-1}(x)-(1-x^{2})U_{n-1}'(x)\right)\\ & =\left[\left(1-x^{2}\right)\left(n\pm1\right)\frac{d}{dx}-x\left(2n\pm\left(n^{2}+1\right)\right)\right]U_{n-1}(x)\\ & =\left[\left(1-x^{2}\right)\left(n\pm1\right)\frac{d}{dx}\mp x\left(n\pm1\right)^{2}\right]U_{n-1}(x) \end{align*}

\begin{align*} \RHS & =\frac{d}{dx}\left(\mp nT_{n\pm1}(x)\right)\\ & =\mp n(n\pm1)U_{n-1\pm1}(x) \end{align*}

これより、

\[ \left[\left(1-x^{2}\right)\left(n\pm1\right)\frac{d}{dx}\mp x\left(n\pm1\right)^{2}\right]U_{n-1}(x)=\mp n(n\pm1)U_{n-1\pm1}(x) \]

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チェビシェフ多項式の昇降演算子

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