チェビシェフ多項式の母関数

チェビシェフ多項式の母関数

通常型母関数

(1)

\[ \sum_{k=0}^{\infty}T_{k}(x)t^{k}=\frac{1-tx}{1-2tx+t^{2}} \]

(2)

\[ \sum_{k=0}^{\infty}U_{k}(x)t^{k}=\frac{1}{1-2tx+t^{2}} \]

指数型母関数

(3)

\[ \sum_{k=0}^{\infty}T_{k}(x)\frac{t^{k}}{k!}=e^{tx}\cos\left(t\sqrt{1-x^{2}}\right) \]

(4)

\[ \sum_{k=0}^{\infty}U_{k}(x)\frac{t^{k}}{k!}=e^{tx}\left\{ \cos\left(t\sqrt{1-x^{2}}\right)+\frac{x}{\sqrt{1-x^{2}}}\sin\left(t\sqrt{1-x^{2}}\right)\right\} \]

(1)

\begin{align*} \sum_{k=0}^{\infty}T_{k}(\cos y)t^{k} & =\sum_{k=0}^{\infty}\cos(ky)t^{k}\\ & =\Re\left(\sum_{k=0}^{\infty}\left(e^{iy}t\right)^{k}\right)\\ & =\Re\left(\frac{1}{1-e^{iy}t}\right)\\ & =\frac{1}{2}\left(\frac{1}{1-e^{-iy}t}+\frac{1}{1-e^{iy}t}\right)\\ & =\frac{1}{2}\left(\frac{1-e^{iy}t+1-e^{-iy}t}{\left(1-e^{iy}t\right)\left(1-e^{-iy}t\right)}\right)\\ & =\frac{1}{2}\left(\frac{2-\left(e^{iy}+e^{-iy}\right)t}{1-\left(e^{iy}+e^{-iy}\right)t+t^{2}}\right)\\ & =\frac{1-t\cos y}{1-2t\cos y+t^{2}} \end{align*}

これより、

\[ \sum_{k=0}^{\infty}T_{k}(x)t^{k}=\frac{1-tx}{1-2tx+t^{2}} \]

(1)-2

\begin{align*} \sum_{k=0}^{\infty}T_{k}(x)t^{k} & =T_{0}(x)+T_{1}(x)t+\sum_{k=2}^{\infty}T_{k}(x)t^{k}\\ & =1+xt+\sum_{k=0}^{\infty}T_{k+2}(x)t^{k+2}\\ & =1+xt+\sum_{k=0}^{\infty}\left(2xT_{k+1}(x)-T_{k}(x)\right)t^{k+2}\\ & =1+xt+2xt\sum_{k=0}^{\infty}T_{k+1}(x)t^{k+1}-t^{2}\sum_{k=0}^{\infty}T_{k}(x)t^{k}\\ & =1+xt+2xt\left\{ \sum_{k=0}^{\infty}T_{k}(x)t^{k}-T_{0}(x)t^{0}\right\} -t^{2}\sum_{k=0}^{\infty}T_{k}(x)t^{k}\\ & =1+xt-2xt+\left(2xt-t^{2}\right)\LHS\\ & =\frac{1-tx}{1-2tx+t^{2}} \end{align*}

(2)

\begin{align*} \sum_{k=0}^{\infty}U_{k}(\cos y)t^{k} & =\frac{1}{\sin y}\sum_{k=0}^{\infty}\sin\left((k+1)y\right)t^{k}\\ & =\frac{1}{\sin y}\sum_{k=0}^{\infty}\left(\sin\left(ky\right)\cos y+\cos(ky)\sin y\right)t^{k}\\ & =\frac{\cos y}{\sin y}\sum_{k=0}^{\infty}\left(\sin\left(ky\right)t^{k}\right)+\sum_{k=0}^{\infty}\left(\cos(ky)t^{k}\right)\\ & =\frac{\cos y}{\sin y}\Im\left(\sum_{k=0}^{\infty}\left(e^{iy}t\right)^{k}\right)+\sum_{k=0}^{\infty}\left(T_{k}(\cos y)t^{k}\right)\\ & =\frac{\cos y}{\sin y}\Im\left(\frac{1}{1-e^{iy}t}\right)+\sum_{k=0}^{\infty}\left(T_{k}(\cos y)t^{k}\right)\\ & =\frac{\cos y}{2i\sin y}\left(\frac{1}{1-e^{iy}t}-\frac{1}{1-e^{-iy}t}\right)+\sum_{k=0}^{\infty}\left(T_{k}(\cos y)t^{k}\right)\\ & =\frac{\cos y}{2i\sin y}\frac{\left(1-e^{-iy}t\right)-\left(1-e^{iy}t\right)}{\left(1-e^{iy}t\right)\left(1-e^{-iy}t\right)}+\sum_{k=0}^{\infty}\left(T_{k}(\cos y)t^{k}\right)\\ & =\frac{\cos y}{2i\sin y}\frac{\left(e^{iy}-e^{-iy}\right)t}{1-\left(e^{iy}+e^{-iy}\right)t+t^{2}}+\sum_{k=0}^{\infty}\left(T_{k}(\cos y)t^{k}\right)\\ & =\frac{t\cos y}{1-2t\cos y+t^{2}}+\frac{1-t\cos y}{1-2t\cos y+t^{2}}\\ & =\frac{1}{1-2t\cos y+t^{2}} \end{align*}

これより、
\[ \sum_{k=0}^{\infty}U_{k}(x)t^{k}=\frac{1}{1-2tx+t^{2}} \]

(2)-2

\begin{align*} \sum_{k=0}^{\infty}U_{k}(x)t^{k} & =U_{0}(x)+U_{1}(x)t+\sum_{k=2}^{\infty}U_{k}(x)t^{k}\\ & =1+2xt+\sum_{k=0}^{\infty}U_{k+2}(x)t^{k+2}\\ & =1+2xt+\sum_{k=0}^{\infty}\left(2xU_{k+1}(x)-U_{k}(x)\right)t^{k+2}\\ & =1+2xt+2xt\sum_{k=0}^{\infty}U_{k+1}(x)t^{k+1}-t^{2}\sum_{k=0}^{\infty}U_{k}(x)t^{k}\\ & =1+2xt+2xt\left\{ \sum_{k=0}^{\infty}U_{k}(x)t^{k}-U_{0}(x)t^{0}\right\} -t^{2}\sum_{k=0}^{\infty}U_{k}(x)t^{k}\\ & =1+2xt-2xt+\left(2xt-t^{2}\right)\LHS\\ & =\frac{1}{1-2tx+t^{2}} \end{align*}

(3)

\begin{align*} \sum_{k=0}^{\infty}T_{k}(x)\frac{t^{k}}{k!} & =\sum_{k=0}^{\infty}\cos(k\cos^{\circ}x)\frac{t^{k}}{k!}\\ & =\frac{1}{2}\sum_{k=0}^{\infty}\left(e^{ik\cos^{\circ}x}+e^{-ik\cos^{\circ}x}\right)\frac{t^{k}}{k!}\\ & =\frac{1}{2}\sum_{k=0}^{\infty}\left(\left(e^{i\cos^{\circ}x}\right)^{k}+\left(e^{-i\cos^{\circ}x}\right)^{k}\right)\frac{t^{k}}{k!}\\ & =\frac{1}{2}\sum_{k=0}^{\infty}\left(\left(x+i\sin\cos^{\circ}x\right)^{k}+\left(x-i\sin\cos^{\circ}x\right)^{k}\right)\frac{t^{k}}{k!}\\ & =\frac{1}{2}\left\{ \sum_{k=0}^{\infty}\left(x+i\sqrt{1-x^{2}}\right)^{k}\frac{t^{k}}{k!}+\sum_{k=0}^{\infty}\left(x-i\sqrt{1-x^{2}}\right)^{k}\frac{t^{k}}{k!}\right\} \\ & =\frac{1}{2}\left\{ e^{t\left(x+i\sqrt{1-x^{2}}\right)}+e^{t\left(x-i\sqrt{1-x^{2}}\right)}\right\} \\ & =\frac{e^{tx}}{2}\left(e^{+it\sqrt{1-x^{2}}}+e^{-it\sqrt{1-x^{2}}}\right)\\ & =e^{tx}\cos\left(t\sqrt{1-x^{2}}\right) \end{align*}

(4)

\begin{align*} \sum_{k=0}^{\infty}U_{k}(x)\frac{t^{k}}{k!} & =\sum_{k=0}^{\infty}\frac{\sin(\left(k+1\right)\cos^{\circ}x)}{\sin\cos^{\circ}x}\frac{t^{k}}{k!}\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\sum_{k=0}^{\infty}\left(e^{i(k+1)\cos^{\circ}x}-e^{-i(k+1)\cos^{\circ}x}\right)\frac{t^{k}}{k!}\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\sum_{k=0}^{\infty}\left(\left(e^{i\cos^{\circ}x}\right)^{k+1}-\left(e^{-i\cos^{\circ}x}\right)^{k+1}\right)\frac{t^{k}}{k!}\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\sum_{k=0}^{\infty}\left(\left(x+i\sin\cos^{\circ}x\right)^{k+1}-\left(x-i\sin\cos^{\circ}x\right)^{k+1}\right)\frac{t^{k}}{k!}\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left\{ \sum_{k=0}^{\infty}\left(x+i\sqrt{1-x^{2}}\right)^{k+1}\frac{t^{k}}{k!}-\sum_{k=0}^{\infty}\left(x-i\sqrt{1-x^{2}}\right)^{k+1}\frac{t^{k}}{k!}\right\} \\ & =\frac{1}{2i\sqrt{1-x^{2}}}\left\{ \left(x+i\sqrt{1-x^{2}}\right)e^{t\left(x+i\sqrt{1-x^{2}}\right)}-\left(x-i\sqrt{1-x^{2}}\right)e^{t\left(x-i\sqrt{1-x^{2}}\right)}\right\} \\ & =\frac{e^{tx}}{2i\sqrt{1-x^{2}}}\left(\left(x+i\sqrt{1-x^{2}}\right)e^{+it\sqrt{1-x^{2}}}-\left(x-i\sqrt{1-x^{2}}\right)e^{-it\sqrt{1-x^{2}}}\right)\\ & =e^{tx}\left\{ \cos\left(t\sqrt{1-x^{2}}\right)+\frac{x}{\sqrt{1-x^{2}}}\sin\left(t\sqrt{1-x^{2}}\right)\right\} \end{align*}

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チェビシェフ多項式の母関数

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