チェビシェフ多項式の級数表示
チェビシェフ多項式の級数表示
(1)
\[ T_{n}(x)=\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k)\left(-1\right)^{k}\left(1-x^{2}\right)^{k}x^{n-2k}\right) \](2)
\[ U_{n-1}(x)=\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k+1)(-1)^{k}\left(1-x^{2}\right)^{k}x^{n-2k-1}\right) \](1)
\begin{align*} 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)\\ & =\frac{1}{2}\sum_{k=0}^{n}\left(C(n,k)\left(i\sqrt{1-x^{2}}\right)^{k}x^{n-k}+C(n,k)\left(-i\sqrt{1-x^{2}}\right)^{k}x^{n-k}\right)\\ & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k)\left(i\sqrt{1-x^{2}}\right)^{2k}x^{n-2k}\right)\\ & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k)\left(-1\right)^{k}\left(1-x^{2}\right)^{k}x^{n-2k}\right) \end{align*}(2)
\begin{align*} 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)\\ & =\frac{1}{2i\sqrt{1-x^{2}}}\sum_{k=0}^{n}\left(C(n,k)\left(i\sqrt{1-x^{2}}\right)^{k}x^{n-k}-C(n,k)\left(-i\sqrt{1-x^{2}}\right)^{k}x^{n-k}\right)\\ & =\frac{1}{i\sqrt{1-x^{2}}}\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k+1)\left(i\sqrt{1-x^{2}}\right)^{2k+1}x^{n-2k-1}\right)\\ & =\sum_{k=0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left(C(n,2k+1)(-1)^{k}\left(1-x^{2}\right)^{k}x^{n-2k-1}\right) \end{align*}
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第1種チェビシェフ多項式と第2種チェビシェフ多項式の関係
\[
nU_{n-1}(x)=T_{n}'(x)
\]
チェビシェフ多項式の母関数
\[
\sum_{k=0}^{\infty}T_{k}(x)t^{k}=\frac{1-tx}{1-2tx+t^{2}}
\]
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超幾何表示
\[
T_{n}(x)=F\left(-n,n;\frac{1}{2};\frac{1-x}{2}\right)
\]