第1種・第2種チェビシェフ多項式の定義
第1種・第2種チェビシェフ多項式の定義
(1)第1種チェビシェフ多項式
\[ T_{n}\left(\cos t\right)=\cos\left(nt\right) \] (2)第2種チェビシェフ多項式
\[ U_{n-1}\left(\cos t\right)=\frac{\sin\left(nt\right)}{\sin t} \]
(1)第1種チェビシェフ多項式
\[ T_{n}\left(\cos t\right)=\cos\left(nt\right) \] (2)第2種チェビシェフ多項式
\[ U_{n-1}\left(\cos t\right)=\frac{\sin\left(nt\right)}{\sin t} \]
第1種チェビシェフ多項式
\begin{align*} T_{0}\left(x\right) & =1\\ T_{1}\left(x\right) & =x\\ T_{2}\left(x\right) & =2x^{2}-1\\ T_{3}\left(x\right) & =4x^{3}-3x\\ T_{4}\left(x\right) & =8x^{4}-8x^{2}+1\\ T_{5}\left(x\right) & =16x^{5}-20x^{3}+5x\\ T_{6}\left(x\right) & =32x^{6}-48x^{4}+18x^{2}-1\\ T_{7}\left(x\right) & =64x^{7}-112x^{5}+56x^{3}-7x\\ T_{8}\left(x\right) & =128x^{8}-256x^{6}+160x^{4}-32x^{2}+1\\ T_{9}\left(x\right) & =256x^{9}-576x^{7}+432x^{5}-120x^{3}+9x\\ T_{10}\left(x\right) & =512x^{10}-1280x^{8}+1120x^{6}-400x^{4}+50x^{2}-1\\ T_{11}\left(x\right) & =1024x^{11}-2816x^{9}+2816x^{7}-1232x^{5}+220x^{3}-11x\\ T_{12}\left(x\right) & =2048x^{12}-6144x^{10}+6912x^{8}-3584x^{6}+840x^{4}-72x^{2}+1\\ T_{13}\left(x\right) & =4096x^{13}-13312x^{11}+16640x^{9}-9984x^{7}+2912x^{5}-364x^{3}+13x\\ T_{14}\left(x\right) & =8192x^{14}-28672x^{12}+39424x^{10}-26880x^{8}+9408x^{6}-1568x^{4}+98x^{2}-1\\ T_{15}\left(x\right) & =16384x^{15}-61440x^{13}+92160x^{11}-70400x^{9}+28800x^{7}-6048x^{5}+560x^{3}-15x \end{align*}
第2種チェビシェフ多項式
\begin{align*} U_{0}\left(x\right) & =1\\ U_{1}\left(x\right) & =2x\\ U_{2}\left(x\right) & =4x^{2}-1\\ U_{3}\left(x\right) & =8x^{3}-4x\\ U_{4}\left(x\right) & =16x^{4}-12x^{2}+1\\ U_{5}\left(x\right) & =32x^{5}-32x^{3}+6x\\ U_{6}\left(x\right) & =64x^{6}-80x^{4}+24x^{2}-1\\ U_{7}\left(x\right) & =128x^{7}-192x^{5}+80x^{3}-8x\\ U_{8}\left(x\right) & =256x^{8}-448x^{6}+240x^{4}-40x^{2}+1\\ U_{9}\left(x\right) & =512x^{9}-1024x^{7}+672x^{5}-160x^{3}+10x\\ U_{10}\left(x\right) & =1024x^{10}-2304x^{8}+1792x^{6}-560x^{4}+60x^{2}-1\\ U_{11}\left(x\right) & =2048x^{11}-5120x^{9}+4608x^{7}-1792x^{5}+280x^{3}-12x\\ U_{12}\left(x\right) & =4096x^{12}-11264x^{10}+11520x^{8}-5376x^{6}+1120x^{4}-84x^{2}+1\\ U_{13}\left(x\right) & =8192x^{13}-24576x^{11}+28160x^{9}-15360x^{7}+4032x^{5}-448x^{3}+14x\\ U_{14}\left(x\right) & =16384x^{14}-53248x^{12}+67584x^{10}-42240x^{8}+13440x^{6}-2016x^{4}+112x^{2}-1\\ U_{15}\left(x\right) & =32768x^{15}-114688x^{13}+159744x^{11}-112640x^{9}+42240x^{7}-8064x^{5}+672x^{3}-16x \end{align*}
\begin{align*} T_{0}\left(x\right) & =1\\ T_{1}\left(x\right) & =x\\ T_{2}\left(x\right) & =2x^{2}-1\\ T_{3}\left(x\right) & =4x^{3}-3x\\ T_{4}\left(x\right) & =8x^{4}-8x^{2}+1\\ T_{5}\left(x\right) & =16x^{5}-20x^{3}+5x\\ T_{6}\left(x\right) & =32x^{6}-48x^{4}+18x^{2}-1\\ T_{7}\left(x\right) & =64x^{7}-112x^{5}+56x^{3}-7x\\ T_{8}\left(x\right) & =128x^{8}-256x^{6}+160x^{4}-32x^{2}+1\\ T_{9}\left(x\right) & =256x^{9}-576x^{7}+432x^{5}-120x^{3}+9x\\ T_{10}\left(x\right) & =512x^{10}-1280x^{8}+1120x^{6}-400x^{4}+50x^{2}-1\\ T_{11}\left(x\right) & =1024x^{11}-2816x^{9}+2816x^{7}-1232x^{5}+220x^{3}-11x\\ T_{12}\left(x\right) & =2048x^{12}-6144x^{10}+6912x^{8}-3584x^{6}+840x^{4}-72x^{2}+1\\ T_{13}\left(x\right) & =4096x^{13}-13312x^{11}+16640x^{9}-9984x^{7}+2912x^{5}-364x^{3}+13x\\ T_{14}\left(x\right) & =8192x^{14}-28672x^{12}+39424x^{10}-26880x^{8}+9408x^{6}-1568x^{4}+98x^{2}-1\\ T_{15}\left(x\right) & =16384x^{15}-61440x^{13}+92160x^{11}-70400x^{9}+28800x^{7}-6048x^{5}+560x^{3}-15x \end{align*}
第2種チェビシェフ多項式
\begin{align*} U_{0}\left(x\right) & =1\\ U_{1}\left(x\right) & =2x\\ U_{2}\left(x\right) & =4x^{2}-1\\ U_{3}\left(x\right) & =8x^{3}-4x\\ U_{4}\left(x\right) & =16x^{4}-12x^{2}+1\\ U_{5}\left(x\right) & =32x^{5}-32x^{3}+6x\\ U_{6}\left(x\right) & =64x^{6}-80x^{4}+24x^{2}-1\\ U_{7}\left(x\right) & =128x^{7}-192x^{5}+80x^{3}-8x\\ U_{8}\left(x\right) & =256x^{8}-448x^{6}+240x^{4}-40x^{2}+1\\ U_{9}\left(x\right) & =512x^{9}-1024x^{7}+672x^{5}-160x^{3}+10x\\ U_{10}\left(x\right) & =1024x^{10}-2304x^{8}+1792x^{6}-560x^{4}+60x^{2}-1\\ U_{11}\left(x\right) & =2048x^{11}-5120x^{9}+4608x^{7}-1792x^{5}+280x^{3}-12x\\ U_{12}\left(x\right) & =4096x^{12}-11264x^{10}+11520x^{8}-5376x^{6}+1120x^{4}-84x^{2}+1\\ U_{13}\left(x\right) & =8192x^{13}-24576x^{11}+28160x^{9}-15360x^{7}+4032x^{5}-448x^{3}+14x\\ U_{14}\left(x\right) & =16384x^{14}-53248x^{12}+67584x^{10}-42240x^{8}+13440x^{6}-2016x^{4}+112x^{2}-1\\ U_{15}\left(x\right) & =32768x^{15}-114688x^{13}+159744x^{11}-112640x^{9}+42240x^{7}-8064x^{5}+672x^{3}-16x \end{align*}
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チェビシェフ多項式の奇遇性
\[
T_{n}(-x)=(-1)^{n}T_{n}(x)
\]
(*)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漸化式
\[
V_{k+1}(x)=2xV_{k}(x)-V_{k-1}(x)
\]
第1種・第2種と第3種チェビシェフ多項式同士の関係
\[
V(-x)=(-1)^{n}W_{n}(x)
\]