ルジャンドル多項式[3]

ルジャンドル多項式について、次式が成り立ちます。
\begin{alignat}{2}
&(1)  P_0(x)=1\\
&(2)  P_1(x)=x\\
&(3)  P_2(x)=\frac{1}{2}(3x^2-1)\\
&(4)  P_3(x)=\frac{1}{2}(5x^3-3x)\\
&(5)  P_4(x)=\frac{1}{8}(35x^4-30x^2+3)\\
&(6)  P_5(x)=\frac{1}{8}(63x^5-70x^3+15x)\\
&(7)  P_6(x)=\frac{1}{16}(231x^6-315x^4+105x^2-5)\\
&(8)  P_7(x)=\frac{1}{16}(429x^7-693x^5+315x^3-35x)\\
&(9)  P_8(x)=\frac{1}{128}(6435x^8-12012x^6+6930x^4-1260x^2+35)\\
\end{alignat}













<証明>

次のロドリグの公式を用います。$$(A)  P_n(x)=\frac{1}{2^nn!}\cdot \frac{d^n}{dx^n}(x^2-1)^n$$










\((A)\) の式で \(n=0,1,2, \cdots 6\) とします。

\begin{alignat}{2}
(1)  P_0(x)&=\frac{1}{2^00!} (x^2-1)^0=1\\
&\\
&\\
(2)  P_1(x)&=\frac{1}{2^11} \cdot \frac{d}{dx}(x^2-1)=\frac{1}{2} \cdot 2x=x\\
&\\
&\\
(3)  P_2(x)&=\frac{1}{2^22!} \cdot \frac{d^2}{dx^2}(x^2-1)^2=\frac{1}{8} \cdot \frac{d^2}{dx^2}(x^4-2x^2+1)\\
&=\frac{1}{8} \cdot \frac{d}{dx}(4x^3-4x)=\frac{1}{8}(12x^2-4)=\frac{1}{2}(3x^2-1) \\
&\\
&\\
(4)  P_3(x)&=\frac{1}{2^33!} \cdot \frac{d^3}{dx^3}(x^2-1)^3=\frac{1}{48} \cdot \frac{d^3}{dx^3}(x^6-3x^4+3x^2-1)\\
&=\frac{1}{48} \cdot \frac{d^2}{dx^2}(6x^5-12x^3+6x)=\frac{1}{48} \cdot \frac{d}{dx}(30x^4-36x^2+6)\\
&=\frac{1}{48}(120x^3-72x)=\frac{1}{2}(5x^3-3x)\\
&\\
&\\
(5)  P_4(x)&=\frac{1}{2^44!} \cdot \frac{d^4}{dx^4}(x^2-1)^4=\frac{1}{16 \cdot 24} \cdot \frac{d^4}{dx^4}(x^8-4x^6+6x^4-4x^2+1)\\
&=\frac{1}{16 \cdot 24} \cdot \frac{d^3}{dx^3}(8x^7-24x^5+24x^3-8x)=\frac{1}{48} \cdot \frac{d^3}{dx^3}(x^7-3x^5+3x^3-x)\\
&=\frac{1}{48} \cdot \frac{d^2}{dx^2}(7x^6-15x^4+9x^2-1)=\frac{1}{48} \cdot \frac{d}{dx}(42x^5-60x^3+18x)\\
&=\frac{1}{8} \cdot \frac{d}{dx}(7x^5-10x^3+3x)=\frac{1}{8}(35x^4-30x^2+3)\\
&\\
&\\
(6)  P_5(x)&=\frac{1}{2^55!} \cdot \frac{d^5}{dx^5}(x^2-1)^5\\
&=\frac{1}{2^55!} \cdot \frac{d^5}{dx^5}(x^{10}-5x^8+10x^6-10x^4+5x^2-1)\\
&=\frac{1}{2^55!} \cdot \frac{d^4}{dx^4}(10x^{9}-40x^7+60x^5-40x^3+10x)\\
&=\frac{1}{2^44!} \cdot \frac{d^4}{dx^4}(x^{9}-4x^7+6x^5-4x^3+x)\\
&=\frac{1}{2^44!} \cdot \frac{d^3}{dx^3}(9x^8-28x^6+30x^4-12x^2+1)\\
&=\frac{1}{2^44!} \cdot \frac{d^2}{dx^2}(72x^7-168x^5+120x^3-24x)\\
&=\frac{1}{16} \cdot \frac{d^2}{dx^2}(3x^7-7x^5+5x^3-x)\\
&=\frac{1}{16} \cdot \frac{d}{dx}(21x^6-35x^4+15x^2-1)\\
&=\frac{1}{16} (126x^5-140x^5+30x)=\frac{1}{8}(63x^5-70x^3+15x)\\
&\\
&\\
(6)  P_6(x)&=\frac{1}{2^66!} \cdot \frac{d^6}{dx^6}(x^2-1)^6\\
&=\frac{1}{2^66!} \cdot \frac{d^6}{dx^6}(x^{12}-6x^{10}+15x^8-20x^6+15x^4-6x^2+1)\\
&=\frac{1}{2^66!} \cdot \frac{d^5}{dx^5}(12x^{11}-60x^{9}+120x^7-120x^5+60x^3-12x)\\
&=\frac{1}{2^6 \cdot 60} \cdot \frac{d^5}{dx^5}(x^{11}-5x^{9}+10x^7-10x^5+5x^3-x)\\
&=\frac{1}{2^6 \cdot 60} \cdot \frac{d^4}{dx^4}(11x^{10}-45x^{8}+70x^6-50x^4+15x^2-1)\\
&=\frac{1}{2^6 \cdot 60} \cdot \frac{d^3}{dx^3}(110x^{9}-360x^{7}+420x^5-200x^3+30x)\\
&=\frac{1}{2^6 \cdot 6} \cdot \frac{d^3}{dx^3}(11x^{9}-36x^{7}+42x^5-20x^3+3x)\\
&=\frac{1}{2^6 \cdot 6} \cdot \frac{d^2}{dx^2}(99x^{8}-252x^{6}+210x^4-60x^2+3)\\
&=\frac{1}{2^6 \cdot 6} \cdot \frac{d}{dx}(792x^{7}-1512x^{5}+840x^3-120x)\\
&=\frac{1}{16} \cdot \frac{d}{dx}(33x^{7}-63x^{5}+35x^3-5x)\\
&=\frac{1}{16}(231x^6-315x^4+105x^2-5)\\
&\\
&\\
(8)  P_7(x)&=\frac{1}{2^77!} \cdot \frac{d^7}{dx^7}(x^2-1)^7\\
&=\frac{1}{2^77!} \cdot \frac{d^7}{dx^7}(x^{14}-7x^{12}+21x^{10}-35x^8+35x^6-21x^4+7x^2-1)\\
&=\frac{1}{2^77!} \cdot \frac{d^6}{dx^6}(14x^{13}-84x^{11}+210x^{9}-280x^7+210x^5-84x^3+14x)\\
&=\frac{1}{2^66!} \cdot \frac{d^6}{dx^6}(x^{13}-6x^{11}+15x^{9}-20x^7+15x^5-6x^3+x)\\
&=\frac{1}{2^66!} \cdot \frac{d^5}{dx^5}(13x^{12}-66x^{10}+135x^{8}-140x^6+75x^4-18x^2+1)\\
&=\frac{1}{2^66!} \cdot \frac{d^4}{dx^4}(156x^{11}-660x^{9}+1080x^{7}-840x^5+300x^3-36x)\\
&=\frac{1}{2^55!} \cdot \frac{d^4}{dx^4}(13x^{11}-55x^{9}+90x^{7}-70x^5+25x^3-3x)\\
&=\frac{1}{2^55!} \cdot \frac{d^3}{dx^3}(143x^{10}-495x^{8}+630x^{6}-350x^4+75x^2-3)\\
&=\frac{1}{2^55!} \cdot \frac{d^2}{dx^2}(1430x^{9}-3960x^{7}+3780x^{5}-1400x^3+150x)\\
&=\frac{1}{2^44!} \cdot \frac{d^2}{dx^2}(143x^{9}-396x^{7}+378x^{5}-140x^3+15x)\\
&=\frac{1}{2^44!} \cdot \frac{d}{dx}(1287x^{8}-2772x^{6}+1890x^{4}-420x^2+15)\\
&=\frac{1}{2^44!} (10296x^7-16632x^5+7560x^3-840x)\\
&=\frac{1}{16}(429x^7-693x^5+315x^3-35x)\\
&\\
&\\
(9)  P_8(x)&=\frac{1}{2^88!} \cdot \frac{d^8}{dx^8}(x^2-1)^8\\
&=\frac{1}{2^88!} \cdot \frac{d^8}{dx^8}(x^{16}-8x^{14}+28x^{12}-56x^{10}+70x^8-56x^6+28x^4-8x^2+1)\\
&=\frac{1}{2^88!} \cdot \frac{d^7}{dx^7}(16x^{15}-112x^{13}+336x^{11}-560x^{9}+560x^7-336x^5+112x^3-16x)\\
&=\frac{1}{2^77!} \cdot \frac{d^7}{dx^7}(x^{15}-7x^{13}+21x^{11}-35x^{9}+35x^7-21x^5+7x^3-x)\\
&=\frac{1}{2^77!} \cdot \frac{d^6}{dx^6}(15x^{14}-91x^{12}+231x^{10}-315x^{8}+245x^6-105x^4+21x^2-1)\\
&=\frac{1}{2^77!} \cdot \frac{d^5}{dx^5}(210x^{13}-1092x^{11}+2310x^{9}-2520x^{7}+1470x^5-420x^3+42x)\\
&=\frac{1}{2^66!} \cdot \frac{d^5}{dx^5}(15x^{13}-78x^{11}+165x^{9}-180x^{7}+105x^5-30x^3+3x)\\
&=\frac{1}{2^66!} \cdot \frac{d^4}{dx^4}(195x^{12}-858x^{10}+1485x^{8}-1260x^{6}+525x^4-90x^2+3)\\
&=\frac{1}{2^66!} \cdot \frac{d^3}{dx^3}(2340x^{11}-8580x^{9}+11880x^{7}-7560x^{5}+2100x^3-180x)\\
&=\frac{1}{2^55!} \cdot \frac{d^3}{dx^3}(195x^{11}-715x^{9}+990x^{7}-630x^{5}+175x^3-15x)\\
&=\frac{1}{2^55!} \cdot \frac{d^2}{dx^2}(2145x^{10}-6435x^{8}+6930x^{6}-3150x^{4}+525x^2-15)\\
&=\frac{1}{2^55!} \cdot \frac{d}{dx}(21450x^{9}-51480x^{7}+41580x^{5}-12600x^{3}+1050x)\\
&=\frac{1}{2^44!} \cdot \frac{d}{dx}(2145x^{9}-5148x^{7}+4158x^{5}-1260x^{3}+105x)\\
&=\frac{1}{2^44!} (19305x^{8}-36036x^{6}+20790x^{4}-3780x^{2}+105)\\
&=\frac{1}{128}(6435x^8-12012x^6+6930x^4-1260x^2+35)
\end{alignat}
\end{alignat}

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