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% $B:n@.G/7nF|(B $B!'(B /02/10/16
% $BEE;R%a!<%k(B $B!'(B g240916@mail.ecc.u-tokyo.ac.jp
%
% \mondai{$B:n@.<T(B}{$B:n@.G/7nF|(B}{$BEE;R%a!<%k(B}{$BLdBjHV9f(B}{$BLdBjJ8(B}{$B2rEz(B}
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\mondai
{240916J $B9b>k(B $B=E9((B}
{/02/10/16}
{g240916@mail.ecc.u-tokyo.ac.jp}
{8-5}
{
%----$BLdBjJ8(B----%
$B%k%8%c%s%I%kB?9`<0(B$P_l(z)(8.50)$$B$O!"%k%8%c%s%I%k$NHyJ,J}Dx<0(B$(8.49)$$B$rK~$?$9$3$H$r<($;!#$^$?!"%k%8%c%s%I%kB?9`<0(B$P_l(z)$B$N(Bl=0,1,2$B$N4X?t7A$r(B(8.50)$$B$h$j5a$a$h!#(B

}
{
%----$B2rEzJ8(B---%
$P_l(z)$$B$r%k%8%c%s%I%k$NHyJ,J}Dx<0$KBeF~$7$F!"$=$N2r$K$J$C$F$$$k$3$H$r<($;$PNI$$!#$D$^$j(B
\begin{equation}
\frac{d}{dz}\left((1-z^2)\frac{d}{dz}\left\{\frac{1}{2^ll!}\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^l\right\}\right)+l(l+1)\left\{\frac{1}{2^ll!}\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^l\right\}=0 \label{eq:8-5-1}
\end{equation}
$B$r<($9!#(B\\
\begin{equation}
(1-z^2)\frac{d}{dz}\left\{\frac{1}{2^ll!}\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^l\right\}
=\frac{1}{2^ll!}(1-z^2)\frac{d^{(l+1)}}{dz^{(l+1)}}(z^2-1)^l \label{eq:8-5-2}
\end{equation}
$B$3$3$G(B
\begin{equation}
(fg)^{\left(n\right)}=\sum^n_{i=o}{}_n C _i f^{\left(i\right)}g^{\left(n-i\right)} \label{eq:8-5-3}
\end{equation}
$B$G$"$k$+$i!"(B$f=(z^2-1)^l,g=(z^2-1)$$B$H$7$F(B
\begin{eqnarray}
\frac{d^{(l+1)}}{dz^{(l+1)}}(z^2-1)^{l+1}&=&(z^2-1)\frac{d^{(l+1)}}{dz^{(l+1)}}(z^2-1)^{l}+2z(l+1)\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^{l}\nonumber \\
&+&2\frac{1}{2}l(l+1)\frac{d^{(l-1)}}{dz^{(l-1)}}(z^2-1)^{l}
\end{eqnarray}
$B$3$l$h$j(B
\begin{eqnarray}
F&=&(z^2-1)\frac{d^{(l+1)}}{dz^{(l+1)}}(z^2-1)^{l} \label{eq:8-5-4} \\
&=&\frac{d^{(l+1)}}{dz^{(l+1)}}(z^2-1)^{l+1}-2z(l+1)\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^{l}-l(l+1)\frac{d^{(l-1)}}{dz^{(l-1)}}(z^2-1)^{l}
\end{eqnarray}
$B$3$N$h$&$K(B$F$$B$r$*$1$P(B
\begin{eqnarray}
\frac{dF}{dz}&=&\frac{d^{(l+2)}}{dz^{(l+2)}}(z^2-1)^{l+1}-2(l+1)\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^{l}\\ 
&-&2z(l+1)\frac{d^{(l+1)}}{dz^{(l+1)}}(z^2-1)^{l}-l(l+1)\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^{l} \nonumber \\
&=&2(l+1)\frac{d^{(l+1)}}{dz^{(l+1)}}z(z^2-1)^{l}-2(l+1)\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^{l}\\ 
&-&2z(l+1)\frac{d^{(l+1)}}{dz^{(l+1)}}(z^2-1)^{l}-l(l+1)\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^{l} \nonumber
\end{eqnarray}
$B$3$3$G$^$?(B(\ref{eq:8-5-3})$B$h$j!"(B$g=z,f=(z^2-1)^l$$B$H$*$1$P(B
\begin{equation}
\frac{d^{(l+1)}}{dz^{(l+1)}}z(z^2-1)^l=z\frac{d^{(l+1)}}{dz^{(l+1)}}(z^2-1)^l+(l+1)\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^l
\end{equation}
$B$h$C$F(B
\begin{equation}
\frac{dF}{dz}=l(l+1)\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^{l} \label{eq:8-5-5}
\end{equation}
$B<($9$N$O(B$(\ref{eq:8-5-2}),(\ref{eq:8-5-4})$B$r(B(\ref{eq:8-5-1})$$B$KBeF~$7$F(B
\begin{equation}
\frac{d}{dz}\left\{\frac{1}{2^ll!}(-F)\right\}+l(l+1)\left\{\frac{1}{2^ll!}\frac{d^{(l)}}{dz^{(l)}}(z^2-1)^l\right\}=0
\end{equation}
$B$3$l$O(B$(\ref{eq:8-5-5})$$B$h$jK~$?$5$l$k!#(B\\
$B0J>e$h$j%k%8%c%s%I%kB?9`<0(B$P_l(z)$$B$O!"%k%8%c%s%I%k$NHyJ,J}Dx<0(B$(8.49)$$B$rK~$?$9!#(B\\
$B$^$?!"(B\\
$P_0(z)=\displaystyle{\frac{1}{2}}$\\
$P_1(z)=\displaystyle{\frac{1}{2}\frac{d}{dz}}(z^2-1)=\displaystyle{z}$\\
$P_3(z)=\displaystyle{\frac{1}{8}\frac{d^2}{dz^2}}(z^2-1)^2=\displaystyle{\frac{1}{8}}\left\{\displaystyle{\frac{d}{dz}}4z(z^2-1)\right\}=\displaystyle{\frac{1}{2}}(z^2-1+2z^2)=\displaystyle{\frac{1}{2}}(3z^2-1)$\\
}


