We note that we have three equations to determine the three coefficients
\( a \), \( b \) and \( c \).
Alternatively, we could have
employed the recursion relation of Eq. (11), resulting in
$$
\begin{equation*}
2L_2(x)=3xL_1(x)-L_0,
\end{equation*}
$$
which leads to Eq. (12).