For the differential equation $(1-x^{2}) y'' - 2 x y' + \lambda y = 0$ it is seen that if $y$ is of the form $\sum a_{n} x^{n}$ the following holds.
\begin{align}
y(x) &= \sum_{n=0}^{\infty} a_{n} \, x^{n} \\
y'(x) &= \sum_{n=0}^{\infty} n \, a_{n} \, x^{n-1} \\
y''(x) &= \sum_{n=0}^{\infty} n(n-1) \, a_{n} \, x^{n-2}
\end{align}
\begin{align}
0 &= (1 - x^{2} ) \, \sum_{n=0}^{\infty} n(n-1) \, a_{n} x^{n-2} - 2 x \, \sum_{n=0}^{\infty} n \, a_{n} \, x^{n-1} + \lambda \sum_{n=0}^{\infty} a_{n} \, x^{n} \\
&= \sum_{n=2}^{\infty} n(n-1) \, a_{n} \, x^{n-2} - \sum_{n=0}^{\infty} (n(n+1) - \lambda) \, a_{n} \, x^{n} \\
&= \sum_{n=0}^{\infty} (n+2)(n+1) \, a_{n+2} \, x^{n} - \sum_{n=0}^{\infty} (n(n+1) - \lambda) \, a_{n} \, x^{n} \\
&= \sum_{n=0}^{\infty} \left[ (n+2)(n+1) \, a_{n+2} - (n(n+1) - \lambda) \, a_{n} \right] \, x^{n}
\end{align}
From this equation the coefficient equation can be obtained. It is
\begin{align}
a_{n+2} = \frac{n(n+1) - \lambda}{(n+1)(n+2)} \, a_{n} \hspace{10mm} n \geq 0.
\end{align}
Now that the recurrence relation has been obtained. Try a few values of $n$ to obtain the first few terms. The first two terms are defined as $a_{0}, a_{1}$ and the remaining are to follow.
\begin{align}
a_{2} &= \frac{- \lambda }{2!} \, a_{0} \\
a_{3} &= \frac{2-\lambda}{2 \cdot 3} \, a_{1} = \frac{(-1) (\lambda - 2)}{3!} \, a_{1} \\
a_{4} &= \frac{6 - \lambda}{3 \cdot 4} \, a_{2} = \frac{(-1)^{2} \lambda (\lambda - 6)}{4!} \, a_{0}
\end{align}
and so on. The solution for $y(x)$ is of the form
\begin{align}
y(x) &= a_{0} \left[1 - \frac{\lambda}{2!} \, x^{2} + \frac{(-1)^{2} \lambda (\lambda -6)}{4!} \, x^{4} + \frac{(-1)^{3} \lambda (\lambda -6)(\lambda -20)}{6!} \, x^{6} + \cdots \right] \\
& \hspace{5mm} + a_{1} \left[x + \frac{(-1)(\lambda - 2)}{ 3!} \, x^{3} + \frac{(-1)^{2} (\lambda - 2)(\lambda - 12)}{5!} \, x^{5} + \cdots \right]
\end{align}
Hint Note that per the recurrence relation, the value of $T(n)$ depends only on $T(n - 5)$, so, e.g., the subsequence $S := [T(1), T(6), T(11), \ldots]$ can be determined without finding any of the intermediate values, $T(2), T(3),$ etc. Computing gives that this subsequence is $[1, 3, 9, \ldots]$---can you find an explicit formula for $S$?
Best Answer
$F(x)=\sum_{i\ge 0}a_ix^i$, where $a_0=1,a_1=2$ and $a_i=3a_{i-2}$ for $i\ge 2$.