= Solution
Substitute the <power series> and collect each power of $\xi$:
$$
\sum_{k\geq0}\left[(k+2)(k+1)a_{k+2}+(\epsilon-1-2k)a_k\right]\xi^k=0.
$$
Matching the <power series> coefficients gives the <recurrence relation>
$$
\boxed{a_{k+2}=\frac{2k+1-\epsilon}{(k+2)(k+1)}a_k,\qquad k=0,1,2,\ldots.}
$$
The <even> and <odd> coefficients form separate sequences, determined by $a_0$ and $a_1$, respectively. For example, $a_2=(1-\epsilon)a_0/2$ and $a_3=(3-\epsilon)a_1/6$. These are the two independent <power series solutions of the Hermite differential equation>.
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