= Solution
For positive integral $m$, the exponent-zero <linear recurrence relation> fixes $a_n=(-1)^na_0/n!$ for $1\leq n<m$, but at $n=m$ its factor $n-m$ vanishes and leaves $a_m$ free. Choose $a_0=1$ and $a_m=0$. All later coefficients then vanish by the same <linear recurrence relation>.
\b[A polynomial solution of degree $m-1$ is]
$$
\boxed{P_{m-1}(x)=\sum_{n=0}^{m-1}\frac{(-1)^nx^n}{n!}.}
$$
For $m=1$ this is the constant solution $1$, in agreement with the <ordinary point> exception. The <terminating Frobenius series> arises because the coefficient equation at the resonant index becomes $0=0$, permitting truncation rather than forcing a logarithm.
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