Question 172
Let $p$ be a fixed prime and let $a \ge 2$ and $e \ge 1$ be fixed integers. Given a function $f: \mathbb{Z}/a\mathbb{Z} \to \mathbb{Z}/p^e\mathbb{Z}$ and an integer $k \ge 0$, the $k$th finite difference, denoted $\Delta^k f$, is the function from $\mathbb{Z}/a\mathbb{Z}$ to $\mathbb{Z}/p^e\mathbb{Z}$ defined recursively by
$$
\begin{aligned}
\Delta^0 f(n) &= f(n) \\
\Delta^k f(n) &= \Delta^{k-1} f(n+1) - \Delta^{k-1} f(n) \quad \text{for } k = 1, 2, \dots
\end{aligned}
$$
Determine the number of functions $f$ such that there exists some $k \ge 1$ for which $\Delta^k f = f$.
$$
\begin{aligned}
\Delta^0 f(n) &= f(n) \\
\Delta^k f(n) &= \Delta^{k-1} f(n+1) - \Delta^{k-1} f(n) \quad \text{for } k = 1, 2, \dots
\end{aligned}
$$
Determine the number of functions $f$ such that there exists some $k \ge 1$ for which $\Delta^k f = f$.