Let $\left(X_{n}\right)_{n \in \mathbb{N}}$ be a sequence of mutually independent Rademacher random variables and $S_{n} = \sum_{j=1}^{n} X_{j}$. For every $n \in \mathbb{N}^{*}$ and every $k \in \mathbb{Z}$, by setting $p_{n}(k) = \mathbb{P}\left(S_{n} = k\right)$, show that
$$\frac{p_{n+1}(k) - p_{n}(k)}{\tau} = \frac{\delta^{2}}{2\tau} \frac{p_{n}(k+1) - 2p_{n}(k) + p_{n}(k-1)}{\delta^{2}}$$