Closed-form formula for some recursively-defined integro-difference sequence of functions

The main purpose of this paper is to derive the closed form solution the sequence $(g_n{)}_{n\in \mathbb{N}}$ of integro-difference equations that is defined recursively as follows: \begin{align*} g_1(x) & = χ_{(-1/2, 1/2)} (x), g_{n+1}(x) & = g_n(x + 1/2)- g_n(x- 1/2) + \int_{x-\frac{1}{2}}^{x + \frac{1}{2}} g_n(s)ds, , n\in $\mathbb{N}$, \end{align*} where $g_1(x)={\chi }_{(-1\mathrm{/}2,1\mathrm{/}2)}(x)$ is the characteristic function of the unit interval $(-1\mathrm{/}2,1\mathrm{/}2)$ has value equal to $1$ on $(-1\mathrm{/}2,1\mathrm{/}2)$ and $0$ elsewhere in $\mathbb{R}$.