In the theory of indefinite sums, anti-differences and finite calculus, the following difference functional equation and its solutions are very important:
$$\bigtriangleup F(x):=F(x+1)-F(x)=f(x) \quad\quad(1),$$
where $\bigtriangleup$ is the forward difference operator when $f$ is given and $F$ is unknown. Also, if $D_f=\mathbb{R}$, then there exists a special solution $F_0(x)$ for equation (1) and we have the general solutions of it as follows: $F=F_0+\lambda$, which $\lambda$ is a one-periodic function.
Now, in my research I deal to the following functional equation $$F(x+1)+F(x)=f(x)$$
but I don't have any knowledge about it and its solution. What is the name of this equation and where can I learn more about it?
