In the theory of indefinite sumsums, antidifferenceanti-differences and finite calculus, the following differencedifference functional equation and obtaining its some special solutions isare very important:
$\bigtriangleup F(x):=F(x+1)-F(x)=f(x) \quad;\quad(1),$$$\bigtriangleup F(x):=F(x+1)-F(x)=f(x) \quad\quad(1),$$
wherewhere $\bigtriangleup$ is the forward difference operator and when $f$ is given and $F$ is unknown. Also Also, I know that 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 $F=F_0+\lambda,$: $F=F_0+\lambda$, which $\lambda$ is a one-periodic function. Now
Now, in my research I deal to the following functional equation
$F(x+1)+F(x)=f(x)$ $$F(x+1)+F(x)=f(x)$$
, butbut I don't have any knowldgeknowledge about it and its soloutionsolution. Any one can help me firstly, what does it call What is the name of this equation and secondly tell me some informationwhere can I learn more about this. Thank you.it?
