Suppose we have two real functions $f(x)$ and $g(x)$, both equal to their Newton series expansion (let's call such function Newton-analytic):
$$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$
$$g(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k g\left (0\right)$$
Is their composition $F(x)=f(g(x))$ also equal to its Newton series expansion (if it converges)?
$$F(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k F\left (0\right)$$