Lets define a discrete analytic function such a function that is equal to its Newton series:
$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$
Is function $g(x)=e^{f(x)}$ also discrete-analytic given the both functions are real-valued and the both expansions converge?
This question arose from the following considerations.
As you know the difference equation
$\Delta y(x) = F(x)$
has multiple solutions that differ only by an arbitrary 1-periodic function $C(x)$:
$y(x)=y_1(x)+C(x)$
At the same time there can be no more than one (up to a constant term) discrete-analytic solution which we can consider to be the natural solution of the equation.
But when considering multiplicative-difference equation $\frac{y(x+1)}{y(x)}=F(x)$ we come to a similar situation, this equation has multiple solutions which differ by an arbitrary 1-periodic factor:
$y(x)=C(x)y_1(x)$
Of these solutions, similarly, no more than one (up to a constant factor) is discrete-analytic which allows us to define the distinguished solution.
But on the other hand the following rule holds for indefinite product and sum:
$\prod_x f(x)= e^{\sum_x \ln f(x)}$
This means that we can obtain the solution to the equation $\frac{y(x+1)}{y(x)}=F(x)$ in the following form:
$y(x)=e^{\sum_x \ln F(x)}$
This allows us to select the distinguished solution by another method, that is taking the natural solution to the sum and taking exponent of it. The result will have a constant factor, but it is unevident whether it will be discrete-analytic or not, and as such, whether the both distinguished solutions coincide.