I’m wondering if a function $f: \mathbb{N} \longrightarrow R$ can be represented as a Newton series given that all its forward differences exist.
The first thing I searched up was a result in complex analysis called Carlson’s Theorem which helps to show when a function in $\mathbb{C}$ is identical to its Newton series.
But the functions I’m concerned with only has domain $\mathbb{N}$. With my naive understanding of Taylor series’s, I know that if a function is infinitely differentiable at a point, then it is identical to its Taylor series at that point.
Can something analogous be true for Newton series’s if I restrict functions to domain $\mathbb{N}$ instead of $\mathbb{C}$ without any extra conditions such as those in Carlson’s Theorem?