if a Newton series exists that p is actually a constant function.
Not exactly so. The solutions are always a set of functions different by a constant term. But among these solutions the one which is equal to its Newton series is unique.
This is due to the Carlson's theorem of 1914.
I have been pointed recently that convergence of Newton series is covered extensively in Gelfond, A. O. Calculus of finite differences. Translated from the Russian. International Monographs on Advanced Mathematics and Physics. Hindustan Publishing Corp., Delhi, 1971
The Russian original is available online:
Page 141 onwards.
Look also for some specific questions: Convergence of Newton series for sin ax
