Let F be meromorphic function,with what properties may it expanded as power series with coefficients of integers in such a form:


$$F=\sum\_0^{\infty}a\_i x^i,a\_i\in \mathcal{N} \bigcup 0,\exists M \space a\_i \leq M^i$$.


and when the coefficients consist of a sequence of computably enumerable relation.


If the question is ambiguous, please tell me.


When may function (meromorphic) be expanded as power series with coefficients of integers



When may function (meromorphic) be expanded as power series with coefficients of integers
This was a large research subject in 1930-s. The key authors are G. Polya, Ch. Pisot
and Raphael Robinson.
The book of Bieberbach, Analytische Fortsetzung (in German, there is a Russian translation) contains a chapter with a survey
of these results.


The general spirit of these results is the following: if you have a Taylor series
with integer coefficients which has an analytic or meromorphic continuation
in sufficiently large region, then the function must be rational, and in certain
cases all such functions can be explicitly described.
But there are too many results to mention them here.


By the way, the question is equivalent, via Borel-Laplace transform to a question about
entire functions which take integer values at positive integers.
So "Integer-values entire functions" is just another name of the same topic.


