When may Function (meromorphic) be expanded as power series with coefficients of integers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T17:52:11Z http://mathoverflow.net/feeds/question/73553 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/73553/when-may-function-meromorphic-be-expanded-as-power-series-with-coefficients-of When may Function (meromorphic) be expanded as power series with coefficients of integers XL 2011-08-24T09:39:08Z 2012-08-04T10:30:49Z <p>Let F be meromorphic function,with what properties may it expanded as power series with coefficients of integers in such a form:</p> <p>$$F=\sum_0^{\infty}a_i x^i,a_i\in \mathcal{N} \bigcup 0,\exists M \space a_i \leq M^i$$.</p> <p>and when the coefficients consist of a sequence of computably enumerable relation.</p> <p>If the question is ambiguous ,please tell me but please do not downvote it.</p> <p>When may Function (meromorphic) be expanded as power series with coefficients of integers</p> http://mathoverflow.net/questions/73553/when-may-function-meromorphic-be-expanded-as-power-series-with-coefficients-of/103930#103930 Answer by Alexandre Eremenko for When may Function (meromorphic) be expanded as power series with coefficients of integers Alexandre Eremenko 2012-08-04T10:30:49Z 2012-08-04T10:30:49Z <p>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.</p> <p>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 any results to mention them here.</p> <p>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.</p>