Mouze, AugustinAugustinMouzeNestoridis, VassiliVassiliNestoridisPapadoperakis, IoannisIoannisPapadoperakisTsirivas, NikolaosNikolaosTsirivas2013-01-142013-01-14Heldermann2012-01Computational Methods and Function Theoryhttp://hdl.handle.net/10197/4035The known proofs for universal Taylor series do not determine a specific universal Taylor series. In the present paper, we isolate a specific universal Taylor series by modifying the proof in [30]. Thus we determine all Taylor coefficients of a specific universal Taylor series on the disc or on a polygonal domain. Furthermore in non simply connected domains, when universal Taylor series exist, we can construct a sequence of specific rational functions converging to a universal function, provided the boundary is good enough. The solution uses an infinite denumerable procedure and a finite number of steps is not sufficient. However we solve a Runge's type problem in a finite number of steps.enUniversal seriesTaylor seriesRunge theoremJournal Article1211731992012-11-23https://creativecommons.org/licenses/by-nc-nd/3.0/ie/