Some arithmetical applications of Newton´s interpolation series

Abstract

The main result of Nopper´s paper [8] in this same journal reads as follows. If a transcendental entire function f has small growth order, then there is at most one rational point w such that all f(n)(w), n = 0,1,..., are rational with denominators increasing sufficiently slowly with n. The principal purpose of our work is to considerably relaxe this arithmetic growth condition, to generalize Nopper´s result to meromorphic functions, and to improve on similar statements of Nikishin [6], and Nopper and Wallisser [10]. From all these assertions, some well-known but not obvious irrationality results can be deduced in a systematic and simple way. In each case, the method of Newton´s interpolation series is essentially used, and it is also explained how this one leads to new and transparent proofs of Straus´ [16] and Schneider´s [13] achievements on integer-valued meromorphic functions.