Some arithmetical applications of Newton's interpolation series

摘要

The main result of Nopper's paper [8] in this same journal reads as follows. If a transeendental 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 meromorphie 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 meromorphie functions.