Irrationality results for values of generalized Tschakaloff series II

The study of irrationality properties of values of the generalized Tschakaloff series f(x) defined by (1.2) below was initiated by Duverney (Portugal. Math. 53(2) (1996) 229; Period. Math. Hungar. 35 (1997) 149), and continued by the authors (J. Number Theory 77 (1999) 155). The present paper proceeds to extend our previous result (Amou and Katsurada, 1999, Theorem). The irrationality of f(α) for any α Q {0} is proved in a quantitative form under fairly general growth conditions on the coefficients of f(x) (Theorem 1), while the same result is shown in a certain ‘limiting’ situation of Theorem 1, at the cost of loosing a quantitative aspect (Theorem 2). The linear independence of certain values of a system of f(x) is also obtained (Theorem 3). The key idea in proving our previous result is a Mahlerʹs transcendence method, due to Loxton and van der Poorten (in: A. Baker, D.W. Masser (Eds.), Transcendence Theory: Advances and Applications, Academic Press, San Diego, 1977, pp. 211–226), applied to an appropriate sequence of functions (see (2.4) and (2.5)). In order to establish Theorems 1 and 2, this method is enhanced by a certain technique which allows us to improve zero estimates for the remainder terms of Padé-type approximations (see Lemmas 3 and 4).