Irrationality results for values of generalized Tschakaloff series II

Masaaki Amou, Masanori Katsurada

Research output: Contribution to journalArticle

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Abstract

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 Mahler's 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).

Original languageEnglish
Pages (from-to)132-155
Number of pages24
JournalJournal of Number Theory
Volume104
Issue number1
DOIs
Publication statusPublished - 2004 Jan

Fingerprint

Irrationality
Series
Theorem
Transcendence
Linear independence
Number theory
Error term
Growth Conditions
Lemma
Limiting
Zero
Coefficient
Approximation
Estimate

Keywords

  • Irrationality
  • Irrationality measure
  • Padé approximation
  • q-difference equation
  • Siegel's lemma

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

Irrationality results for values of generalized Tschakaloff series II. / Amou, Masaaki; Katsurada, Masanori.

In: Journal of Number Theory, Vol. 104, No. 1, 01.2004, p. 132-155.

Research output: Contribution to journalArticle

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