Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-12T15:01:46.593Z Has data issue: false hasContentIssue false
  • English
  • Français

Liouville sequences*

Published online by Cambridge University Press:  22 January 2016

Jaroslav Hančl
Jaroslav Hančl*
Affiliation:
Department of Mathematics, University of Ostrava, Dvořákova 7, 701 03 Ostrava 1 Czech Republic hancl@osu.cz
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

The new concept of a Liouville sequence is introduced in this paper by mean of the related Liouville series. Main results are two criteria for when certain sequences are Liouville. Several applications are presented. A counterexample is included for the case that we substantially weaken the hypotheses in the main results.

Keywords

11J28

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 172 , 2003, pp. 173 - 187
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2003

Footnotes

*

Supported by the grant 201/01/0471 of the Czech Grant Agency

References

[1] Adams, W. W., On the algebraic independence of certain Liouville numbers, J. Pure Appl. Algebra, 13 (1978), 41–47.Google Scholar
[2] Adams, W. W., The algebraic independence of certain Liouville continued fractions, Proc. Amer. Math. Soc., 95, no.4, (1985), 512–516.Google Scholar
[3] Bundschuh, P., A criterion for algebraic independence with some applications, Osaka J. Math., 25 (1988), 849–858.Google Scholar
[4] Erdös, P., Representation of real numbers as sums and products of Liouville numbers, Michigan Math. J., 9 (1962), 59–60.Google Scholar
[5] Erdös, P., Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci., 10 (1975), 1–7.Google Scholar
[6] Nishioka, K., Proof of Masser’s conjecture on the algebraic independence of values of Liouville series, Proc. Japan Acad. Ser., A 62 (1986), 219–222.Google Scholar
[7] Nishioka, K., Mahler functions and transcendence, Lecture Notes in Mathematics 1631, Springer (1996).Google Scholar
[8] Pass, R., Results concerning the algebraic independence of sets of Liouville numbers, Thesis Univ. of Maryland, College Park, 1978.Google Scholar
[9] Petruska, G., On strong Liouville numbers, Indag. Math., N.S., 3(2) (1992), 211–218.Google Scholar
[10] Schlickewei, H. P. and der Poorten, A. J. van, The growth conditions for recurrence sequences, Macquarie University Math. Rep., 82–0041, North Ryde, Australia.Google Scholar