Skip to main content Accessibility help
×
Home

TRANSCENDENCE OVER MEROMORPHIC FUNCTIONS

  • MICHAEL COONS (a1) and YOHEI TACHIYA (a2)

Abstract

In this short note, considering functions, we show that taking an asymptotic viewpoint allows one to prove strong transcendence statements in many general situations. In particular, as a consequence of a more general result, we show that if $F(z)\in \mathbb{C}[[z]]$ is a power series with coefficients from a finite set, then $F(z)$ is either rational or it is transcendental over the field of meromorphic functions.

Copyright

Corresponding author

Footnotes

Hide All

The research of M. Coons was supported by ARC grant DE140100223 and the research of Y. Tachiya was supported by JSPS, Grant-in-Aid for Young Scientists (B), 15K17504.

Footnotes

References

Hide All
[1] Allouche, J.-P., ‘Transcendence of formal power series with rational coefficients’, Theoret. Comput. Sci. 218(1) (1999), 143160.
[2] Allouche, J.-P. and Shallit, J., Automatic Sequences (Cambridge University Press, Cambridge, 2003).
[3] Bézivin, J.-P., ‘Sur une classe d’équations fonctionnelles non linéaires’, Funkcial. Ekvac. 37(2) (1994), 263271.
[4] Borwein, P. and Coons, M., ‘Transcendence of power series for some number theoretic functions’, Proc. Amer. Math. Soc. 137(4) (2009), 13031305.
[5] Borwein, P., Erdélyi, T. and Littmann, F., ‘Polynomials with coefficients from a finite set’, Trans. Amer. Math. Soc. 360(10) (2008), 51455154.
[6] Coons, M., ‘An asymptotic approach in Mahler’s method’, Preprint, 2015, available at arXiv:1511.07534, 15 pages.
[7] Duffin, R. J. and Schaeffer, A. C., ‘Power series with bounded coefficients’, Amer. J. Math. 67 (1945), 141154.
[8] Fatou, P., ‘Séries trigonométriques et séries de Taylor’, Acta Math. 30(1) (1906), 335400.
[9] Knopp, K., ‘Über Lambertsche Reihen’, J. reine angew. Math. 142 (1913), 283315.
[10] Randé, B., Équations fonctionnelles de Mahler et applications aux suites p-régulières (Institut National de Recherche en Informatique et en Automatique (INRIA), Rocquencourt, 1992); Thèse, Université de Bordeaux I, Talence, 1992.
[11] Stein, E. M. and Shakarchi, R., Complex Analysis, Princeton Lectures in Analysis, II (Princeton University Press, Princeton, NJ, 2003).
[12] Titchmarsh, E. C., The Theory of Functions (Oxford University Press, Oxford, 1958), reprint of the second (1939) edition.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

TRANSCENDENCE OVER MEROMORPHIC FUNCTIONS

  • MICHAEL COONS (a1) and YOHEI TACHIYA (a2)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.