Hostname: page-component-7479d7b7d-c9gpj Total loading time: 0 Render date: 2024-07-12T09:18:57.876Z Has data issue: false hasContentIssue false
  • English
  • Français

On the transcendency of the solutions of a special class of functional equations

Published online by Cambridge University Press:  17 April 2009

Kurt Mahler
Kurt Mahler
Affiliation:
Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
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.

Let a(z) and b(w) be two rational functions in z or W with algebraic coefficients, where a(0) = 0 and let

Assume that 0 < |z| < 1, that z is not a pole of for n ≥ 0, that w is neither a pole nor a zero of b(w, n) for n ≥ 1, and that the series

for fixed w is a transcendental function of z. Then, if z and W are algebraic numbers, f(z, w) is a transcendental number.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 13 , Issue 3 , December 1975 , pp. 389 - 410
Copyright
Copyright © Australian Mathematical Society 1975

References

[1]Mahler, Kurt, “Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen”, Math. Ann. 101 (1929), 342–366.CrossRefGoogle Scholar
[2]Mahler, Kurt, “Über das Verschvinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen”, Math. Ann. 103 (1930), 573–587.CrossRefGoogle Scholar
[3]Mahler, Kurt “Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen”, Math. Z. 32 (1930), 545–585.CrossRefGoogle Scholar
[4]Mahler, K., “Remarks on a paper by W. Schwarz”, J. Number Theory 1 (1969), 512–521.CrossRefGoogle Scholar
[5]Mignotte, Maurice, “Quelques problèmes d'effectivité en théorie des nombres” (DSc Thèses, L'Université de Paris XIII, 1974).Google Scholar
[6]Schmidt, Wolfgang M., “Simultaneous approximation to algebraic numbers by rationals”, Acta Math. 125 (1970), 189–201.Google Scholar