Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-06T06:09:00.987Z Has data issue: false hasContentIssue false

Arithmetic properties of certain functions in several variables III

Published online by Cambridge University Press:  17 April 2009

J.H. Loxton
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales.
A.J. van der Poorten
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales.
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.

We obtain a general transcendence theorem for the solutions of a certain type of functional equation. A particular and striking consequence of the general result is that, for any irrational number w, the function

takes transcendental values at all algebraic points α with 0 < |α| < 1.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1977

References

[1]Böhmer, P.E., “Über die Transzendenz gewisser dyadischer Brüche”, Math. Ann. 96 (1927), 367377.CrossRefGoogle Scholar
[2]Cassels, J.W.S., An introduction to diophantine approximation (Cambridge Tracts in Mathematics, 45. Cambridge University Press, 1957).Google Scholar
[3]Cijsouw, P.L. and Tijdeman, R., “On the transcendence of certain power series of algebraic numbers”, Acta Arith. 23 (1973), 301305.CrossRefGoogle Scholar
[4]Dienes, P., The Taylor series: an introduction to the theory of functions of a complex variable (Clarendon, Oxford, 1931; reprinted Dover, New York, 1957).Google Scholar
[5]Gantmacher, F.R., Applications of the theory of matrices (translated and revised by Brenner, J.L., Brishaw, D.W., and Evanusa, S.. Interscience, New York, London, 1959).Google Scholar
[6]Hecke, E., “Über analytische Funktionen und die Verteilung von Zahlen mod. eins”, Abh. Math. Sem. Univ. Hamburg 1 (1922), 5476.CrossRefGoogle Scholar
[7]Loxton, J.H. and van der Poorten, A.J., “Arithmetic properties of certain functions in several variables”, J. Number Theory (to appear).Google Scholar
[8]Loxton, J.H. and van der Poorten, A.J., “Arithmetic properties of certain functions in several variables II”, submitted.Google Scholar
[9]Mahler, Kurt, “Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen”, Math. Ann. 101 (1929), 342366.CrossRefGoogle Scholar
[10]Mahler, Kurt, “Über das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen”, Math. Ann. 103 (1930), 573587.CrossRefGoogle Scholar
[11]Mahler, Kurt, “Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen”, Math. Z. 32 (1930), 545585.CrossRefGoogle Scholar
[12]Mahler, K., “Remarks on a paper of W. Schwarz”, J. Number Theory 1 (1969), 512521.CrossRefGoogle Scholar
[13]Mahler, Kurt, “On the transcendency of the solutions of a special class of functional equations”, Bull. Austral. Math. Soc. 13 (1975), 389410.CrossRefGoogle Scholar
[14]Mahler, Kurt, “On the transcendency of the solutions of a special class of functional equations: Corrigendum”, Bull. Austral. Math. Soc. 14 (1976), 477478.CrossRefGoogle Scholar
[15]Mignotte, Maurice, “Quelques problèmes d'effectivité en théorie des nombres” (DSc Thèses, L'Université de Paris XIII, 1974).Google Scholar
[16]Newman, Morris, “Irrational power series”, Proc. Amer. Math. Soc. 11 (1960), 699702.CrossRefGoogle Scholar
[17]Petersson, Hans, “Über Potenzreihen mit ganzen algebraischen Koeffizienten”, Abh. Math. Sem. Univ. Hamburg 8 (1931), 315322.CrossRefGoogle Scholar
[18]Schneider, Theodor, Einführung in die transzendenten Zahlen (Die Grundlehren der Mathematischen Wissenschaften, 81. Springer-Verlag, Berlin, Göttingen, Heidelberg, 1957).CrossRefGoogle Scholar
[19]Schwarz, Wolfgang, “Remarks on the irrationality and transcendence of certain series”, Math. Scand. 20 (1967), 269274.CrossRefGoogle Scholar
[20]Schwarz, Wolfgang, “Über Potenzreihen, die irrationale Funktionen darstellen. II”, Überblicke Math. 7 (1974), 732.Google Scholar
[21]Scott, W.T. and Wall, H.S., “Continued fraction expansions for arbitrary power series”, Ann. of Math. (2) 41 (1940), 328349.CrossRefGoogle Scholar
[22]Waldschmidt, Michael, Nombres transcendants (Lecture Notes in Mathematics, 402. Springer-Verlag, Berlin, Heidelberg, New York, 1974).CrossRefGoogle Scholar
[23]Wall, H.S., Analytic theory of continued fractions (Van Nostrand, Toronto, New York, London, 1948).Google Scholar
[24]Wallisser, Rolf, “Eine Bemerkung über irrationale Werte und Nichtfortsetzbarkeit von Potenzreihen mit ganzzahligen Koeffizienten”, Colloq. Math. 23 (1971), 141144.CrossRefGoogle Scholar