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Algebraic independence properties of the Fredholm series

Published online by Cambridge University Press:  09 April 2009

J. H. Loxton  and
A. J. van der Poorten
J. H. Loxton
Affiliation:
School of Mathematics University of New South WalesKensington New South Wales 2033Australia
A. J. van der Poorten
Affiliation:
School of Mathematics University of New South WalesKensington New South Wales 2033Australia
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Abstract

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We consider algebraic independence properties of series such as We show that the functions fr(z) are algebraically independent over the rational functions Further, if αrs (r = 2, 3, 4, hellip; s = 1, 2, 3, hellip) are algebraic numbers with 0 < |αrs|, we obtain an explicit necessary and sufficient condition for the algebraic independence of the numbers fr(αrs) over the rationals.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 26 , Issue 1 , August 1978 , pp. 31 - 45
Copyright
Copyright © Australian Mathematical Society 1978

References

Cijsouw, P. L. and Tijdeman, R. (1973), “On the transcendence of certain power series of algebraic numbers”, Acta Arith. 23, 301–305.CrossRefGoogle Scholar
Dienes, P. (1957), The Taylor Series (Dover, New York).Google Scholar
Kubota, K. K. (1976), “An application of Kronecker's theorem to transcendence theory”, Sém. de Théorie des Nombres de Bordeaux, Année 1975/1976, no. 25.Google Scholar
Kubota, K. K. (1977a), “On the algebraic independence of holomorphic solutions of certain functional equations and their values”, Math. Ann. 227, 9–50.CrossRefGoogle Scholar
Kubota, K. K. (1977b), Algebraic Independence Properties of Solutions to Linear Functional Equations, Transcendence Theory: Advances and Applications, edited by Baker, A. and Masser, D. W. (Academic Press, London), Ch. 16, 227–229.Google Scholar
Loxton, J. H. and van der Poorten, A. J. (1977a), “Arithmetic properties of certain functions in several variables”, J. Number Theory 9, 87–106.CrossRefGoogle Scholar
Loxton, J. H. and van der Poorten, A. J. (1977b), “Arithmetic properties of certain functions in several variables III”, Bull. Austral. Math. Soc. 16, 15–47.CrossRefGoogle Scholar
Loxton, J. H. and van der Poorten, A. J. (1977c), Transcendence and Algebraic Independence by a Method of Mahler, Transcendence Theory: Advances and Applications, edited by Baker, A. and Masser, D. W. (Academic Press, London), Ch. 15, 211–226.Google Scholar
Loxton, J. H. and van der Poorten, A. J. (to appear, a), “A class of hypertranscendental functions’, Aequationes Math.Google Scholar
Loxton, J. H. and van der Poorten, A. J. (to appear, b), “Arithmetic properties of certain functions in several variables II”, J. Austral. Math. Soc.Google Scholar
Mahler, K. (1929), “Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen”, Math. Ann. 101, 342–366.CrossRefGoogle Scholar
Mahler, K. (1930a), “Über das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen”, Math. Ann. 103, 573–587.CrossRefGoogle Scholar
Mahler, K. (1930b), “Arithmetische Eigenschaften einer Klasse transzendentaltranszendenter Funktionen”, Math. Z. 32, 545–585.CrossRefGoogle Scholar
Mahler, K. (1965), “Arithmetic properties of lacunary power series with integral coefficients”, J. Austral. Math. Soc. 5, 56–64.CrossRefGoogle Scholar
Ritt, J. F. (1927), “A factorisation theory for functions ”, Trans. Amer. Math. Soc, 29, 584–596.Google Scholar
Schneider, Th. (1957), Einführung in die Transzendenten Zahlen (Springer, Berlin).CrossRefGoogle Scholar