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Analytic continuation of power series whose coefficients belong to an algebraic number field

Published online by Cambridge University Press:  26 February 2010

Stephen Gerig
Stephen Gerig
Affiliation:
The University of Western Australia, Perth, Australia.
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Extract

An old theorem of Pólya and Carlson [2] states that, if the power series has rational integer coefficients, positive radius of convergence, and can be continued analytically to a region that contains points outside the closed unit disc, then the function that the power series represents is rational. This result has been extended in a number of ways (cf. e.g. Petersson [4]). The present note gives a new extension based on a recent theorem of Güting [3]. My thanks to Professors Henry Helson and Raphael Robinson for introducing me to this subject.

Type
Research Article
Information
Mathematika , Volume 16 , Issue 2 , December 1969 , pp. 167 - 169
Copyright
Copyright © University College London 1969

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References

1.Bieberbach, Ludwig, Analytische Fortzetzung (Springer-Verlag, 1955).CrossRefGoogle Scholar
2.Carlson, F., “Über Potenzreihen mit ganzzahligen Koeffizienten”, Math. Zeit., 9 (1921), 1–13.CrossRefGoogle Scholar
3.Güting, R., “Approximation of algebraic numbers by algebraic numbers”, Mich. Math. Journal, 8 (1961), 149–159.CrossRefGoogle Scholar
4.Petersson, H., “Über Potenzreihen mit ganzen algebraischen Zahlenkoeffizienten”, Abh. Math. Sent. Hamburg, 8 (1931), 315–322.CrossRefGoogle Scholar