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ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES

Published online by Cambridge University Press:  11 November 2014

PETER BUNDSCHUH*
Affiliation:
Mathematisches Institut, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany email pb@math.uni-koeln.de
KEIJO VÄÄNÄNEN
Affiliation:
Department of Mathematical Sciences, University of Oulu, PO Box 3000, 90014 Oulun Yliopisto, Finland email keijo.vaananen@oulu.fi
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Abstract

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This paper considers algebraic independence and hypertranscendence of functions satisfying Mahler-type functional equations $af(z^{r})=f(z)+R(z)$, where $a$ is a nonzero complex number, $r$ an integer greater than 1, and $R(z)$ a rational function. Well-known results from the scope of Mahler’s method then imply algebraic independence over the rationals of the values of these functions at algebraic points. As an application, algebraic independence results on reciprocal sums of Fibonacci and Lucas numbers are obtained.

Type
Research Article
Copyright
© 2014 Australian Mathematical Publishing Association Inc. 

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