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ALGEBRAIC INDEPENDENCE OF CERTAIN MAHLER FUNCTIONS AND OF THEIR VALUES

Part of: Sequences and sets Differential and difference algebra Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  11 November 2014

PETER BUNDSCHUH  and
KEIJO VÄÄNÄNEN
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
*
✉pb@math.uni-koeln.de
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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.

Keywords

algebraic independence of functionsalgebraic independence of numbershypertranscendenceMahler’s methodreciprocal sums of Fibonacci and Lucas numbers

MSC classification

Primary: 11J91: Transcendence theory of other special functions
Secondary: 11J81: Transcendence (general theory) 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 12H05: Differential algebra 39B32: Equations for complex functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 98 , Issue 3 , June 2015 , pp. 289 - 310
Copyright
© 2014 Australian Mathematical Publishing Association Inc. 

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