# Algebraic independence of the generating functions of Stern’s sequence and of its twist

• [1] Mathematisches Institut Universität zu Köln Weyertal 86-90 50931 Köln, Germany
• [2] Department of Mathematical Sciences University of Oulu P. O. Box 3000 90014 Oulu, Finland
• Volume: 25, Issue: 1, page 43-57
• ISSN: 1246-7405

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## Abstract

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Very recently, the generating function $A\left(z\right)$ of the Stern sequence ${\left({a}_{n}\right)}_{n\ge 0}$, defined by ${a}_{0}:=0,{a}_{1}:=1,$ and ${a}_{2n}:={a}_{n},{a}_{2n+1}:={a}_{n}+{a}_{n+1}$ for any integer $n>0$, has been considered from the arithmetical point of view. Coons [8] proved the transcendence of $A\left(\alpha \right)$ for every algebraic $\alpha$ with $0<|\alpha |<1$, and this result was generalized in [6] to the effect that, for the same $\alpha$’s, all numbers $A\left(\alpha \right),{A}^{\prime }\left(\alpha \right),{A}^{\prime \prime }\left(\alpha \right),...$ are algebraically independent. At about the same time, Bacher [4] studied the twisted version $\left({b}_{n}\right)$ of Stern’s sequence, defined by ${b}_{0}:=0,{b}_{1}:=1,$ and ${b}_{2n}:=-{b}_{n},{b}_{2n+1}:=-\left({b}_{n}+{b}_{n+1}\right)$ for any $n>0$.The aim of our paper is to show the analogs on the generating function $B\left(z\right)$ of $\left({b}_{n}\right)$ of the above-mentioned arithmetical results on $A\left(z\right)$, to prove the algebraic independence of $A\left(z\right),B\left(z\right)$ over the field $ℂ\left(z\right)$, to use this fact to conclude that, for any complex $\alpha$ with $0<|\alpha |<1$, the transcendence degree of the field $ℚ\left(\alpha ,A\left(\alpha \right),B\left(\alpha \right)\right)$ over $ℚ$ is at least 2, and to provide rather good upper bounds for the irrationality exponent of $A\left(r/s\right)$ and $B\left(r/s\right)$ for integers $r,s$ with $0<|r|<s$ and sufficiently small $\left(log|r|\right)/\left(logs\right)$.

## How to cite

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Bundschuh, Peter, and Väänänen, Keijo. "Algebraic independence of the generating functions of Stern’s sequence and of its twist." Journal de Théorie des Nombres de Bordeaux 25.1 (2013): 43-57. <http://eudml.org/doc/275690>.

@article{Bundschuh2013,
abstract = {Very recently, the generating function $A(z)$ of the Stern sequence $(a_n)_\{n\ge 0\}$, defined by $a_0:=0, a_1:=1,$ and $a_\{2n\}:=a_n, a_\{2n+1\}:=a_n+a_\{n+1\}$ for any integer $n&gt;0$, has been considered from the arithmetical point of view. Coons [8] proved the transcendence of $A(\alpha )$ for every algebraic $\alpha$ with $0&lt;|\alpha |&lt;1$, and this result was generalized in [6] to the effect that, for the same $\alpha$’s, all numbers $A(\alpha ),A^\{\prime\}(\alpha ),A^\{\prime\prime\}(\alpha ),\ldots$ are algebraically independent. At about the same time, Bacher [4] studied the twisted version $(b_n)$ of Stern’s sequence, defined by $b_0:=0, b_1:=1,$ and $b_\{2n\}:=-b_n, b_\{2n+1\}:=-(b_n+b_\{n+1\})$ for any $n&gt;0$.The aim of our paper is to show the analogs on the generating function $B(z)$ of $(b_n)$ of the above-mentioned arithmetical results on $A(z)$, to prove the algebraic independence of $A(z), B(z)$ over the field $\mathbb\{C\}(z)$, to use this fact to conclude that, for any complex $\alpha$ with $0&lt;|\alpha |&lt;1$, the transcendence degree of the field $\mathbb\{Q\}(\alpha ,A(\alpha ),B(\alpha ))$ over $\mathbb\{Q\}$ is at least 2, and to provide rather good upper bounds for the irrationality exponent of $A(r/s)$ and $B(r/s)$ for integers $r, s$ with $0&lt;|r|&lt;s$ and sufficiently small $(\log |r|)/(\log s)$.},
affiliation = {Mathematisches Institut Universität zu Köln Weyertal 86-90 50931 Köln, Germany; Department of Mathematical Sciences University of Oulu P. O. Box 3000 90014 Oulu, Finland},
author = {Bundschuh, Peter, Väänänen, Keijo},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Stern sequence; transcendence; algebraic independence; Mahler's method; irrationality measure; measure of algebraic independence; hyper transcendental functions},
language = {eng},
month = {4},
number = {1},
pages = {43-57},
publisher = {Société Arithmétique de Bordeaux},
title = {Algebraic independence of the generating functions of Stern’s sequence and of its twist},
url = {http://eudml.org/doc/275690},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Bundschuh, Peter
AU - Väänänen, Keijo
TI - Algebraic independence of the generating functions of Stern’s sequence and of its twist
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/4//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 1
SP - 43
EP - 57
AB - Very recently, the generating function $A(z)$ of the Stern sequence $(a_n)_{n\ge 0}$, defined by $a_0:=0, a_1:=1,$ and $a_{2n}:=a_n, a_{2n+1}:=a_n+a_{n+1}$ for any integer $n&gt;0$, has been considered from the arithmetical point of view. Coons [8] proved the transcendence of $A(\alpha )$ for every algebraic $\alpha$ with $0&lt;|\alpha |&lt;1$, and this result was generalized in [6] to the effect that, for the same $\alpha$’s, all numbers $A(\alpha ),A^{\prime}(\alpha ),A^{\prime\prime}(\alpha ),\ldots$ are algebraically independent. At about the same time, Bacher [4] studied the twisted version $(b_n)$ of Stern’s sequence, defined by $b_0:=0, b_1:=1,$ and $b_{2n}:=-b_n, b_{2n+1}:=-(b_n+b_{n+1})$ for any $n&gt;0$.The aim of our paper is to show the analogs on the generating function $B(z)$ of $(b_n)$ of the above-mentioned arithmetical results on $A(z)$, to prove the algebraic independence of $A(z), B(z)$ over the field $\mathbb{C}(z)$, to use this fact to conclude that, for any complex $\alpha$ with $0&lt;|\alpha |&lt;1$, the transcendence degree of the field $\mathbb{Q}(\alpha ,A(\alpha ),B(\alpha ))$ over $\mathbb{Q}$ is at least 2, and to provide rather good upper bounds for the irrationality exponent of $A(r/s)$ and $B(r/s)$ for integers $r, s$ with $0&lt;|r|&lt;s$ and sufficiently small $(\log |r|)/(\log s)$.
LA - eng
KW - Stern sequence; transcendence; algebraic independence; Mahler's method; irrationality measure; measure of algebraic independence; hyper transcendental functions
UR - http://eudml.org/doc/275690
ER -

## References

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