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

Peter Bundschuh^{[1]}; Keijo Väänänen^{[2]}

- [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

Journal de Théorie des Nombres de Bordeaux (2013)

- Volume: 25, Issue: 1, page 43-57
- ISSN: 1246-7405

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topBundschuh, 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 $n>0$, has been considered from the arithmetical point of view. Coons [8] proved the transcendence of $A(\alpha )$ 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(\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 $n>0$.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 $0<|\alpha |<1$, 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 $0<|r|<s$ 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 $n>0$, has been considered from the arithmetical point of view. Coons [8] proved the transcendence of $A(\alpha )$ 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(\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 $n>0$.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 $0<|\alpha |<1$, 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 $0<|r|<s$ 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 -

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