Algebraic independence of the generating functions of Stern’s sequence and of its twist
Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 1, p. 43-57

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| and sufficiently small $\left(log|r|\right)/\left(logs\right)$.

Très récemment, la fonction génératrice $A\left(z\right)$ de la suite ${\left({a}_{n}\right)}_{n\ge 0}$ de Stern, définie par ${a}_{0}\phantom{\rule{-0.166667em}{0ex}}:=\phantom{\rule{-0.166667em}{0ex}}0,{a}_{1}\phantom{\rule{-0.166667em}{0ex}}:=\phantom{\rule{-0.166667em}{0ex}}1,$ et ${a}_{2n}\phantom{\rule{-0.166667em}{0ex}}:=\phantom{\rule{-0.166667em}{0ex}}{a}_{n},{a}_{2n+1}\phantom{\rule{-0.166667em}{0ex}}:={a}_{n}+{a}_{n+1}$ pour tout entier $n>0$, a été considérée du point de vue arithmétique. Coons [8] a montré la transcendance de $A\left(\alpha \right)$ pour tout $\alpha$ algébrique avec $0<|\alpha |<1$, et ce résultat fut généralisé dans [6] de sorte que, pour les mêmes $\alpha$, les nombres $A\left(\alpha \right),{A}^{\prime }\left(\alpha \right),{A}^{\prime \prime }\left(\alpha \right),...$ sont algébriquement indépendants. À peu près au même temps, Bacher [4] a étudié la version tordue $\left({b}_{n}\right)$ de la suite de Stern, définie par ${b}_{0}:=0,{b}_{1}:=1,$ et ${b}_{2n}:=-{b}_{n},{b}_{2n+1}:=-\left({b}_{n}+{b}_{n+1}\right)$ pour tout $n>0$.

Les objectifs principaux du présent travail sont d’établir les analogues sur la fonction génératrice $B\left(z\right)$ de $\left({b}_{n}\right)$ des résultats arithmétiques mentionnés plus haut concernant $A\left(z\right)$, de démontrer l’indépendance algébrique de $A\left(z\right),B\left(z\right)$ sur le corps $ℂ\left(z\right)$, d’utiliser ce fait pour en déduire que, pour tout nombre complexe $\alpha$ avec $0<|\alpha |<1$, le degré de transcendance du corps $ℚ\left(\alpha ,A\left(\alpha \right),B\left(\alpha \right)\right)$ sur $ℚ$ est au moins 2, et de fournir des majorations assez bonnes pour l’exposant d’irrationalité de $A\left(r/s\right)$ et de $B\left(r/s\right)$, où $r,s$ sont des entiers avec $0<|r| et $\left(log|r|\right)/\left(logs\right)$ suffisamment petit.

@article{JTNB_2013__25_1_43_0,
author = {Bundschuh, Peter and V\"a\"an\"anen, Keijo},
title = {Algebraic independence of the generating functions of Stern's sequence and of its twist},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {25},
number = {1},
year = {2013},
pages = {43-57},
doi = {10.5802/jtnb.824},
mrnumber = {3063829},
zbl = {1268.11096},
language = {en},
url = {http://www.numdam.org/item/JTNB_2013__25_1_43_0}
}

Bundschuh, Peter; Väänänen, Keijo. Algebraic independence of the generating functions of Stern’s sequence and of its twist. Journal de théorie des nombres de Bordeaux, Volume 25 (2013) no. 1, pp. 43-57. doi : 10.5802/jtnb.824. http://www.numdam.org/item/JTNB_2013__25_1_43_0/

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