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

•  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
• 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  proved the transcendence of $A\left(\alpha \right)$ for every algebraic $\alpha$ with $0<|\alpha |<1$, and this result was generalized in  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  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 $\left(log|r|\right)/\left(logs\right)$.

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