Algebraic independence of the generating functions of Stern's sequence and of its twist
type: Item
instance of: scholarly article
title: Algebraic independence of the generating functions of Stern's sequence and of its twist
zbMATH Open document ID: 1268.11096
author: Peter Bundschuh
author: Keijo Väänänen
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2013-06-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
review text: The sequence \((a_0,a_1,a_2,\dots)\) defined by \(a_0=0\), \(a_1=1\), \(a_{2n}=a_n\) and \(a_{2n+1}=a_n+a_{n+1}\) for \(n\geq 1\) was introduced by \textit{M. A. Stern} [J. Reine Angew. Math. 55, 193--220 (1858; ERAM 055.1457cj)]. The generating function \(A(z)=\sum_{n\geq 0} a_{n+1}z^n\) satisfies \(A(z)=p(z)A(z^2)\) and \(A(z)=\prod_{k=0}^\infty p(z^{2^k})\) with \(p(z)=1+z+z^2\). A twisted version \((b_0,b_1,b_2,\dots)\) was introduced by \textit{R. Bacher} [``Twisting the Stern sequence'', \url{arXiv:1005.5627}] and \textit{J.-P. Allouche} [``On the Stern sequence and its twisted version'', Integers 12, A58 (2012), \url{http://arxiv.org/abs/1202.4171}], where \(b_0=0\), \(b_1=1\), \(b_{2n}=-b_n\) and \(b_{2n+1}=-b_n-b_{n+1}\) for \(n\geq 1\). The generating function \(B(z)=\sum_{n\geq 0} b_{n+1}z^n\) satisfies \(B(z)=2-p(z)B(z^2)\). In [Int. J. Number Theory 6, No. 1, 211--217 (2010; Zbl 1250.11015)], \textit{M.~Coons} proved that \(A(\alpha)\) is transcendental for \(\alpha\) algebraic with \(0<|\alpha|<1\). In [Int. J. Number Theory 8, No. 2, 361--376 (2012; Zbl 1288.11070)], the first author proved that the numbers \(A(\alpha),A'(\alpha),A''(\alpha),\dots\) are algebraically independent for the same values of \(\alpha\). Here, the authors prove the same result for \(B\) in place of \(A\). Further, they prove that for any complex number \(\alpha\) with \(0<|\alpha|<1\), the transcendence degree over \(\mathbb Q\) of the field \(\mathbb Q(\alpha,A(\alpha),B(\alpha))\) is at least two. Furthermore, for algebraic \(\alpha\) with \(0<|\alpha|<1\), they produce a measure of algebraic independence of the two numbers \(A(\alpha)\), \(B(\alpha)\). Finally, for rational \(\alpha=r/s\) with \(0<|r|<s\), they produce irrationality measures for \(A(r/s)\) and for \(B(r/s)\).
reviewed by: Michel Waldschmidt
Mathematics Subject Classification ID: 11J81
Mathematics Subject Classification ID: 11J82
zbMATH DE Number: 6173996
zbMATH Keywords: Stern sequence
zbMATH Keywords: transcendence
zbMATH Keywords: algebraic independence
zbMATH Keywords: Mahler's method
zbMATH Keywords: irrationality measure
zbMATH Keywords: measure of algebraic independence
zbMATH Keywords: hyper transcendental functions
MaRDI profile type: Publication
OpenAlex ID: W2314190161
cites work: Irrationality measures for some automatic real numbers
cites work: On the Stern sequence and its twisted version
cites work: Algebraic independence of the values of certain functions at a transcendental number
cites work: On the rational approximation to the Thue-Morse-Mahler numbers
cites work: TRANSCENDENCE AND ALGEBRAIC INDEPENDENCE OF SERIES RELATED TO STERN'S SEQUENCE
cites work: THE TRANSCENDENCE OF SERIES RELATED TO STERN'S DIATOMIC SEQUENCE
cites work: Q4539513
cites work: On the algebraic independence of holomorphic solutions of certain functional equations and their values
cites work: A note on differentially algebraic solutions of first order linear difference equations
cites work: New approach in Mahler's method.
cites work: Mahler functions and transcendence
DOI: 10.5802/JTNB.824
Recommended article: Algebraic independence of the values of Mahler functions associated with a certain continued fraction expansion
Recommended article: Q4963607
Recommended article: Arithmetical properties of infinite products generated by binary recurrences
Recommended article: Algebraic independence of the generating functions of the stern polynomials and their twisted analogues
Recommended article: A generalization of the binomial series.
Recommended article: Q3099312
Recommended article: Transcendental values of the incomplete gamma function and related questions
Recommended article: Transcendency of some constants related to integer sequences of polynomial iterations
Recommended article: ПЕРИОДИЧЕСКИЕ И НЕПЕРИОДИЧЕСКИЕ КОНЕЧНЫЕ ПОСЛЕДОВАТЕЛЬНОСТИ
Recommended article: Q4841799
description: scientific article; zbMATH DE number 6173996