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

Abstract

top
Very recently, the generating function A ( z ) of the Stern sequence ( a n ) n ≥ 0 , defined by a 0 : = 0 , a 1 : = 1 , and a 2 n : = a n , a 2 n + 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 ( α ) for every algebraic α with 0 < | α | < 1 , and this result was generalized in [6] to the effect that, for the same α ’s, all numbers A ( α ) , A ′ ( α ) , A ′ ′ ( α ) , ... 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 2 n : = - b n , b 2 n + 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 ℂ ( z ) , to use this fact to conclude that, for any complex α with 0 < | α | < 1 , the transcendence degree of the field ℚ ( α , A ( α ) , B ( α ) ) over 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 ) .

How to cite

top

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

top
  1. B. Adamczewski and T. Rivoal, Irrationality measures for some automatic real numbers. Math. Proc. Cambridge Phil. Soc. 147 (2009), 659–678. Zbl1205.11080MR2557148
  2. J.-P. Allouche, On the Stern sequence and its twisted version. Integers 12 (2012), A58. Zbl1283.11026MR2955582
  3. M. Amou, Algebraic independence of the values of certain functions at a transcendental number. Acta Arith. 59 (1991), 71–82. Zbl0735.11031MR1133238
  4. R. Bacher, Twisting the Stern sequence. Preprint, 2010, available at http://arxiv.org/abs/1005.5627 
  5. Y. Bugeaud, On the rational approximation to the Thue-Morse-Mahler numbers. Ann. Inst. Fourier (Grenoble) 61 (2011), 2065–2076. Zbl1271.11074MR2961848
  6. P. Bundschuh, Transcendence and algebraic independence of series related to Stern’s squence. Int. J. Number Theory 8 (2012), 361–376. Zbl1288.11070MR2890484
  7. F. Carlson, Über Potenzreihen mit ganzen Koeffizienten. Math. Z. 9 (1921), 1–13. Zbl48.1208.02MR1544447
  8. M. Coons, The transcendence of series related to Stern’s diatomic sequence. Int. J. Number Theory 6 (2010), 211–217. Zbl1250.11015MR2641723
  9. P. Corvaja and U. Zannier, Some new applications of the subspace theorem. Compos. Math. 131 (2002), 319–340. Zbl1010.11038MR1905026
  10. K. K. Kubota, On the algebraic independence of holomorphic solutions of certain functional equations and their values. Math. Ann. 227 (1977), 9–50. Zbl0359.10030MR498423
  11. K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Math. Ann. 101 (1929), 342–366. Zbl55.0115.01MR1512537
  12. Ke. Nishioka, A note on differentially algebraic solutions of first order linear difference equations. Aequationes Math. 27 (1984), 32–48. Zbl0542.12012MR758857
  13. Ku. Nishioka, New approach in Mahler’s method. J. Reine Angew. Math. 407 (1990), 202–219. Zbl0694.10035MR1048535
  14. Ku. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Math. 1631. Springer, Berlin, 1996. Zbl0876.11034MR1439966
  15. M. A. Stern, Über eine zahlentheoretische Funktion. J. Reine Angew. Math. 55 (1858), 193–220. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.