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(z) of the Stern sequence (a n ) n0 , 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(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 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 (z), to use this fact to conclude that, for any complex α with 0<|α|<1, the transcendence degree of the field (α,A(α),B(α)) over 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|)/(logs).

Très récemment, la fonction génératrice A(z) de la suite (a n ) n0 de Stern, définie par a 0 :=0,a 1 :=1, et a 2n :=a n ,a 2n+1 :=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(α) pour tout α algébrique avec 0<|α|<1, et ce résultat fut généralisé dans [6] de sorte que, pour les mêmes α, les nombres A(α),A (α),A (α),... sont algébriquement indépendants. À peu près au même temps, Bacher [4] a étudié la version tordue (b n ) de la suite de Stern, définie par b 0 :=0,b 1 :=1, et b 2n :=-b n ,b 2n+1 :=-(b n +b n+1 ) pour tout n>0.

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