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

Peter Bundschuh, Keijo Väänänen (2013)

Journal de Théorie des Nombres de Bordeaux

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...

Algebraic leaves of algebraic foliations over number fields

Jean-Benoît Bost (2001)

Publications Mathématiques de l'IHÉS

We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field K embedded in C , a smooth algebraic variety X over K , equipped with a K - rational point P , and F an algebraic subbundle of the its tangent bundle T X , defined over K . Assume moreover that the vector bundle F is involutive, i.e., closed under Lie bracket. Then it defines an holomorphic foliation of the analytic manifold X ( C ) , and one may consider its leaf F through P . We prove...

All Liouville Numbers are Transcendental

Artur Korniłowicz, Adam Naumowicz, Adam Grabowski (2017)

Formalized Mathematics

In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real...

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