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Very recently, the generating function $A (z)$ of the Stern sequence ${(a_{n})}_{n \geq 0}$ , 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 independence of the numbers $\prod_{K = 0}^{\infty} (1 - p^{- 2^{K}})$ with $p$ prime

Kenneth K. Kubota (1975/1976)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

Algebraic Independence of the Values of Certain Series by Mahler's Method.

Paul-Georg Becker (1992)

Monatshefte für Mathematik

Algebraic Independence of the Values of Elliptic Function at Algebraic Points.

G. Chudnovsky (1980)

Inventiones mathematicae

Algebraic independence results for reciprocal sums of Fibonacci numbers

Carsten Elsner, Shun Shimomura, Iekata Shiokawa (2011)

Acta Arithmetica

Algebraic integers whose conjugates lie near the unit circle

Cameron L. Stewart (1978)

Bulletin de la Société Mathématique de France

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...

Algebraic points of abelian functions in two variables

Alex Bijlsma (1982)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Algebraic relations between special values of multiple sine functions

Hidekazu Tanaka (2009)

Acta Arithmetica

Algebraic relations for reciprocal sums of Fibonacci numbers

Carsten Elsner, Shun Shimomura, Iekata Shiokawa (2007)

Acta Arithmetica

Algebraic values of G-functions.

F. Beukers (1993)

Journal für die reine und angewandte Mathematik

Algebraische Abhängigkeit von Wurzeln

Carl Siegel (1972)

Acta Arithmetica

Algebraische Unabhängigkeit P-adischer Zahlen.

Peter Bundschuh, Rolf Wallisser (1976)

Mathematische Annalen

Algébricité de valeurs spéciales de fonctions L.

P. Colmez (1989)

Inventiones mathematicae

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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