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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 values of certain functions at a transcendental number

Masaaki Amou (1991)

Acta Arithmetica

Algebraic independence of the values of generalized Mahler functions

Thomas Töpfer (1995)

Acta Arithmetica

Algebraic independence of the values of Mahler functions satisfying implicit functional equations

Bernd Greuel (2000)

Acta Arithmetica

An arithmetic criterion for the values of the exponential function

Damien Roy (2001)

Acta Arithmetica

Approximation diophantienne et distances ultramétriques non standard

Sandra Delaunay (2005)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Approximation measures for logarithms of algebraic numbers

Francesco Amoroso, Carlo Viola (2001)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Approximations to $q$ -logarithms and $q$ -dilogarithms, with applications to $q$ -zeta values.

Zudilin, W. (2005)

Journal of Mathematical Sciences (New York)

Arithmetical investigations of a certain infinite product

Peter Bundschuh, Keijo Väänänen (1994)

Compositio Mathematica

Birational transformations and values of the Riemann zeta-function

Carlo Viola (2003)

Journal de théorie des nombres de Bordeaux

In his proof of Apery’s theorem on the irrationality of $ζ (3)$ , Beukers [B] introduced double and triple integrals of suitable rational functions yielding good sequences of rational approximations to $ζ (2)$ and $ζ (3)$ . Beukers’ method was subsequently improved by Dvornicich and Viola, by Hata, and by Rhin and Viola. We give here a survey of our recent results ([RV2] and [RV3]) on the irrationality measures of $ζ (2)$ and $ζ (3)$ based upon a new algebraic method involving birational transformations and permutation groups...

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A Gel'fond type criterion in degree two

A geometric proof to Cantor's theorem and an irrationality measure for some Cantor's series.

A new irrationality measure for ζ(3)

A note to the transcendence of special infinite series

A powerful determinant.

À propos de la série $\sum_{n = 1}^{+ \infty} \frac{x^{n}}{q^{n} - 1}$

A remark on trigonometric sums

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

Algebraic independence of the values of certain functions at a transcendental number

Algebraic independence of the values of generalized Mahler functions

Algebraic independence of the values of Mahler functions satisfying implicit functional equations

An arithmetic criterion for the values of the exponential function

Approximation diophantienne et distances ultramétriques non standard

Approximation measures for logarithms of algebraic numbers

Approximations to $q$ -logarithms and $q$ -dilogarithms, with applications to $q$ -zeta values.

Arithmetical investigations of a certain infinite product

Birational transformations and values of the Riemann zeta-function

Complément à : «Sur l’approximation de $π$ par des nombres algébriques particuliers»

Continued Fractions for Some Alternating Series.

Convergents and irrationality measures of logarithms.

Currently displaying 1 – 20 of 81