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A class of transcendental numbers with explicit g-adic expansion and the Jacobi-Perron algorithm

Jun-ichi Tamura (1992)

Acta Arithmetica

In this paper, we give transcendental numbers φ and ψ such that (i) both φ and ψ have explicit g-adic expansions, and simultaneously, (ii) the vector $^{t} (φ, ψ)$ has an explicit expression in the Jacobi-Perron algorithm (cf. Theorem 1). Our results can be regarded as a higher-dimensional version of some of the results in [1]-[5] (see also [6]-[8], [10], [11]). The numbers φ and ψ have some connection with algebraic numbers with minimal polynomials x³ - kx² - lx - 1 satisfying (1.1) k ≥ l ≥0, k + l ≥ 2 (k,l...

A generalisation of an identity of Lehmer

Sanoli Gun, Ekata Saha, Sneh Bala Sinha (2016)

Acta Arithmetica

We prove an identity involving generalised Euler-Briggs constants, Euler's constant, and linear forms in logarithms of algebraic numbers. This generalises and gives an alternative proof of an identity of Lehmer (1975). Further, this identity facilitates the investigation of the (conjectural) transcendental nature of generalised Euler-Briggs constants. Earlier investigations of similar type by the present authors involved the interplay between additive and multiplicative characters. This in turn...

A generalization of Sturmian sequences: Combinatorial structure and transcendence

Rebecca Risley, Luca Zamboni (2000)

Acta Arithmetica

A generalization of the method of global relation.

Chirskiĭ, V.G. (2005)

Journal of Mathematical Sciences (New York)

Algebraic independence measures of the values of Mahler functions.

Kumiko Nishioka (1991)

Journal für die reine und angewandte Mathematik

Algebraic Independence of Reciprocal Sums of Binary Recurrences.

Kumiko Nishioka (1997)

Monatshefte für Mathematik

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

Algebraic independence of the values of power series generated by linear recurrences

Taka-Aki Tanaka (1996)

Acta Arithmetica

Algebraic independence results for reciprocal sums of Fibonacci numbers

Carsten Elsner, Shun Shimomura, Iekata Shiokawa (2011)

Acta Arithmetica

Algebraic relations for reciprocal sums of Fibonacci numbers

Carsten Elsner, Shun Shimomura, Iekata Shiokawa (2007)

Acta Arithmetica

An axiomatization of Nesterenko's method and applications on Mahler functions II

Thomas Töpfer (1995)

Compositio Mathematica

Applications of differential algebra to algebraic independence of arithmetic functions

Wai Yan Pong (2016)

Acta Arithmetica

We generalize and unify the proofs of several results on algebraic independence of arithmetic functions and Dirichlet series by using a theorem of Ax on the differential Schanuel conjecture. Along the way, we find counter-examples to some results in the literature.

Arithmetical properties of the values of functions satisfying certain functional equations of Poincaré

Masaaki Amou, Masanori Katsurada, Keijo Väänänen (2001)

Acta Arithmetica

Counting rational points on a certain exponential-algebraic surface

Jonathan Pila (2010)

Annales de l’institut Fourier

We study the distribution of rational points on a certain exponential-algebraic surface and we prove, for this surface, a conjecture of A. J. Wilkie.

Des résultats d'irrationalité pour deux fonctions particulières.

Richard Choulet (2001)

Collectanea Mathematica

Diophantine approximations related to rational values of G-functions

Makoto Nagata (2003)

Acta Arithmetica

D'une mesure d'approximation simultanee a une mesure d'irrationalite: le cas de Γ(1/4) et Γ(1/3)

Sylvain Bruiltet (2002)

Acta Arithmetica

Effective estimates for global relations on Euler-type series

Daniel Bertrand, Vladimir Chirskii, Johan Yebbou (2004)

Annales de la Faculté des sciences de Toulouse : Mathématiques

Explicit algebraic dependence formulae for infinite products related with Fibonacci and Lucas numbers

Hajime Kaneko, Takeshi Kurosawa, Yohei Tachiya, Taka-aki Tanaka (2015)

Acta Arithmetica

Let d ≥ 2 be an integer. In 2010, the second, third, and fourth authors gave necessary and sufficient conditions for the infinite products $\prod_{\binom{k = 1}{U_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) / (U_{d^{k}}))$ (i=1,...,m) or $\prod_{\binom{k = 1}{V_{d^{k}} \neq - a_{i}}}^{\infty} (1 + (a_{i}) (V_{d^{k}})$ (i=1,...,m) to be algebraically dependent, where $a_{i}$ are non-zero integers and $U_{n}$ and $V_{n}$ are generalized Fibonacci numbers and Lucas numbers, respectively. The purpose of this paper is to relax the condition on the non-zero integers $a_{1}, . . ., a_{m}$ to non-zero real algebraic numbers, which gives new cases where the infinite products above are algebraically dependent....

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