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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...

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 k = 1 U d k - a i ( 1 + ( a i ) / ( U d k ) ) (i=1,...,m) or k = 1 V d k - a i ( 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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