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

Acta Arithmetica (1992)

- Volume: 61, Issue: 1, page 51-67
- ISSN: 0065-1036

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topJun-ichi Tamura. "A class of transcendental numbers with explicit g-adic expansion and the Jacobi-Perron algorithm." Acta Arithmetica 61.1 (1992): 51-67. <http://eudml.org/doc/206451>.

@article{Jun1992,

abstract = {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 ∈ ℤ).
In the special case k = l = 1, our Theorems 1-3 have been shown in [15] by a different method using the theory of representation of numbers by Fibonacci numbers of third degree.},

author = {Jun-ichi Tamura},

journal = {Acta Arithmetica},

keywords = {transcendence; Jacobi-Perron expansion; linear independence; rational approximation; -adic expansion},

language = {eng},

number = {1},

pages = {51-67},

title = {A class of transcendental numbers with explicit g-adic expansion and the Jacobi-Perron algorithm},

url = {http://eudml.org/doc/206451},

volume = {61},

year = {1992},

}

TY - JOUR

AU - Jun-ichi Tamura

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

JO - Acta Arithmetica

PY - 1992

VL - 61

IS - 1

SP - 51

EP - 67

AB - 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 ∈ ℤ).
In the special case k = l = 1, our Theorems 1-3 have been shown in [15] by a different method using the theory of representation of numbers by Fibonacci numbers of third degree.

LA - eng

KW - transcendence; Jacobi-Perron expansion; linear independence; rational approximation; -adic expansion

UR - http://eudml.org/doc/206451

ER -

## References

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- [3] P. Bundschuh, Über eine Klasse reeller transzendenter Zahlen mit explizit angebbarer g-adischer und Kettenbruch-Entwicklung, J. Reine Angew. Math. 318 (1980), 110-119. Zbl0425.10038
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- [6] J. H. Loxton and A. J. van der Poorten, Arithmetic properties of certain functions in several variables, III, Bull. Austral. Math. Soc. 16 (1977), 15-47. Zbl0339.10028
- [7] K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann. 101 (1929), 342-366. Zbl55.0115.01
- [8] D. Masser, A vanishing theorem for power series, Invent. Math. 67 (1982), 275-296. Zbl0481.10034
- [9] E. M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality, Nauka, Moscow 1988, 168-175 (in Russian).
- [10] K. Nishioka, Evertse theorem in algebraic independence, Arch. Math. (Basel) 53 (1989), 159-170. Zbl0676.10024
- [11] K. Nishioka, I. Shiokawa and J. Tamura, Arithmetical properties of certain power series, J. Number Theory, to appear. Zbl0770.11039
- [12] V. I. Parusnikov, The Jacobi-Perron algorithm and simultaneous approximation of functions, Mat. Sb. 114 (156) (2) (1981), 322-333 (in Russian). Zbl0461.30003
- [13] A. Salomaa, Jewels of Formal Language Theory, Pitman, 1981. Zbl0487.68063
- [14] A. Salomaa, Computation and Automata, Cambridge Univ. Press, 1985. Zbl0565.68046
- [15] J. Tamura, Transcendental numbers having explicit g-adic and Jacobi-Perron expansions, in: Séminaire de Théorie des Nombres de Bordeaux, to appear. Zbl0763.11029

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