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

Jun-ichi Tamura

Acta Arithmetica (1992)

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

Abstract

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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.

How to cite

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Jun-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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  2. [2] P. E. Böhmer, Über die Transzendenz gewisser dyadischer Brüche, Math. Ann. 96 (1927), 367-377; Berichtigung, Proc. Amer. Math. Soc. 735. Zbl52.0188.02
  3. [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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  5. [5] J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc. 63 (1977), 29-32. Zbl0326.10030
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  7. [7] K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann. 101 (1929), 342-366. Zbl55.0115.01
  8. [8] D. Masser, A vanishing theorem for power series, Invent. Math. 67 (1982), 275-296. Zbl0481.10034
  9. [9] E. M. Nikishin and V. N. Sorokin, Rational Approximations and Orthogonality, Nauka, Moscow 1988, 168-175 (in Russian). 
  10. [10] K. Nishioka, Evertse theorem in algebraic independence, Arch. Math. (Basel) 53 (1989), 159-170. Zbl0676.10024
  11. [11] K. Nishioka, I. Shiokawa and J. Tamura, Arithmetical properties of certain power series, J. Number Theory, to appear. Zbl0770.11039
  12. [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. [13] A. Salomaa, Jewels of Formal Language Theory, Pitman, 1981. Zbl0487.68063
  14. [14] A. Salomaa, Computation and Automata, Cambridge Univ. Press, 1985. Zbl0565.68046
  15. [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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