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1992 | 61 | 1 | 51-67
Tytuł artykułu

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

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
61
Numer
1
Strony
51-67
Opis fizyczny
Daty
wydano
1992
otrzymano
1990-11-30
poprawiono
1991-06-18
Twórcy
  • Faculty of General Education International Junior College Ekoda 4-15-1, Nakano-ku Tokyo 165, Japan
Bibliografia
  • [1] W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
  • [2] P. E. Böhmer, Über die Transzendenz gewisser dyadischer Brüche, Math. Ann. 96 (1927), 367-377; Berichtigung, Proc. Amer. Math. Soc. 735.
  • [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.
  • [4] L. V. Danilov, Some classes of transcendental numbers, Mat. Zametki 12 (2) (1972), 149-154 (in Russian); English transl.: Math. Notes 12 (1972), 524-527.
  • [5] J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc. 63 (1977), 29-32.
  • [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.
  • [7] K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann. 101 (1929), 342-366.
  • [8] D. Masser, A vanishing theorem for power series, Invent. Math. 67 (1982), 275-296.
  • [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.
  • [11] K. Nishioka, I. Shiokawa and J. Tamura, Arithmetical properties of certain power series, J. Number Theory, to appear.
  • [12] V. I. Parusnikov, The Jacobi-Perron algorithm and simultaneous approximation of functions, Mat. Sb. 114 (156) (2) (1981), 322-333 (in Russian).
  • [13] A. Salomaa, Jewels of Formal Language Theory, Pitman, 1981.
  • [14] A. Salomaa, Computation and Automata, Cambridge Univ. Press, 1985.
  • [15] J. Tamura, Transcendental numbers having explicit g-adic and Jacobi-Perron expansions, in: Séminaire de Théorie des Nombres de Bordeaux, to appear.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-aav61i1p51bwm
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