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Mat. Tr., 2007, Volume 10, Number 1, Pages 97–131 (Mi mt30)  

This article is cited in 4 scientific papers (total in 4 papers)

Continued Fractions, the Group $\mathrm{GL}(2,\mathbb Z)$, and Pisot Numbers

V. N. Berestovskiia, Yu. G. Nikonorovb

a Omsk Branch of Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Science
b Rubtsovsk Industrial Intitute, Branch of Altai State Technical University

Abstract: The properties of continued fractions, generalized golden sections, and generalized Fibonacci and Lucas numbers are proved on the ground of the properties of subsemigroups of the group of invertible integer matrices. Some properties of special recurrent sequences are studied. A new proof of the Pisot-Vijayaraghavan theorem is given. Some connections between continued fractions and Pisot numbers are considered. Some unsolved problems are stated.

Key words: continued fractions, Pisot numbers, recurrent sequences, generalized Fibonacci and Lucas numbers.

Full text: PDF file (365 kB)
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English version:
Siberian Advances in Mathematics, 2007, 17:4, 268–290

Bibliographic databases:

UDC: 511.26
Received: 26.01.2006

Citation: V. N. Berestovskii, Yu. G. Nikonorov, “Continued Fractions, the Group $\mathrm{GL}(2,\mathbb Z)$, and Pisot Numbers”, Mat. Tr., 10:1 (2007), 97–131; Siberian Adv. Math., 17:4 (2007), 268–290

Citation in format AMSBIB
\Bibitem{BerNik07}
\by V.~N.~Berestovskii, Yu.~G.~Nikonorov
\paper Continued Fractions, the Group $\mathrm{GL}(2,\mathbb Z)$, and Pisot Numbers
\jour Mat. Tr.
\yr 2007
\vol 10
\issue 1
\pages 97--131
\mathnet{http://mi.mathnet.ru/mt30}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2485367}
\elib{http://elibrary.ru/item.asp?id=9483455}
\transl
\jour Siberian Adv. Math.
\yr 2007
\vol 17
\issue 4
\pages 268--290
\crossref{https://doi.org/10.3103/S1055134407040025}


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    This publication is cited in the following articles:
    1. T. A. Kozlovskaya, “Konkho-spirali na poverkhnosti konusa”, Vestn. NGU. Ser. matem., mekh., inform., 11:2 (2011), 65–76  mathnet
    2. N. M. Dobrovolskii, N. N. Dobrovolskii, “O minimalnykh mnogochlenakh ostatochnykh drobei dlya algebraicheskikh irratsionalnostei”, Chebyshevskii sb., 16:3 (2015), 147–182  mathnet  elib
    3. N. M. Dobrovol'skii, I. N. Balaba, I. Yu. Rebrova, N. N. Dobrovol'skii, “On Lagrange algorithm for reduced algebraic irrationalities”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2016, no. 2, 27–39  mathnet
    4. N. M. Dobrovolskii, N. N. Dobrovolskii, D. K. Sobolev, V. N. Soboleva, “Klassifikatsiya chisto-veschestvennykh algebraicheskikh irratsionalnostei”, Chebyshevskii sb., 18:2 (2017), 98–128  mathnet  crossref  elib
  • Математические труды Siberian Advances in Mathematics
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