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2000 | 84/85 | 1 | 109-123
Tytuł artykułu

'The mother of all continued fractions'

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We give the relationship between regular continued fractions and Lehner fractions, using a procedure known as insertion}. Starting from the regular continued fraction expansion of any real irrational x, when the maximal number of insertions is applied one obtains the Lehner fraction of x. Insertions (and singularizations) show how these (and other) continued fraction expansions are related. We also investigate the relation between Lehner fractions and the Farey expansion (also known as the full continued fraction), and obtain the ergodic system underlying the Farey expansion.
Rocznik
Tom
Numer
1
Strony
109-123
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-05-25
poprawiono
1999-09-22
Twórcy
autor
  • Universiteit Utrecht, Fac. Wiskunde en Informatica and MRI, Budapestlaan 6, P.O. Box 80.000, 3508 TA Utrecht, the Netherlands
  • Technische Universiteit Delft, ITS (SSOR) Thomas Stieltjes Institute for Mathematics, Mekelweg 4, 2628 CD Delft, the Netherlands
Bibliografia
  • [AF] R. L. Adler and L. Flatto, The backward continued fraction map and geodesic flow, Ergodic Theory Dynam. Systems 4 (1984), 487-492.
  • [B] W. Bosma, Approximation by mediants, Math. Comp. 54 (1990), 421-434.
  • [BY] G. Brown and Q. H. Yin, Metrical theory for Farey continued fractions, Osaka J. Math. 33 (1996), 951-970.
  • [C] A. L. Cauchy, Oeuvres, Gauthier-Villars, Paris, 1890-1895.
  • [G] J. R. Goldman, Hurwitz sequences$,$ the Farey process$,$ and general continued fractions, Adv. Math. 72 (1988), 239-260.
  • [I] S. Ito, Algorithms with mediant convergents and their metrical theory, Osaka J. Math. 26 (1989), 557-578.
  • [Kh] A. Ya. Khintchine, Metrische Kettenbruchproblem, Compositio Math. 1 (1935), 361-382.
  • [K] C. Kraaikamp, A new class of continued fraction expansions, Acta Arith. 57 (1991), 1-39.
  • [L] J. Lehner, Semiregular continued fractions whose partial denominators are $1$ or $2$, in: The Mathematical Legacy of Wilhelm Magnus: Groups, Geometry and Special Functions (Brooklyn, NY, 1992), Contemp. Math. 169, Amer. Math. Soc., Providence, RI, 1994, 407-410.
  • [R] A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 472-493.
  • [Rich] I. Richards, Continued fractions without tears, Math. Mag. 54 (1981), 163-171.
  • [Z] D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, Berlin, 1981.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv84i1p109bwm
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