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Modifications of continued fractions, a survey

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Analytic Theory of Continued Fractions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 932))

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References

  1. J. D. De Pree and W. J. Thron, On sequences of Moebius transformations, Math. Zeitschr. 80 (1962), 184–193.

    Article  MathSciNet  MATH  Google Scholar 

  2. John Gill, Infinite compositions of Möbius transformations, Trans. Amer. Math. Soc. 176(1973) 479–487.

    MathSciNet  MATH  Google Scholar 

  3. John Gill, The use of attractive fixed points in accelerating the convergence of limit periodic continued fractions, Proc. Amer. Math. Soc. 47 (1975) 119–126.

    Article  MathSciNet  MATH  Google Scholar 

  4. John Gill, Modifying factors for sequences of linear fractional transformations, Kgl. Norske Vid. Selsk. Skr. (Trondheim) (1978) No. 3, 1–7.

    MATH  Google Scholar 

  5. John Gill, Enhancing the convergence region of a sequence of bilinear transformations, Math. Scand. 43 (1978), 74–80.

    MathSciNet  MATH  Google Scholar 

  6. John Gill, Convergence acceleration for continued fractions K(an/1) with lim an=0, these Lecture Notes.

    Google Scholar 

  7. John Gill, Truncation error analysis for continued fractions K(an/1) where , these Lecture Notes.

    Google Scholar 

  8. J. W. L. Glaisher, On the transformation of continued products into continued fractions, Proc. London Math. Soc. 5(1873/4), 85.

    MathSciNet  Google Scholar 

  9. H. Hamburger, Über eine Erweiterung des Stielt jesschen Momentenproblems I, Math. Ann. 81(1920) 235–319.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. Hamel, Eine charakteristische Eigenschaft beschränkter analytischer Funktionen, Math. Ann. 78 (1918), 257–269.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Hamel, Über einen limitärperiodischen Kettenbruch, Arch. d. Math. u. Phys. 27 (1918) 37–43.

    MATH  Google Scholar 

  12. T. L. Hayden, Continued fraction approximation to functions, Numer. Math. 7 (1965) 292–309.

    Article  MathSciNet  MATH  Google Scholar 

  13. Rolf M. Hovstad, Solution of a convergence problem in the theory of T-fractions, Proc. Amer. Math. Soc. 48 (1975) 337–343.

    MathSciNet  MATH  Google Scholar 

  14. Lisa Jacobsen, A method for convergence acceleration of continued fractions K(an/1), these Lecture Notes.

    Google Scholar 

  15. Lisa Jacobsen, Some periodic sequences of circular convergence regions, these Lecture Notes.

    Google Scholar 

  16. William B. Jones and W. J. Thron, Sequences of meromorphic functions corresponding to a formal Laurent series, SIAM J. Math. Anal. 10 (1979), 1–17.

    Article  MathSciNet  MATH  Google Scholar 

  17. William B. Jones and W. J. Thron, Continued Fractions: Analytic theory and applications Encyclopedia of Mathematics and its Applications vol 11, Addison Wesley, Reading, Mass. 1980.

    MATH  Google Scholar 

  18. M. Mandell and Arne Magnus, On convergence of sequences of linear fractional transformations, Math. Zeitschr. 115 (1970), 11–17.

    Article  MathSciNet  MATH  Google Scholar 

  19. Oskar Perron, Über einen Satz des Herrn Poincaré, J. reine angew. Math. 136 (1909), 17–37.

    MathSciNet  MATH  Google Scholar 

  20. T. E. Phipps, Jr., A continued fraction representation of eigenvalues, SIAM Rev. 13 (1971), 390–395.

    Article  Google Scholar 

  21. T. E. Phipps, Jr., A new approach to evaluation of infinite processes, NOLTR 71–36, Naval Ordnance Laboratory, White Oak, Silver Spring, Maryland 1971.

    Google Scholar 

  22. S. Pincherle, Sur la génération de systemes récurrents au moyen d’une equation linéaire differentielle, Acta Math. 16 (1893), 341–363.

    Article  MathSciNet  Google Scholar 

  23. George Piranian and W. J. Thron, Convergence properties of sequences of linear fractional transformations, Michigan Math. J. 4 (1957) 129–135.

    Article  MathSciNet  MATH  Google Scholar 

  24. Henri Poincaré, Sur les equations linéaires aux différentielles ordinaires et aux différences finies, Amer. J. Math. 7 (1885), 203–258.

    Article  MathSciNet  MATH  Google Scholar 

  25. I. Schur, Über Potenzreihen die im Innern des Einheitskreises beschränkt sind, J. reine angew. Math. 147 (1917) 205–232.

    MathSciNet  MATH  Google Scholar 

  26. O. Szasz, Collected mathematical papers, Cincinnati, 1955.

    Google Scholar 

  27. W. J. Thron, Some properties of continued fractions 1+d0z+K(z/(1+dnz)), Bull. Amer. Math. Soc. 54 (1948), 206–28.

    Article  MathSciNet  MATH  Google Scholar 

  28. W. J. Thron, Convergence of sequences of linear fractional transformations and of continued fractions, J. Indian Math. Soc. 27 (1963), 103–127.

    MathSciNet  MATH  Google Scholar 

  29. W. J. Thron, A priori truncation error estimates for Stieltjes fractions, Christoffel Memorial volume, Birkhäuser, Basel (1981).

    Book  MATH  Google Scholar 

  30. W. J. Thron and Haakon Waadeland, Accelerating convergence of limit periodic continued fractions K(an/1), Numer. Math. 34 (1980) 155–170.

    Article  MathSciNet  MATH  Google Scholar 

  31. W. J. Thron and Haakon Waadeland, Analytic continuation of functions defined by means of continued fractions, Math. Scand 47 (1980) 72–90.

    MathSciNet  MATH  Google Scholar 

  32. W. J. Thron and Haakon Waadeland, Convergence questions for limit periodic continued fractions, Rocky Mountain J. Math. 11 (1981), 641–657.

    Article  MathSciNet  MATH  Google Scholar 

  33. W. J. Thron and Haakon Waadeland, On a certain transformation of continued fractions, these Lecture Notes.

    Google Scholar 

  34. Haakon Waadeland, On T-fractions of functions, holomorphic and bounded in a circular disk, Norske Vid. Selsk. Skr. (Trondheim) (1964), No. 8, 1–19.

    Google Scholar 

  35. Haakon Waadeland, A convergence property of certain T-fraction expansions, Norske Vid. Selsk. Skr. (Trondheim) (1966), No. 9, 1–22.

    MathSciNet  MATH  Google Scholar 

  36. Haakon Waadeland, On T-fractions of certain functions with a first order pole at the point of infinity, Norske Vid. Selsk. Forh. 40 (1967), No. 1.

    Google Scholar 

  37. Haakon Waadeland, T-fractions from a different point of view, Rocky Mountain J. Math. 4 (1974) 391–393.

    Article  MathSciNet  MATH  Google Scholar 

  38. Haakon Waadeland, General T-fractions corresponding to functions satisfying certain boundedness conditions, J. Approximation Theory 26 (1979) 317–328.

    Article  MathSciNet  MATH  Google Scholar 

  39. Haakon Waadeland, Limit periodic general T-fractions and holomorphic functions, J. Approximation Theory 27 (1979) 329–345.

    Article  MathSciNet  MATH  Google Scholar 

  40. H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910) 220–269.

    Article  MathSciNet  MATH  Google Scholar 

  41. P. Wynn, Converging factors for continued fractions, Numer. Math. 1 (1959) 272–320.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Author notes

  1. W. J. Thron

    Present address: Department of Mathematics, University of Colorado, Campus Box 426, 80309, Boulder, Colorado, U.S.A.

  2. Haakon Waadeland

    Present address: Institutt for Matematikk og Statistilsk, Universitetet i Trondheim, 7055, Dragvoll, Norway

Authors and Affiliations

    Authors
    1. W. J. Thron
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    2. Haakon Waadeland
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    Editor information

    William B. Jones W. J. Thron Haakon Waadeland

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    © 1982 Springer-Verlag

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    Thron, W.J., Waadeland, H. (1982). Modifications of continued fractions, a survey. In: Jones, W.B., Thron, W.J., Waadeland, H. (eds) Analytic Theory of Continued Fractions. Lecture Notes in Mathematics, vol 932. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0093304

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    • DOI: https://doi.org/10.1007/BFb0093304

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    • Print ISBN: 978-3-540-11567-0

    • Online ISBN: 978-3-540-39276-7

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