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Linear fractional transformations and nonlinear leaping convergents of some continued fractions

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Abstract

For \(\alpha _0 = \left[ a_0, a_1, \ldots \right] \) an infinite continued fraction and \(\sigma \) a linear fractional transformation, we study the continued fraction expansion of \(\sigma (\alpha _0)\) and its convergents. We provide the continued fraction expansion of \(\sigma (\alpha _0)\) for four general families of continued fractions and when \(\left| \det \sigma \right| = 2\). We also find nonlinear recurrence relations among the convergents of \(\sigma (\alpha _0)\) which allow us to highlight relations between convergents of \(\alpha _0\) and \(\sigma (\alpha _0)\). Finally, we apply our results to some special and well-studied continued fractions, like Hurwitzian and Tasoevian ones, giving a first study about leaping convergents having steps provided by nonlinear functions.

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References

  1. 1.

    Astels, S.: Sums of numbers with small partial quotients. Proc. Am. Math. Soc. 130, 637–642 (2002)

  2. 2.

    Davis, C.S.: On some simple continued fractions connected with \(e\). J. Lond. Math. Soc. 20, 194–198 (1945)

  3. 3.

    Diviš, B.: On the sums of continued fractions. Acta Arith. 22, 157–173 (1973)

  4. 4.

    Elsner, C.: On arithmetic properties of the convergents of Euler’s number. Colloq. Math. 79, 133–145 (1999)

  5. 5.

    Elsner, C., Komatsu, T.: A recurrence formula for leaping convergents of non-regular continued fractions. Linear Algebra Appl. 428, 824–833 (2008)

  6. 6.

    Elsner, C., Komatsu, T.: On the residue classes of integer sequences satisfying a linear three-term recurrence formula. Linear Algebra Appl. 429, 933–947 (2008)

  7. 7.

    Fowler, D.H.: The Mathematics of Plato’s Academy: A New Reconstruction, 2nd edn. Oxford Science Publications, New York (1999)

  8. 8.

    Gosper, R.W.: Continued fraction arithmetic. https://perl.plover.com/classes/cftalk/INFO/gosper.txt

  9. 9.

    Beeler, M., Gosper, R.W., Schroeppel, R.: “HAKMEM”, Tech. Rep. No. 239, Artificial Intelligence Lab., MIT, Cambridge, MA (1972). https://www.inwap.com/pdp10/hbaker/hakmem/hakmem.html or https://w3.pppl.gov/~hammett/work/2009/AIM-239-ocr.pdf

  10. 10.

    Hall, M.: On the sum and product of continued fractions. Ann. Math. 48, 966–993 (1947)

  11. 11.

    Komatsu, T.: On Hurwitz and Tasoev’s continued fractions. Acta Arith. 107, 161–177 (2003)

  12. 12.

    Komatsu, T.: Arithmetical properties of the leaping convergents of \(e^{1/s}\). Tokyo J. Math. 27, 1–12 (2004)

  13. 13.

    Komatsu, T.: Hurwitz and Tasoev continued fractions. Monatsh. Math. 145, 47–60 (2005)

  14. 14.

    Komatsu, T.: Some combinatorial properties of the leaping convergents. Integers 7(2), 21 (2007)

  15. 15.

    Komatsu, T.: Hurwitz continued fractions with confluent hypergeometric functions. Czechoslov. Math. J. 57, 919–932 (2007)

  16. 16.

    Komatsu, T.: More on Hurwitz and Tasoev continued fractions. Sarajevo J. Math. 4, 155–180 (2008)

  17. 17.

    Komatsu, T.: Some combinatorial properties of the leaping convergents, II. Applications of Fibonacci Numbers (Proc. of the 12th Int. Conf. on Fibonacci Numbers and Their Applications). Congr. Numer. 200, 187–196 (2010)

  18. 18.

    Komatsu, T.: Leaping convergents of Hurwitz continued fractions. Discuss. Math. Gen. Algebra Appl. 31, 101–121 (2011)

  19. 19.

    Komatsu, T.: Leaping convergents of Tasoev continued fractions. Discuss. Math. Gen. Algebra Appl. 31, 201–216 (2011)

  20. 20.

    Komatsu, T.: Some exact algebraic expressions for the tails of Tasoev continued fractions. J. Aust. Math. Soc. 92, 179–193 (2012)

  21. 21.

    Lagarias, J.C., Shallit, J.O.: Linear fractional transformation of continued fractions with bounded partial quotients. J. Theor. Nombr. Bordx. 9, 267–279 (1997)

  22. 22.

    Liardet, P., Stambul, P.: Algebraic computations with continued fractions. J. Number Theory 73(1), 92–121 (1998)

  23. 23.

    Lee, K.: Continued fractions for linear fractional transformations of power series. Finite Fields Th. App. 11, 45–55 (2005)

  24. 24.

    Lehmer, D.H.: Continued fractions containing arithmetic progressions. Scripta Math. 29, 17–24 (1973)

  25. 25.

    Lehmer, D.N.: Arithmetical theory of certain Hurwitz continued fractions. Am. J. Math. 40(4), 375–390 (1918)

  26. 26.

    Mc Laughlin, J.: Some new families of Tasoevian and Hurwitzian continued fractions. Acta Arith. 135(3), 247–268 (2008)

  27. 27.

    Matthews, K.R., Walters, R.F.C.: Some properties of the continued fraction expansion of \((m/n)e^{1/q}\). Proc. Camb. Philos. Soc. 67, 67–74 (1970)

  28. 28.

    Panprasitwech, O., Laohakosol, V., Chaichana, T.: Linear fractional transformations of continued fractions with bounded partial quotients in the field of formal series. East-West J. Math. 11, 185–194 (2009)

  29. 29.

    Raney, G.N.: On continued fractions and finite automata. Math. Ann. 206, 265–284 (1973)

  30. 30.

    Rockett, A.M., Szüsz, P.: Continued Fractions. World Scientific Publishing Co. Pte. Ltd., Singapore (1992)

  31. 31.

    Vardi, I.: Code and pseudocode. Math J. 6(2), 66–71 (1996)

  32. 32.

    Walters, R.F.C.: Alternative derivation of some regular continued fractions. J. Aust. Math. Soc. 8, 205–212 (1968)

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Correspondence to Nadir Murru.

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Havens, C., Barbero, S., Cerruti, U. et al. Linear fractional transformations and nonlinear leaping convergents of some continued fractions. Res. number theory 6, 11 (2020). https://doi.org/10.1007/s40993-020-0187-5

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