Linear fractional transformations and nonlinear leaping convergents of some continued fractions


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.

This is a preview of subscription content, access via your institution.


  1. 1.

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

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

  3. 3.

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

    MathSciNet  Article  Google Scholar 

  4. 4.

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

    MathSciNet  Article  Google Scholar 

  5. 5.

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

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    Google Scholar 

  8. 8.

    Gosper, R.W.: Continued fraction arithmetic.

  9. 9.

    Beeler, M., Gosper, R.W., Schroeppel, R.: “HAKMEM”, Tech. Rep. No. 239, Artificial Intelligence Lab., MIT, Cambridge, MA (1972). or

  10. 10.

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

    MathSciNet  Article  Google Scholar 

  11. 11.

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

    MathSciNet  Article  Google Scholar 

  12. 12.

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

    MathSciNet  Article  Google Scholar 

  13. 13.

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

    MathSciNet  Article  Google Scholar 

  14. 14.

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

    MATH  Google Scholar 

  15. 15.

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

    MathSciNet  Article  Google Scholar 

  16. 16.

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

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  18. 18.

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

    MathSciNet  Article  Google Scholar 

  19. 19.

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

    MathSciNet  Article  Google Scholar 

  20. 20.

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

    MathSciNet  Article  Google Scholar 

  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)

    MathSciNet  Article  Google Scholar 

  22. 22.

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

    MathSciNet  Article  Google Scholar 

  23. 23.

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

    MathSciNet  Article  Google Scholar 

  24. 24.

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

    MathSciNet  MATH  Google Scholar 

  25. 25.

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

    MathSciNet  Article  Google Scholar 

  26. 26.

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

    MathSciNet  Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  29. 29.

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

    MathSciNet  Article  Google Scholar 

  30. 30.

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

    Google Scholar 

  31. 31.

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

    Google Scholar 

  32. 32.

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

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Nadir Murru.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation