Skip to main content
SpringerLink
Log in

q-Blossoming and Hermite-Padé approximants to the q-exponential function

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

We derive explicit expressions for the type II Hermite-Padé approximants to the q-exponential function using the q-blossom of its partial sums. In particular, this derivation leads to an interpretation of the Padé approximants as control points of a q-Bézier curve defined in terms of the partial sums of the q-exponential function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Ait-Haddou, R., Goldman, R.: Best polynomial degree reduction on q-lattices with applications to q-orthogonal polynomials. App. Math Comp. 266, 267–276 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bergamaschi, L., Vianello, M.: M, Computation of the exponential operator for large, sparse, symmetric matrices. Numer. Linear Algebra 7, 27–46 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borwein, P. B.: Padé approximants for the q-elementary functions. Constr. Approx. 4(1), 391–402 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Budakçi, G., Dişibüyük, Ç., Goldman, R., Oruç, H.: Extending fundamental formulas from classical B-splines to quantum B-splines. J. Comput. Appl. Math. 282, 17–33 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chudnovsky, D. V., Chudnovsky, G. V.: Padé and Rational Approximations to Systems of Functions and Their Arithmetic Applications Lecture Notes in Mathematics, vol. 1052. Springer, Berlin (1984)

  6. Exton, M.: Q-Hypergeometric Functions and Applications. Ellis Horwood, Chichester (1983)

    MATH  Google Scholar 

  7. Hermite, C.: Sur la généralisation des fractions continues algébriques. Annali Math. Pura Appl. Ser. A 2–21, 289–308 (1893)

    Article  MATH  Google Scholar 

  8. Iserles, A., Powell, M. D. J.: On the A-acceptability of rational approximations that interpolate the exponential function. IMA J. Numer. Anal. 1, 241–251 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jackson, F. H.: A basic sine and cosine with symbolic solutions of certain differential equations. In: Proceedings of the Edinburgh Math. Soc., vol. 22, pp 28–39 (1904)

  10. Kac, V., Cheung, P.: Quantum calculus, universitext series, IX. springer verlag (2002)

  11. Mahler, K.: Application of some formulae by Hermite to the approximation of exponentials and logarithms. Math. Ann. 168, 200–227 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  12. Maier, W.: Potenzreihen irrationalen Grenzwertes. J. Reine angew Math. 156, 93–148 (1927)

    MathSciNet  MATH  Google Scholar 

  13. Matala-aho, T.: Type II hermite-padé approximations of generalized hypergeometric series. Constr. Approx. 33, 289–312 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mazure, M.-L.: Blossoming: a geometrical approach. Constr. Approx. 15, 33–68 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Oruç, H., Phillips, G. M.: (2003) q-Bernstein polynomials and bézier curves. J. Comp. Appl. Math. 151, 1–12 (2003)

    Article  MATH  Google Scholar 

  16. Padé Oeuvres, H. In: Brézinski, C. (ed.) . Albert Blanchard, Paris (1984)

  17. Pottmann, H.: The geometry of Tchebycheffian splines. Comput. Aided Geom. Design 10, 181–210 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ramshaw, L.: Blossoms are polar forms. Comput. Aided Geom. Design 6(4), 323–358 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sablonnière, P.: An algorithm for the computation of hermite-padé approximations to the exponential function: divided differences and hermite-padé forms. Numerical Algorithms 33(1-4), 443–452 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sablonnière, P.: B-splines and Hermite-Padá approximants to the exponential function. J. Comput. Appl. Math. 219(2), 509–517 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Sidge, R. B.: EXPOKIT: Software package for computing matrix exponentials. ACM Trans. Math. Software 24, 130–156 (1998)

    Article  MATH  Google Scholar 

  22. Simeonov, P., Zafiris, V., Goldman, R.: q-Blossoming A new approach to algorithms and identities for q-Bernstein bases and q-bézier curves. Journal of Approximation Theory 164(1), 77–104 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Simeonov, P., Goldman, R.: Quantum B-splines. BIT Numerical Mathematics 53(1), 193–223 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wanner, G., Hairer, E., Nersett, S. P.: Order stars and stability theorems. BIT 18, 475–489 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. A-803 Mihogaoka 19, Ibaraki City, Osaka, 567-0047, Japan

    Rachid Ait-Haddou

Authors
  1. Rachid Ait-Haddou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Rachid Ait-Haddou.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ait-Haddou, R. q-Blossoming and Hermite-Padé approximants to the q-exponential function. Numer Algor 76, 53–66 (2017). https://doi.org/10.1007/s11075-016-0242-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-016-0242-4

Keywords

Mathematics Subject Classification (2010)

Search

Navigation