Advertisement

Results in Mathematics

, Volume 64, Issue 1–2, pp 13–35 | Cite as

Linear Graininess Time Scales and Ladder Operators of Orthogonal Polynomials

  • Galina FilipukEmail author
  • Stefan Hilger
Open Access
Article

Abstract

In this paper we introduce linear graininess (LG) time scales. We further study orthogonal polynomials (OPs) with the weight function supported on LG time scales and derive the raising and lowering ladder operators by using the time scales calculus. We also derive a second order dynamic equation satisfied by these polynomials. The notion of an LG time scale encompasses the cases of the reals, the h-equidistant grid, the q-grid and, more general, a mixed (q, h)-grid. This allows a unified treatment of the ladder operators theory for classical OPs on these time scales. Moreover we will explain, why exclusively LG time scales provide the right framework for general OP theory.

Mathematics Subject Classification (2000)

26E70 33C47 

Keywords

Time scales linear graininess time scales ladder operators orthogonal polynomials 

Notes

Acknowledgements

GF was supported by the DAAD scholarship for one month to visit the Katholische Universität Eichstätt where this paper was started. The support of MNiSzW Iuventus Plus grant Nr 0124/IP3/2011/71 is also gratefully acknowledged.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

References

  1. 1.
    Ismail M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, vol. 98. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  2. 2.
    Chen Y., Ismail M.E.H.: Ladder operators and differential equations for orthogonal polynomials. J. Phys. A Math. Gen. 30, 7817–7829 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Ismail M.E.H., Nikolova I., Simeonov P.: Difference equations and discriminants for discrete orthogonal polynomials. Ramanujan J. 8, 475–502 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Ismail M.E.H.: Difference equations and quantized discriminants for q-orthogonal polynomials. Adv. Appl. Math. 30, 562–589 (2003)zbMATHCrossRefGoogle Scholar
  5. 5.
    Ismail M.E.H., Simeonov P.: Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials. J. Math. Anal. Appl. 376, 259–274 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Chen Y., Ismail M.E.H.: Ladder operators for q-orthogonal polynomials. J. Math. Anal. Appl. 345, 1–10 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Ismail M.E.H., Simeonov P.: q-difference operators for orthogonal polynomials. J. Comput. Appl. Math. 233, 749–761 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Ey K., Ruffing A.: Fixing the ladder operators on two types of Hermite functions. Dyn. Syst. Appl. 12, 115–129 (2003)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Bohner M., Guseinov G.: The h-Laplace and q-Laplace transforms. J. Math. Anal. Appl. 365, 75–92 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hilger S.: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Bohner M., Peterson A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)CrossRefGoogle Scholar
  12. 12.
    Bohner, M., Peterson, A. (eds.): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)zbMATHGoogle Scholar
  13. 13.
    Agarwal R., Bohner M., O’Regan D., Peterson A.: Dynamic equations on time scales: a survey. J. Comput. Appl. Math. 141, 1–26 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Lakshmikantham V., Sivasundaram S., Kayamakcalan B.: Dynamic Systems on Measure Chains. Mathematics and its Applications, vol. 370. Kluwer Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  15. 15.
    Time-scale calculus. http://en.wikipedia.org/wiki/Time_scale_calculus. Accessed 20 Jan 2012
  16. 16.
    Time-scales. http://www.timescales.org. Accessed 20 Jan 2012
  17. 17.
    Spedding, V.: Taming nature’s numbers. Cover story. New Scientist: the global science and technology weekly, 19 July 2003Google Scholar
  18. 18.
    Filipuk, G., Van Assche, W., Zhang, L.: Ladder operators and differential equations for multiple orthogonal polynomials. http://arxiv.org/abs/1204.5058. Accessed 1 May 2012
  19. 19.
    Durán A.J., Ismail M.E.H.: Differential coefficients of orthogonal matrix polynomials. J. Comput. Appl. Math. 190, 424–436 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Grünbaum F.A., de la Iglesia M.D., Martínez-Finkelshtein A.: Properties of matrix orthogonal polynomials via their Riemann–Hilbert characterization. SIGMA Symmetry Integrability Geom. Methods Appl. 7(098), 31 (2011)Google Scholar
  21. 21.
    Hilger S.: The category of ladders. Results Math. 57, 335–364 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics, Informatics and MechanicsUniversity of WarsawWarsawPoland
  2. 2.Mathematisch–Geographische FakultätKatholische Universität Eichstätt-IngolstadtEichstättGermany

Personalised recommendations