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Advances in Applied Clifford Algebras

, Volume 24, Issue 4, pp 981–994 | Cite as

A Matrix Recurrence for Systems of Clifford Algebra-Valued Orthogonal Polynomials

  • I. Cação
  • M. I. Falcão
  • H. R. MalonekEmail author
Article

Abstract

Recently, the authors developed a matrix approach to multivariate polynomial sequences by using methods of Hypercomplex Function Theory (Matrix representations of a basic polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory, 12 (2012), no. 2, 371-391). This paper deals with an extension of that approach to a recurrence relation for the construction of a complete system of orthogonal Clifford-algebra valued polynomials of arbitrary degree. At the same time the matrix approach sheds new light on results about systems of Clifford algebra-valued orthogonal polynomials obtained by Gürlebeck, Bock, Lávička, Delanghe et al. during the last five years. In fact, it allows to prove directly some intrinsic properties of the building blocks essential in the construction process, but not studied so far.

Keywords

Clifford Analysis generalized Appell polynomials recurrence relations 

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References

  1. 1.
    Appell P.: Sur une classe de polynômes. Ann. Sci. École Norm. Sup. 9(no. 2), 119–144 (1880)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Bock S, Gürlebeck K.: On a Generalized Appell System and Monogenic Power Series. Math. Methods Appl. Sci. 33(no. 4), 394–411 (2010)zbMATHMathSciNetGoogle Scholar
  3. 3.
    S. Bock, K. Gürlebeck, Lávisčka, and Souček, V., Gelfand-Tetslin bases for spherical monogenics in dimension 3. Rev. Mat. Iberoam., 28 no. 4 (2012), 1165–1192.Google Scholar
  4. 4.
    F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis. Pitman, Boston- London-Melbourne, 1982.Google Scholar
  5. 5.
    Cação I, Gürlebeck K, Malonek H. R.: Special monogenic polynomials and L2-approximation. Adv. Appl. Clifford Algebr. 11, 47–60 (2001)CrossRefzbMATHGoogle Scholar
  6. 6.
    I. Cação, Constructive Approximation by Monogenic polynomials. Ph.D thesis, Univ. Aveiro, 2004.Google Scholar
  7. 7.
    I. Cação, K. Gürlebeck, and S. Bock, Complete orthonormal systems of spherical monogenics - a constructive approach. In Methods of Complex and Clifford Analysis, Son LH, Tutschke W, Jain S (eds). SAS International Publications, 2004.Google Scholar
  8. 8.
    I. Cação, M. I. Falcão, and H. R. Malonek, Laguerre Derivative and Monogenic Laguerre Polynomials: An Operational Approach. Math. Comput. Model. 53 no. 5-6 (2011), 1084–1094.Google Scholar
  9. 9.
    I. Cação, M. I. Falcão, and H. R. Malonek, On Generalized Hypercomplex Laguerre-Type Exponentials and Applications. In Computational Science and Its Applications - ICCSA 2011, B. Murgante et al.(eds.), Lecture Notes in Computer Science, vol. 6784, Springer-Verlag, Berlin, Heidelberg, (2011), 271– 286.Google Scholar
  10. 10.
    Cação I, Falcão M. I, Malonek H. R.: Matrix representations of a basic polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory, 12(no. 2), 371–391 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Cação I, Gürlebeck K, Bock S.: On derivatives of spherical monogenics. Complex Variables and Elliptic Equations, 51 no 8(11), 847–869 (2006)CrossRefGoogle Scholar
  12. 12.
    I. Cação and H. R. Malonek, Remarks on some properties of monogenic polynomials, In Proceedings of ICNAAM 2006, Simos, T.E. et al. (eds.); Weinheim: Wiley-VCH., (2006) 596-599.Google Scholar
  13. 13.
    I. Cação and H. R. Malonek, On Complete Sets of Hypercomplex Appell Polynomials. In AIP Conference Proceedings, Simos, T.E. et al. (eds.) vol. 1048, 2008, 647–650.Google Scholar
  14. 14.
    Carlson B. C: Polynomials Satisfying a Binomial Theorem. J. Math. Anal. Appl. 32, 543–558 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    R. Delanghe, F. Sommen, and Souček, V., Clifford algebra and spinor-valued functions. Kluwer Academic Publishers, Dordrecht, 1992.Google Scholar
  16. 16.
    Eelbode D.: Monogenic Appell sets as representations of the Heisenberg algebra. Adv. Appl. Clifford Algebra 22(no. 4), 1009–1023 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    M. I. Falcão and H. R. Malonek, Generalized Exponentials Through Appell Sets in \({\mathbb{R}^{n+1}}\) and Bessel Functions. In AIP Conference Proceedings, Simos, T.E. et al. (eds.), vol. 936, 2007, 738–741.Google Scholar
  18. 18.
    J.E. Gilbert and M.A.M. Murray, Clifford algebras and Dirac operators in harmonic analysis. Cambridge University Press, Cambridge, 1991.Google Scholar
  19. 19.
    K. Gürlebeck and H. R. Malonek, A Hypercomplex Derivative of Monogenic Functions in \({\mathbb{R}^{n+1}}\) and Its Applications. Complex Variables Theory Appl. 39 (1999), 199–228.Google Scholar
  20. 20.
    K. Gürlebeck, K. Habetha, and W. Sprößig, Holomorphic Functions in the Plane and n-Dimensional Space. Translated from the 2006 German original. Birkhäuser Verlag, Basel, 2008,Google Scholar
  21. 21.
    Hahn W.: Über die Jacobischen Polynome und zwei verwandte Polynomklassen. Math. Z. 39, 634–638 (1935)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Lávička R: Canonical Bases for sl(2, c)-Modules of Spherical Monogenics in Dimension 3. Archivum Mathematicum 46, 339–349 (2010)Google Scholar
  23. 23.
    Lávička R.: Complete Orthogonal Appell Systems for Spherical Monogenics. Complex Anal. Oper. Theory, 6, 477–489 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Malonek H. R.: A New Hypercomplex Structure of the Euclidean Space \({\mathbb{R}^{n+1}}\) and the Concept of Hypercomplex Differentiability. Complex Variables 14, 25–33 (1990)CrossRefGoogle Scholar
  25. 25.
    H. R. Malonek, Selected Topics in Hypercomplex Function Theory. In Clifford Algebras and Potential Theory, 7, S.-L. Eriksson (ed.), University of Joensuu, (2004), 111–150.Google Scholar
  26. 26.
    H. R. Malonek and M. I. Falcão, Special Monogenic Polynomials–Properties and Applications. In AIP Conference Proceedings, Th. E. Simos et al. (eds.) vol. 936, (2007), 764–767.Google Scholar
  27. 27.
    Peña Peña D.: Shifted Appell Sequences in Clifford Analysis. Results. Math. 63, 1145–1157 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    E. Rainville, Special Functions, Macmillan, New York, 1965.Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  1. 1.Center for Research and Development in Mathematics and Applications, Department of MathematicsUniversity of AveiroAveiroPortugal
  2. 2.Centre of Mathematics, Department of Mathematics and ApplicationsUniversity of MinhoBragaPortugal

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