On an Hypercomplex Generalization of Gould-Hopper and Related Chebyshev Polynomials

  • I. Cação
  • H. R. Malonek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6784)


An operational approach introduced by Gould and Hopper to the construction of generalized Hermite polynomials is followed in the hypercomplex context to build multidimensional generalized Hermite polynomials by the consideration of an appropriate basic set of monogenic polynomials. Directly related functions, like Chebyshev polynomials of first and second kind are constructed.


Hypercomplex function theory exponential operators generalized Hermite polynomials Chebyshev polynomials 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bock, S., Gürlebeck, K.: On a generalized Appell system and monogenic power series. Math. Methods Appl. Sci. 33(4), 394–411 (2010)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Brackx, F.: The exponential function of a quaternion variable. Applicable Anal. 8(3), 265–276 (1978/1979)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brackx, F., De Schepper, N., Sommen, F.: Clifford algebra-valued orthogonal polynomials in Euclidean space. J. Approx. Theory 137(1), 108–122 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis. Pitman, Boston (1982)zbMATHGoogle Scholar
  5. 5.
    Cação, I., Falcão, M.I., Malonek, H.R.: Laguerre derivative and monogenic Lag uerre polynomials: an operational approach. Math. Comput. Modelling 53, 1084–1094 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Cação, I., Malonek, H.: On complete sets of hypercomplex Appell polynomials. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) AIP Conference Proceedings, vol. 1048, pp. 647–650 (2008)Google Scholar
  7. 7.
    Dattoli, G.: Laguerre and generalized Hermite polynomials: the point of view of the operational method. Integral Transforms. Spec. Funct. 15(2), 93–99 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Falcão, M.I., Cruz, J., Malonek, H.R.: Remarks on the generation of monogenic functions. In: 17th Inter. Conf. on the Appl. of Computer Science and Mathematics on Architecture and Civil Engineering, Weimar (2006)Google Scholar
  9. 9.
    Falcão, M.I., Malonek, H.R.: Generalized exponentials through Appell sets in ℝn + 1 and Bessel functions. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) AIP Conference Proceedings, vol. 936, pp. 738–741 (2007)Google Scholar
  10. 10.
    Gould, H.W., Hopper, A.: Operational formulas connected with two generalizations of Hermite polynomials. Duke Math. J. 29, 51–62 (1962)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Gürlebeck, K., Malonek, H.: A hypercomplex derivative of monogenic functions in ℝn + 1 and its applications. Complex Variables Theory Appl. 39, 199–228 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Lávička, R.: Canonical bases for sl(2,c)-modules of spherical monogenics in dimension 3. Archivum Mathematicum Tomus 46, 339–349 (2010)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Malonek, H.: A new hypercomplex structure of the euclidean space ℝm + 1 and the concept of hypercomplex differentiability. Complex Variables 14, 25–33 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Malonek, H.: Selected topics in hypercomplex function theory. In: Eriksson, S.L. (ed.) Clifford Algebras and Potential Theory, University of Joensuu, vol. 7, pp. 111–150 (2004)Google Scholar
  15. 15.
    Malonek, H.R., Falcão, M.I.: Special monogenic polynomials|properties and applications. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) AIP Conference Proceedings, vol. 936, pp. 764–767 (2007)Google Scholar
  16. 16.
    Riordan, J.: Combinatorial identities. John Wiley & Sons Inc., New York (1968)zbMATHGoogle Scholar
  17. 17.
    Sommen, F.: A product and an exponential function in hypercomplex function theory. Appl. Anal. 12, 13–26 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Sommen, F.: Special functions in Clifford analysis and axial symmetry. J. Math. Anal. Appl. 130(1), 110–133 (1988)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • I. Cação
    • 1
  • H. R. Malonek
    • 1
  1. 1.Departamento de MatemáticaUniversidade de AveiroPortugal

Personalised recommendations