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Monogenic Polynomials of Four Variables with Binomial Expansion

  • Carla Cruz
  • M. Irene Falcão
  • Helmuth R. Malonek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8579)

Abstract

In the recent past one of the main concern of research in the field of Hypercomplex Function Theory in Clifford Algebras was the development of a variety of new tools for a deeper understanding about its true elementary roots in the Function Theory of one Complex Variable. Therefore the study of the space of monogenic (Clifford holomorphic) functions by its stratification via homogeneous monogenic polynomials is a useful tool. In this paper we consider the structure of those polynomials of four real variables with binomial expansion. This allows a complete characterization of sequences of 4D generalized monogenic Appell polynomials by three different types of polynomials. A particularly important case is that of monogenic polynomials which are simply isomorphic to the integer powers of one complex variable and therefore also called pseudo-complex powers.

Keywords

monogenic polynomials Appell sequences pseudo-complex powers 

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References

  1. 1.
    Abul-Ez, M., Constales, D.: Basic sets of polynomials in Clifford analysis. Complex Variables, Theory Appl. 14, 177–185 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Álvarez-Peña, C., Porter, M.: Appell bases for monogenic functions of three variables. Adv. in Appl. Clifford Algebras 23, 547–560 (2013)CrossRefzbMATHGoogle Scholar
  3. 3.
    Appell, P.: Sur une classe de polynômes. Ann. Sci. École Norm. Sup. 9(2), 119–144 (1880)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Bock, S., Gürlebeck, K.: On a generalized Appell system and monogenic power series. Math. Methods Appl. Sci. 33(4), 394–411 (2010)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Brackx, F., Delanghe, R., Sommen, F.: Clifford analysis. Pitman, Boston (1982)zbMATHGoogle Scholar
  6. 6.
    Cação, I., Falcão, M.I., Malonek, H.R.: Laguerre derivative and monogenic Laguerre polynomials: An operational approach. Math. Comput. Model. 53(5-6), 1084–1094 (2011)CrossRefzbMATHGoogle Scholar
  7. 7.
    Cação, I., Falcão, M.I., Malonek, H.R.: Matrix representations of a special polynomial sequence in arbitrary dimension. Comput. Methods Funct. Theory 12, 371–391 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cação, I., Malonek, H.R.: On complete sets of hypercomplex Appell polynomials. In: Simos, T., et al. (eds.) AIP Conference Proceedings, vol. 1048, pp. 647–650 (2008)Google Scholar
  9. 9.
    Carlson, B.C.: Polynomials satisfying a binomial theorem. J. Math. Anal. Appl. 32, 543–558 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Cruz, C., Falcão, M.I., Malonek, H.R.: On the structure of generalized Appell sequences of paravector valued homogeneous monogenic polynomials. In: Simos, T., et al. (eds.) AIP Conference Proceedings, vol. 1479(1), pp. 283–286 (2012)Google Scholar
  11. 11.
    Cruz, C., Falcão, M.I., Malonek, H.R.: On pseudo-complex bases for monogenic polynomials. In: Sivasundaram, S. (ed.) AIP Conference Proceedings, vol. 1493, pp. 350–355 (2012)Google Scholar
  12. 12.
    Cruz, C., Falcão, M.I., Malonek, H.R.: A note on totally regular variables and appell sequences in hypercomplex function theory. In: Murgante, B., Misra, S., Carlini, M., Torre, C.M., Nguyen, H.-Q., Taniar, D., Apduhan, B.O., Gervasi, O. (eds.) ICCSA 2013, Part I. LNCS, vol. 7971, pp. 293–303. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Cruz, C., Falcão, M.I., Malonek, H.R.: Monogenic pseudo-complex power functions and their applications. Mathematical Methods in the Applied Sciences (2013), doi:10.1002/mma.2931Google Scholar
  14. 14.
    Delanghe, R.: On regular-analytic functions with values in a Clifford algebra. Math. Ann. 185, 91–111 (1970)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Delanghe, R., Lávička, R., Souček, V.: The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces. Math. Methods Appl. Sci. 7(35), 745–757 (2012)CrossRefGoogle Scholar
  16. 16.
    Delanghe, R., Sommen, F., Soucek, V.: Clifford algebra and spinor-valued functions. A function theory for the Dirac operator. Kluwer Academic Publishers (1992)Google Scholar
  17. 17.
    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
  18. 18.
    Falcão, M.I., Malonek, H.R.: Generalized exponentials through Appell sets in ℝn + 1 and Bessel functions. In: Simos, T., et al. (eds.) AIP Conference Proceedings, vol. 936, pp. 738–741 (2007)Google Scholar
  19. 19.
    Fueter, R.: Über Funktionen einer Quaternionenvariablen. Atti Congresso Bologna 2, 145 (1930)MathSciNetGoogle Scholar
  20. 20.
    Fueter, R.: Analytische Funktionen einer Quaternionenvariablen. Comment. Math. Helv. 4, 9–20 (1932)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Fueter, R.: Functions of a Hyper Complex Variable. Lecture Notes written and supplemented by E. Bareiss, Fall Semester 1948/49. Univ. Zürich (1950)Google Scholar
  22. 22.
    Gürlebeck, K., Malonek, H.R.: A hypercomplex derivative of monogenic functions in ℝn + 1 and its applications. Complex Variables Theory Appl. 39, 199–228 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Gürlebeck, K., Sprößig, W.: Quaternionic and Clifford Calculus for Physicists and Engineers. John Wiley & Sons (1997)Google Scholar
  24. 24.
    Gürlebeck, K., Habetha, K., Sprößig, W.: Holomorphic functions in the plane and n-dimensional space. Birkhäuser Verlag, Basel (2008)Google Scholar
  25. 25.
    Lávička, R.: Canonical bases for sl(2,c)-modules of spherical monogenics in dimension 3. Archivum Mathematicum 46, 339–349 (2010)Google Scholar
  26. 26.
    Malonek, H.R.: Rudolf Fueter and his motivation for hypercomplex function theory. Advances in Applied Clifford Algebras 11, 219–229 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Malonek, H.R.: A new hypercomplex structure of the euclidean space ℝm + 1 and the concept of hypercomplex differentiabilit. Complex Variables 14, 25–33 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Malonek, H.R.: 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
  29. 29.
    Malonek, H.R., De Almeida, R.: A note on a generalized Joukowski transformation. Appl. Math. Lett. 23(10), 1174–1178 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Malonek, H.R., Falcão, M.I.: Special monogenic polynomials—properties and applications. In: Simos, T., et al. (eds.) AIP Conference Proceedings, vol. 936, pp. 764–767 (2007)Google Scholar
  31. 31.
    Malonek, H.R., Falcão, M.I.: 3D-mappings by means of monogenic functions and their approximation. Math. Methods Appl. Sci. 33, 423–430 (2010)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Malonek, H.R., Falcão, M.I.: On paravector valued homogeneous monogenic polynomials with binomial expansion. Advances in Applied Clifford Algebras 22(3, SI), 789–801 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Stein, E.M., Weiss, G.: Generalization of the Cauchy-Riemann equations and representations of the rotation group. Amer. J. Math. 90, 163–196 (1968)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Carla Cruz
    • 1
  • M. Irene Falcão
    • 2
  • Helmuth R. Malonek
    • 1
    • 3
  1. 1.Centro de Investigação e Desenvolvimento em Matemática e AplicaçõesUniversidade de AveiroPortugal
  2. 2.Departamento de Matemática e Aplicações and Centro de MatemáticaUniversidade do MinhoPortugal
  3. 3.Departamento de MatemáticaUniversidade de AveiroPortugal

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