Further Results on the Inverse Base of Axially Monogenic Polynomials

  • Mohamed AbdallaEmail author
  • Mahmoud Abul-Ez
  • Aida Al-Ahmadi


The main goal of this paper is to investigate the convergence properties of the inverse base of axially monogenic polynomials. These convergence properties proceed from the investigation of the relation between the effectiveness in closed balls, open balls as well as effectiveness for integral functions. The obtained results are the natural generalization of the original ones in complex setting to higher dimensions. In the meantime our results cover some open questions concerning the Clifford inverse bases.


Axially monogenic function Bases of polynomials Inverse base Convergence properties 

Mathematics Subject Classification

30G35 41A10 



The authors are very grateful to the anonymous referees for many valuable comments and suggestions which helped to improve the paper.


  1. 1.
    Abul-Ez, M., Constales, D.: Basic sets of polynomials in Clifford analysis. Complex Variables 14, 177–185 (1990)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Abul-Ez, M., Constales, D.: Linear substitution for basic sets of polynomials in Clifford analysis. Portug. Mathe. 48, 143–154 (1991)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Abul-Ez, M.: Inverse sets of polynomials in Clifford analysis. Arch. der Math. 58, 561–567 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Abul-Ez, M.: Product simple sets of polynomials in Clifford analysis. Rivista dimatematica della Univ. Parma. 3, 283–293 (1994)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Abul-Ez, M.: Hadamard product of bases of polynomials in Clifford analysis. Complex Variables 43, 109–128 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Abul-Ez, M.: Bessel polynomial expansions in spaces of holomorphic functions. J. Math. Anal. Appl. 221, 177–190 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Abul-Ez, M., Constales, D.: On convergence properties of basic series representing special monogenic functions. Arch. Math. 81, 62–71 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Abul-Ez, M.A., Zayad, M.: Similar transposed bases of polynomials in clifford analysis. Appl. Math. Inform. Sci. 4, 63–78 (2010)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Abul-Ez, M., Saleem, M., Zayed, M.: On the representation near a point of Clifford valued functions by infinite series of polynomials. In: 9th International Conference on Clifford Algebras, Weimar, Germany, 15–20 July, (2011)Google Scholar
  10. 10.
    AbulEz, M., Constales, D., Morais, J., Zayed, M.: Hadamard three-hyperballs type theorem and overconvergence of special monogenic simple series. J. Math. Anal. Appl. 412, 426–434 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Arfaoui, S., Rezgui, I., Ben Mabrouk, A.: Harmonic wavelet analysis on the sphere, spheroidal wavelets. Degryuter, (2016), ISBN 978-11-048188-4Google Scholar
  12. 12.
    Aloui, L., Hassan, G.F.: Hypercomplex derivative bases of polynomials in Clifford analysis. Math. Meth. Appl. Sci. 33, 350–357 (2010)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Aloui, L., Abul-Ez, M.A., Hassan, G.F.: On the order of the difference and sum bases of polynomials in Clifford setting, complex variables and elliptic equations. An Int. J. 55, 1117–1130 (2010)zbMATHGoogle Scholar
  14. 14.
    Aloui, L., Abul-Ez, M.A., Hassan, G.F.: Bernoulli special monogenic polynomials with the difference and sum polynomial bases. Complex Variables and Elliptic Equ. 59, 631–650 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Common, A.K., Sommen, F.: Axial monogenic functions from holomorphic functions. J. Math. Anal. Appl. 179, 610–629 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Cação, I., Falcão, M.I., Malonek, H.: Laguerre derivative and monogenic Leguerre polynomials: an operational approach. Math. Comput. Model. 53, 1084–1094 (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    Cannon, B.: On the convergence of series of polynomials. Proc. Lond. Math. Soc. 43, 364 (1937)MathSciNetGoogle Scholar
  18. 18.
    Cannon, B.: On the convergence of series of polynomials. Proc. Lond. Math. Soc. 43, 348–365 (1938)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Cannon, B.: On convergence properties of basic series. J. Lond. Math. Soc. 14, 51–62 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Cannon, B.: On the representation of integral functions by general basic series. Math. Zeit. 45, 185–208 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Cnops, J., Abul-Ez, M.: Basis transforms in nuclear Frechet spaces. Simon Stevin. 67, 145–156 (1993)MathSciNetzbMATHGoogle Scholar
  22. 22.
    De Schepper, N.: The generalized Clifford–Gegenbauer polynomials revisited. Adv. Appl. Clifford Alg. 19, 253–268 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Eweida, M.T.: On the effectiveness at a point of product and reciprocal sets of polynomials. Lond. Math. Soc. Ser. 2(51), 81–89 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Falcão, M., Malonek, H.: Generalized exponentials through Appell sets in \({\mathbb{R}}^{n+1}\) and Bessel functions.In: AIP-Proceedings, 738–741 (2007)Google Scholar
  25. 25.
    Falcão, M., Malonek, H.: Special monogenic polynomials - properties and applications. In: AlP-Proceedings, 764–767 (2007)Google Scholar
  26. 26.
    Gürlebeck, N.: On appell sets and the Fueter-Sce mapping. Adv. Appl. Clifford Alg. 19, 51–61 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Halim, E.: On the effectiveness in a closed circle of simple sets of polynomials and associated sets, proc. Math. Phys. Soc. Egypt. 5, 31–39 (1953)MathSciNetGoogle Scholar
  28. 28.
    Hassan, G.F.: A note on the growth order of the inverse and product bases of special monogenic polynomials. Math. Meth. Appl. Sci. 35, 286–292 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hassan, G.F., Aloui, L.: Bernoulli and Euler polynomials in Clifford analysis. Adv. Appl. Clifford Alg. 25, 351–376 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hassan, G. F., Aloui, L., Bakali, A.: Basic sets of special monogenic polynomials in Frchet modules, J. Complex Anal., Article ID 2075938, 11 pages, (2017). doi: 10.1155/2017/2075938
  31. 31.
    Lounesto, P., Bergh, P.: Axially symmetric vector fields and their complex potentials. Complex Variables 2, 139–150 (1983)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Malonek HR, H., Falcao, M.: Special monogenic polynomials. properties and applications, In: Simos TE, Psihoyios G, Tsitouras Ch (eds) Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, vol. 936. American Institute of Physics: Melville, 764–767 (2007)Google Scholar
  33. 33.
    Mikhail, M.N.: Basic sets of polynomials and their reciprocal, product and quotient sets. Duke Math. J. 20, 459–480 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Mikhail, M.N.: Simple basic sets of polynomials. Am. J. Math. 67, 647–653 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Mursi, M., Makar, R.: Sur la base inverse \(\text{d}^{^{\prime }}\text{ une }\) base de polynomes. Bullet. des Sc. Math. \(2^{e}\) 71, 47–51 (1947)Google Scholar
  36. 36.
    Newns, W.F.: On the representation of analytic functions by infinite series, philosophical transactions of the Royal Society of London. Ser. A. Math. Phys. Sci. 245, 429–468 (1953)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Nassif, M.: On the effectiveness at the origin of a product and inverse sets of polynomials. J. Lond. Math. Soc. 26, 232–238 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sommen, F.: Special functions in Clifford analysis and axial symmetry. J. Math. Anal. Appl. 130, 110–133 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sommen, F.: Plane elliptic systems and monogenic functions in symmetric domains. Suppl. Rend. Circ. Mat. Palermo. 6, 259–269 (1984)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Saleem, M.A., Abul-Ez, M., Zayed, M.: On polynomial series expansions of Cliffordian functions. Math. Meth. Appl. Sci. 35, 134–143 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Tantawi, A.: On a particular case of the multiplication and inversion of basic sets. Proc. Math. Phys. Soc. Egypt., 4, (1950)Google Scholar
  42. 42.
    Whittaker, J.: Sur les séries de base de polynômes quelconques. Avec la col laboration de C. Gattegno. (Collection de monographies sur la theorie des fonctions) Paris: Gauthier-Villars. VI., (1949)Google Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2017

Authors and Affiliations

  • Mohamed Abdalla
    • 1
    Email author
  • Mahmoud Abul-Ez
    • 2
  • Aida Al-Ahmadi
    • 3
  1. 1.Department of Mathematics, Faculty of ScienceSouth Valley UniversityQenaEgypt
  2. 2.Department of Mathematics, Faculty of ScienceSohag UniversitySohagEgypt
  3. 3.Department of Mathematics, Faculty of Education for GirlsTabuk UniversityTabukSaudi Arabia

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