Advertisement

A family of complex Appell polynomial sets

  • H. M. SrivastavaEmail author
  • Paolo Emilio Ricci
  • Pierpaolo Natalini
Original Paper
  • 19 Downloads

Abstract

In the present sequel to a recent work by Srivastava et al. (Rocky Mt J Math 49 (in press), 2019), the authors propose to show that the real and imaginary parts of a general set of complex Appell polynomials can be represented in terms of the Chebyshev polynomials of the first and second kind. Furthermore, by applying a general technique based upon the monomiality principle and quasi-monomial sets [see, for details, Ben Cheikh (Appl Math Comput 141:63–76, 2003), Dattoli (in: Cocolicchio, Dattoli, Srivastava (eds), Advanced special functions and applications (Proceedings of the Melfi School on advanced topics in mathematics and physics; Melfi, May 9–12), Aracne Editrice, Rome, 2000) and Steffensen (Acta Math 73:333–366, 1941)], the differential equations satisfied by the Bernoulli, Euler and Genocchi polynomials are derived.

Keywords

Appell polynomials Sheffer polynomials Shift operators Boas–Buck polynomial set Complex Appell polynomials Complex Euler polynomials Complex Genocchi polynomials Monomiality principle and quasi-monomial sets Generating functions Differential equations Chebyshev polynomials of the first and second kind Taylor series expansions 

Mathematics Subject Classification

Primary 11B83 33D99 Secondary 26C05 

Notes

References

  1. 1.
    Ben Cheikh, Y.: Some results on quasi-monomiality. Appl. Math. Comput. 141, 63–76 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Boas, R.P., Buck, R.C.: Polynomials defined by generating relations. Am. Math. Mon. 63, 626–632 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Boas, R.P., Buck, R.C.: Polynomial Expansions of Analytic Functions. Springer, Berlin (1958)CrossRefzbMATHGoogle Scholar
  4. 4.
    Brenke, W.C.: On generating functions of polynomial systems. Am. Math. Mon. 52, 297–301 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bretti, G., Natalini, P., Ricci, P.E.: New sets of Euler-type polynomials. J. Anal. Number Theory 6(2), 51–54 (2018)CrossRefGoogle Scholar
  6. 6.
    Dattoli, G.: Hermite–Bessel and Laguerre–Bessel functions: a by-product of the monomiality principle. In: Cocolicchio, D., Dattoli, G., Srivastava, H.M. (eds) Advanced Special Functions and Applications (Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics; Melfi, May 9–12, 1999), pp. 147–164. Aracne Editrice, Rome (2000)Google Scholar
  7. 7.
    Dattoli, G., Ricci, P. E., Srivastava, H. M. (Editors).: Advanced Special Functions and Related Topics in Probability and in Differential Equations, Proceedings of the Melfi School on Advanced Topics in Mathematics and Physics (Melfi; June 24–29, 2001), Appl. Math. Comput. 141(1), 1–230 (2003)Google Scholar
  8. 8.
    Lu, D.-Q., Srivastava, H.M.: Some series identities involving the generalized Apostol type and related polynomials. Comput. Math. Appl. 62, 3591–3602 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Luo, Q.-M., Srivastava, H.M.: Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials. J. Math. Anal. Appl. 308, 290–302 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Luo, Q.-M., Srivastava, H.M.: Some relationships between the Apostol–Bernoulli and Apostol–Euler polynomials. Comput. Math. Appl. 51, 631–642 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Luo, Q.-M., Srivastava, H.M.: Some generalizations of the Apostol–Genocchi polynomials and the Stirling numbers of the second kind. Appl. Math. Comput. 217, 5702–5728 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Masjed-Jamei, M., Koepf, W.: Symbolic computation of some power-trigonometric series. J. Symbol. Comput. 80, 273–284 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ricci, P.E., Bretti, G., Natalini, P.: New sets of Hahn-type polynomials. J. Anal. (2018) (submitted)Google Scholar
  14. 14.
    Roman, S.M.: The Umbral Calculus. Academic Press, New York (1984)zbMATHGoogle Scholar
  15. 15.
    Sheffer, I.M.: Some properties of polynomials sets of zero type. Duke Math. J. 5, 590–622 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press (Ellis Horwood Limited), Chichester (1984)zbMATHGoogle Scholar
  17. 17.
    Srivastava, H.M., Masjed-Jamei, M., Beyki, M.R.: Some new generalizations and applications of the Apostol–Bernoulli, Apostol–Euler and Apostol–Genocchi polynomials. Rocky Mt. J. Math. 49 (in press) (2019)Google Scholar
  18. 18.
    Steffensen, J.F.: The poweroid, an extension of the mathematical notion of power. Acta Math. 73, 333–366 (1941)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Royal Academy of Sciences, Madrid 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.Department of Medical Research, China Medical University HospitalChina Medical UniversityTaichungTaiwan, ROC
  3. 3.Dipartimento di MatematicaInternational Telematic University UniNettunoRomeItaly
  4. 4.Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreRomeItaly

Personalised recommendations