Advertisement

Arabian Journal of Mathematics

, Volume 8, Issue 1, pp 63–77 | Cite as

q-Difference equations for the 2-iterated q-Appell and mixed type q-Appell polynomials

  • H. M. Srivastava
  • Subuhi Khan
  • Mumtaz RiyasatEmail author
Open Access
Article
  • 144 Downloads

Abstract

In this article, the authors establish the recurrence relations and q-difference equations for the 2-iterated q-Appell polynomials. The recurrence relations and the q-difference equations for the 2-iterated q-Bernoulli polynomials, the q-Euler polynomials and the q-Genocchi polynomials are also derived. An analogous study of certain mixed type q-special polynomials is also presented.

Mathematics Subject Classification

33D45 33D99 33E20 

Notes

References

  1. 1.
    Al-Salam, W.A.: \(q\)-Appell polynomials. Ann. Mat. Pura Appl. 4(77), 31–45 (1967)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Al-Salam, W.A.: \(q\)-Bernoulli numbers and polynomials. Math. Nachr. 17, 239–260 (1959)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andrews, G.E.; Askey, R.; Roy, R.: Special Functions of Encyclopedia. Mathematics and its Applications. Cambridge University Press, Cambridge (1999)Google Scholar
  4. 4.
    Appell, P.: Sur une classe de polynômes. Ann. Sci. École Norm. Sup. 9(2), 119–144 (1880)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cheikh, Y.B.; Douak, K.: On the classical \(d\)-orthogonal polynomials defined by certain generating functions, I. Bull. Belg. Math. Soc. Simon Stevin 7, 107–124 (2000)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Costabile, F.A.; Longo, E.: A determinantal approach to Appell polynomials. J. Comput. Appl. Math. 234(5), 1528–1542 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Douak, K.: The relation of the \(d\)-orthogonal polynomials to the Appell polynomials. J. Comput. Appl. Math. 70(2), 279–295 (1996)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ernst, T.: \(q\)-Bernoulli and \(q\)-Euler polynomials, an umbral approach. Int. J. Differ. Equ. 1(1), 31–80 (2006). (0973-6069)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ernst, T.: A Comprehensive Treatment of \(q\)-Calculus, p. xvi + 491. Birkhäuser/Springer Basel AG, Basel (2012)CrossRefGoogle Scholar
  10. 10.
    He, M.X.; Ricci, P.E.: Differential equation of Appell polynomials via the factorization method. J. Comput. Appl. Math. 139, 231–237 (2002)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Keleshteri, M.E.; Mahmudov, N.I.: A \(q\)-umbral approach to \(q\)-Appell polynomials. arXiv:1505.05067
  12. 12.
    Keleshteri, M.E.; Mahmudov, N.I.: On the class of \(2 D ~q\)-Appell polynomials. arXiv:1512.03255v1
  13. 13.
    Khan, S.; Al-Saad, M.W.M.; Khan, R.: Laguerre-based Appell polynomials: properties and applications. Math. Comput. Model. 52(1–2), 247–259 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Khan, S.; Raza, N.: 2-Iterated Appell polynomials and related numbers. Appl. Math. Comput. 219(17), 9469–9483 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Khan, S.; Raza, N.: General-Appell polynomials within the context of monomiality principle. Int. J. Anal. 2013 (2013).  https://doi.org/10.1155/2013/328032
  16. 16.
    Khan, S.; Riyasat, M.: Differential and integral equations for the 2-iterated Appell polynomials. J. Comput. Appl. Math. 306, 116–132 (2016)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Khan, S.; Riyasat, M.: Determinant approach to the 2-Iterated \(q\)-Appell and mixed type \(q\)-Appell polynomials. arXiv:1606.04265
  18. 18.
    Khan, S.; Yasmin, G.; Khan, R.; Hassan, N.A.M.: Hermite-based Appell polynomials: properties and applications. J. Math. Anal. Appl. 351(2), 756–764 (2009)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Mahmudov, N.I.: On a class of \(q\)-Bernoulli and \(q\)-Euler polynomials. Adv. Differ. Equ. 108, 11 (2013)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Mahmudov, N.I.: Difference equations of \(q\)-Appell polynomials. Appl. Math. Comput. 245, 539–543 (2014)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Mahmudov, N.I.; Keleshteri, M.E.: On a class of generalized \(q\)-Bernoulli and \(q\)-Euler polynomials. Adv. Differ. Equ. 115, 1–10 (2013)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mahmudov, N.I.; Keleshteri, M.E.: \(q\)-Extensions for the Apostol type polynomials. J. Appl. Math. 2014 (2014).  https://doi.org/10.1155/2014/868167
  23. 23.
    Maroni, P.: L’orthogonalité et les récurrences de polynômes d’ordre supérieur á deux. Ann. Fac. Sci. Toulouse 10, 105–139 (1989)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Özarslan, M.A.: Hermite-based unified Apostol-Bernoulli, Euler and Genocchi polynomials. Adv. Differ. Equ. 116, 1–13 (2013)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Özarslan, M.A.; Yilmaz, B.: A set of finite order differential equations for the Appell polynomials. J. Comput. Appl. Math. 259, 108–116 (2014)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Roman, S.: The theory of the umbral calculus I. J. Math. Anal. Appl. 87, 58–115 (1982)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Roman, S.: The Umbral Calculus. Academic Press, New York (1984)zbMATHGoogle Scholar
  28. 28.
    Roman, S.: More on the umbral calculus, with emphasis on the \(q\)-umbral calculus. J. Math. Anal. Appl. 107, 222–254 (1985)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Srivastava, H.M.: Some characterizations of Appell and \(q\)-Appell polynomials. Ann. Mat. Pura Appl. 4(130), 321–329 (1982)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Srivastava, H.M.: Some generalizations and basic (or \(q\)-) extensions of the Bernoulli, Euler and Genocchi polynomials. Appl. Math. Inform. Sci. 5, 390–444 (2011)MathSciNetGoogle Scholar
  31. 31.
    Srivastava, H.M.; Choi, J.: Zeta and \(q\)-Zeta Functions and Associated Series and Integrals. Elsevier Science Publishers, Amsterdam (2012)zbMATHGoogle Scholar
  32. 32.
    Srivastava, H.M.; Özarslan, M.A.; Yilmaz, B.: Some families of differential equations associated with the Hermite-based Appell polynomials and other classes of Hermite-based polynomials. Filomat 28(4), 695–708 (2014)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zaghouani, A.: On some basic \(d\)-orthogonal polynomial sets. Georgian Math. J. 12, 583–593 (2005)MathSciNetzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • H. M. Srivastava
    • 1
    • 2
  • Subuhi Khan
    • 3
  • Mumtaz Riyasat
    • 3
    Email author
  1. 1.Department of Mathematics and StatisticsUniversity of VictoriaVictoriaCanada
  2. 2.China Medical UniversityTaichungTaiwan, ROC
  3. 3.Department of MathematicsAligarh Muslim UniversityAligarhIndia

Personalised recommendations