Extended Forms of Certain Hybrid Special Polynomials Related to Appell Sequences

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Abstract

The use of integral transforms and operational methods is a fairly useful tool to deal with new families of special polynomials. In this article, the extended Laguerre–Gould–Hopper–Appell polynomials are introduced by means of generating function and determinant definition. Their quasi-monomial properties are also established. Examples of some members belonging to the extended Laguerre–Gould–Hopper–Appell polynomials are considered, and their contour and 3-D plots are drawn.

Keywords

Appell sequences Euler’s integral Operational methods 

Mathematics Subject Classification

26A33 33B10 33C45 

Notes

Acknowledgements

The authors are thankful to the Reviewer(s) for several useful comments and suggestions towards the improvement of this paper.

References

  1. 1.
    Aceto, L., Malonek, H.R., Tomaz, G.: A unified matrix approach to the representation of Appell polynomials. Integral Transforms Spec. Funct. 26(6), 426–441 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Al-Gonah, Ahmed Ali: Some results involving special polynomials and integral transforms. Palest. J. Math. 4(2), 327–334 (2015)MathSciNetMATHGoogle Scholar
  3. 3.
    Appell, P.: Sur une classe de polynômes. Ann. Sci. École. Norm. Sup. 9(2), 119–144 (1880)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Anshelevich, M.: Appell polynomials and their relatives III. Conditionally free theory. Ill. J. Math. 53(1), 39–66 (2009)MathSciNetMATHGoogle Scholar
  5. 5.
    Artioli, M., Dattoli, G.: The Geometry of Hermite Polynomials. http://demonstrations.wolfram.com/TheGeometryOfHermitePolynomials/. Wolfram Demonstrations Project Published, March 4 (2015)
  6. 6.
    Brackx, F., De Schepper, H., Lávička, R., Soucek, V.: Gel’fand-Tsetlin procedure for the construction of orthogonal bases in Hermitean Clifford analysis. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) Numerical Analysis and Applied Mathematics-ICNAAM 2010, AIP Conference Proceedings (1281), pp. 1508–1511. Melville, American Institute of Physics (AIP) (2010)Google Scholar
  7. 7.
    Cesarano, C.: Generalized Chebyshev polynomials. Hacet. J. Math. Stat. 43(5), 731–740 (2014)MathSciNetMATHGoogle Scholar
  8. 8.
    Cesarano, C.: Integral representations and new generating functions of Chebyshev polynomials. Hacet. J. Math. Stat. 44(3), 535–546 (2015)MathSciNetMATHGoogle Scholar
  9. 9.
    Cesarano, C.: Operational methods and new identities for hermite polynomials. Math. Model. Nat. Phenom. 12(3), 44–50 (2017)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Costabile, F.A., Dell’Accio, F., Gualtieri, M.I.: A new approach to Bernoulli polynomials. Rend. Mat. Appl. 26(1), 1–12 (2006)MathSciNetMATHGoogle Scholar
  11. 11.
    Costabile, F.A., Longo, E.: A determinantal approach to Appell polynomials. J. Comput. Appl. Math. 234(5), 1528–1542 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Dattoli, G.: Hermite-Bessel and Laguerre–Bessel functions: a by-product of the monomiality principle. Advanced special functions and applications (Melfi, 1999). In: Proceedings of Melfi School Advanced Topics in Mathematics and Physics, Aracne, Rome, vol. 1, pp. 147–164 (2000)Google Scholar
  13. 13.
    Dattoli, G., Ricci, P.E., Cesarano, C., Vázquez, L.: Special polynomials and fractional calculas. Math. Comput. Model. 37, 729–733 (2003)CrossRefMATHGoogle Scholar
  14. 14.
    Dattoli, G., Ottaviani, P.L., Torre, A., Vázquez, L.: Evolution operator equations: integration with algebraic and finite-difference methods. Applications to physical problems in classical and quantum mechanics and quantum field theory. Riv. Nuovo Cimento Soc. Ital. Fis. 20(2), 1–133 (1997)MathSciNetGoogle Scholar
  15. 15.
    Khan, Subuhi, Al-Gonah, Ahmed Ali: Operational methods and Laguerre–Gould Hopper polynomials. Appl. Math. Comput. 218, 9930–9942 (2012)MathSciNetMATHGoogle Scholar
  16. 16.
    Khan, S., Al-Saad, M.W., Khan, R.: Laguerre-based Appell polynomials: properties and applications. Math. Comput. Model. 52(1–2), 247–259 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Khan, S., Raza, N.: General-Appell polynomials within the context of monomiality principle. Int. J. Anal. 1–11 (2013)Google Scholar
  18. 18.
    Khan, Subuhi, Raza, Nusrat, Ali, Mahvish: Finding mixed families of special polynomials associated with Appell sequences. J. Math. Anal. Appl. 447, 398–418 (2017)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Lávička, R.: Canonical bases for sl(2, C)-modules of spherical monogenics in dimension 3. Arch. Math. Tomus. 46, 339–349 (2010)MathSciNetMATHGoogle Scholar
  20. 20.
    Malonek, H.R., Falcão, M.I.: 3D-mappings using monogenic functions. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) Numerical Analysis and Applied Mathematics-ICNAAM 2006, pp. 615–619. Wiley, Weinheim (2006)Google Scholar
  21. 21.
    Salminen, P.: Optimal stopping, Appell polynomials and Wiener–Hopf factorization. Int. J. Probab. Stoch. Process. 83(4–6), 611–622 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press, New York (1984)MATHGoogle Scholar
  23. 23.
    Steffensen, J.F.: The poweroid, an extension of the mathematical notion of power. Acta Math. 73, 333–366 (1941)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Weinberg, S.T.: The quantum theory of fields, vol. 1. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  25. 25.
    Yang, Y., Youn, H.: Appell polynomial sequences: a linear algebra approach. JP J. Algebra Number Theory Appl. 13(1), 65–98 (2009)MathSciNetMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.Department of MathematicsAligarh Muslim UniversityAligarhIndia

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