Matrix Approach to Frobenius-Euler Polynomials

  • Graça Tomaz
  • Helmuth R. Malonek
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8579)


In the last two years Frobenius-Euler polynomials have gained renewed interest and were studied by several authors. This paper presents a novel approach to these polynomials by treating them as Appell polynomials. This allows to apply an elementary matrix representation based on a nilpotent creation matrix for proving some of the main properties of Frobenius-Euler polynomials in a straightforward way.


Appell polynomials Frobenius-Euler polynomials matrix representation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, New York (1972)Google Scholar
  2. 2.
    Aceto, L., Trigiante, D.: The Matrices of Pascal and Other Greats. Amer. Math. Monthly 108(3), 232–245 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Appell, P.: Sur une classe de polynômes. Ann. Sci. École Norm. Supér. 9, 119–144 (1880)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Araci, S., Acikgoz, M.: A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials. Adv. Stud. Contemp. Math. 22(3), 399–406 (2012)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Avram, F., Taqqu, M.S.: Noncentral Limit Theorems and Appell Polynomial. Ann. Probab. 15(2), 767–775 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Boas, R.P., Buck, R.C.: Polynomial expansions of analytic functions. Springer, Berlin (1964)CrossRefzbMATHGoogle Scholar
  7. 7.
    Call, G.S., Velleman, J.: Pascal’s Matrices. Amer. Math. Monthly 100, 372–376 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Carlitz, L.: The product of two eulerian polynomials. Math. Magazine. 36(1), 37–41 (1963)CrossRefzbMATHGoogle Scholar
  9. 9.
    Carlson, B.C.: Polynomials Satisfying a Binomial Theorem. J. Math. Anal. Appl. 32, 543–558 (1970)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chen, S., Cai, Y., Luo, Q.-M.: An extension of generalized Apostol-Euler polynomials. Adv. Difference Equ. 61 (2013), doi:10.1186/1687-1847-2013-61Google Scholar
  11. 11.
    Choi, J., Kim, D.S., Kim, T., Kim, Y.H.: A Note on Some Identities of Frobenius-Euler Numbers and Polynomials. Int. J. Math. Math. Sci (2012), doi:10.115/2012/861797Google Scholar
  12. 12.
    Costabile, F.A., Longo, E.: The Appell interpolation problem. J. Comp. Appl. Math. 236, 1024–1032 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Fairlie, D.B., Veselov, A.P.: Faulhaber and Bernoulli polynomials and solitons. Physica D 152 153, 47–50 (2000)Google Scholar
  14. 14.
    Frobenius, G.: Uber die Bernoulli’schen Zahlen und die Euler’schen Polynome. Sitzungsberichte der Preussischen Akademie der Wissenschaften, 809–847 (1910)Google Scholar
  15. 15.
    Grosset, M.P., Veselov, A.P.: Elliptic Faulhaber polynomials and Lamé densities of states. Int. Math. Res. Not., Article ID 62120, 31 (2006), doi: 10.1155/IMRN/2006/62120 Google Scholar
  16. 16.
    Khan, S., Raza, N.: 2-iterated Appell polynomials and related numbers. Appl. Math. Comput. 219, 9469–9483 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Kim, D.S., Kim, T.: Some new identities of Frobenius-Euler numbers and polynomials. J. Inequal. Appl. 307 (2012), doi:10.1186/1029-242X-2012-307Google Scholar
  18. 18.
    Kim, D.S., Kim, T., Lee, S.-H., Rim, S.-H.: A note on the higher-order Frobenius-Euler polynomials and Sheffer sequences. Adv. Difference Equ. 41 (2013), doi:10.1186/1687-1847-2013-41Google Scholar
  19. 19.
    Kurt, B., Simsek, Y.: On the generalized Apostol-type Frobenius-Euler polynomials. Adv. Difference Equ. 2013, 1 (2013), doi:10.1186/1687-1847-2013-1CrossRefMathSciNetGoogle Scholar
  20. 20.
    Malonek, H., Tomaz, G.: Laguerre polynomials in several hypercomplex variables and their matrix representation. In: Murgante, B., Gervasi, O., Iglesias, A., Taniar, D., Apduhan, B.O. (eds.) ICCSA 2011, Part III. LNCS, vol. 6784, pp. 261–270. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Srivastava, H.M., Pintér, Á.: Remarks on Some Relationships Between the Bernoulli and Euler Polynomials. Appl. Math. Lett. 17, 375–380 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Tomaz, G., Malonek, H.R.: Special Block Matrices and Multivariate Polynomials. In: Simos, T.E., Psihoyios, G., Tsitouras, C. (eds.) Numerical Analysis and Applied Mathematics (ICNAAM 2010). AIP Conference Proceedings, Melville, New York, vol. 1281, pp. 1515–1518 (2010)Google Scholar
  23. 23.
    Tomaz, G.: Polinómios de Appell multidimensionais e sua representação matricial. PhD-Thesis, University of Aveiro (2012)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Graça Tomaz
    • 1
    • 2
  • Helmuth R. Malonek
    • 2
  1. 1.Instituto Politécnico da GuardaGuardaPortugal
  2. 2.Centro de Investigação e Desenvolvimento em Matemática e AplicaçõesUniversidade de AveiroAveiroPortugal

Personalised recommendations