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Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials

  • A. Bayad
  • T. Kim
Article

Abstract

In this paper, we give relations involving values of q-Bernoulli, q-Euler, and Bernstein polynomials. Using these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler, and Bernstein polynomials.

Keywords

Bernstein Polynomial Usual Convention Interesting Identity Stein Polynomial Araci 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    M. Acikgoz and S. Araci, “A Study on the Integral of the Product of Several Type Berstein Polynomials,” IST Trans. Appl. Math.-Model. Simulat., 11 (2), 10–14 (2010).Google Scholar
  2. 2.
    A. Bayad, T. Kim, B. Lee, and S.-H. Rim, “Some Identities on the Bernstein Polynomials Associated with q-Euler Polynomials” (Communicated).Google Scholar
  3. 3.
    S. Bernstein, “Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités,” Commun. Soc. Math. Kharkow 13, 1–2 (1912).Google Scholar
  4. 4.
    I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the Higher-Order w-q-Genocchi Numbers,” Adv. Stud. Contemp. Math. 19, 39–57 (2009).MathSciNetGoogle Scholar
  5. 5.
    H. W. Gould, “A Theorem Concerning the Bernstein Polynomials,” Math. Mag., 31(5), 259–264 (1958).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    N. K. Govil and V. Gupta, “Convergence of q-Meyer-König-Zeller-Durrmeyer Operators,” Adv. Stud. Contemp. Math. 19, 97–108 (2009).MathSciNetMATHGoogle Scholar
  7. 7.
    L.C. Jang, W-J. Kim, and Y. Simsek, “A Study on the p-Adic Integral Representation on ℤp Associated with Bernstein and Bernoulli Polynomials,” Adv. Difference Equ., (2010), Article ID 163219, 6pp.Google Scholar
  8. 8.
    K. I. Joy, “Bernstein Polynomials,” On-Line Geometric Modelling Notes, http://en.wikipedia.org/wiki/Bernsteinpolynomials.
  9. 9.
    T. Kim, “Barnes-Type Multiple q-Zeta Functions and q-Euler Polynomials,” J. Phys. A: Math. Theor. 43 255201, (2010), 11p.ADSCrossRefGoogle Scholar
  10. 10.
    T. Kim, “An Analogue of Bernoulli Numbers and Their Congruences,” Rep. Fac. Sci. Engrg. Saga Univ. Math. 22(2), 21–26 (1994).MathSciNetMATHGoogle Scholar
  11. 11.
    T. Kim, “On the Symmetries of the q-Bernoulli Polynomials,” Abstr. Appl. Anal., (2008), Article ID 914367, 7pp.Google Scholar
  12. 12.
    T. Kim, “A New Approach to p-Adic q-L-Functions,” Adv. Stud. Contemp. Math. 12(1), 61–72 (2006).MathSciNetMATHGoogle Scholar
  13. 13.
    T. Kim, “q-Bernoulli Numbers and Polynomials Associated with Gaussian Binomial Coefficients,” Russ. J. Math. Phys. 15, 51–59 (2008).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    T. Kim, J. Choi, and Y. H. Kim, “q-Bernstein Polynomials Associated with q-Stirling Numbers and Carlitz’s q-Bernoulli Numbers,” Absr. Appl. Anal. (2010), Article ID 150975, 11.Google Scholar
  15. 15.
    T. Kim, “Some Identities on the q-Euler Polynomials of Higner Order and q-Stirling Numbers by Fermionic p-Adic Integral on ℤp,” Russ. J. Math. Phys. 16, 484–491 (2009).MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    T. Kim, “Note on the Euler q-Zeta Functions,” J. Number Theory 129, 1798–1804 (2009).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    T. Kim and B. Lee, “Some Identities of the Frobenius-Euler Polynomials,” Abstr. Appl. Anal. (2009), Article ID 639439.Google Scholar
  18. 18.
    T. Kim, J. Choi, Y.H. Kim, and C. S. Ryoo, “On the Fermionic p-Adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials,” J. Inequal. and Appl. (2010), Article ID 864249, 12.Google Scholar
  19. 19.
    B. A. Kupershmidt, “Reflection Symmetries of q-Bernoulli Polynomials,” J. Nonlinear Math. Phys. 12, 412–422 (2005).MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    G. G. Lorentz, Bernstein Polynomials (2nd edition) (printed in USA 1986).Google Scholar
  21. 21.
    S-H. Rim, J-H. Jin, E-J. Moon, and S-J. Lee, “On Multiple Interpolation Functions of the q-Genocchi Polynomials,” J. Inequal. Appl., (2010), Article ID 351419, 13.Google Scholar
  22. 22.
    S. Zorlu, H. Aktuglu, and M. A. Ozarslan, “An Estimation to the Solution of an Initial Value Problem via q-Bernstein Polynomials,” J. Comput. Anal. Appl. 12(3), 637–645 (2010).MathSciNetMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2011

Authors and Affiliations

  1. 1.Département de mathématiques Université d’Evry Val d’EssonneEvry CedexFrance
  2. 2.Kwangwoon UniversitySeoulRepublic of Korea

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