Table of Contents Author Guidelines Submit a Manuscript
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 343981, 15 pages
http://dx.doi.org/10.1155/2012/343981
Research Article

Euler Basis, Identities, and Their Applications

1Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 11 June 2012; Accepted 9 August 2012

Academic Editor: Yilmaz Simsek

Copyright © 2012 D. S. Kim and T. Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. Araci, D. Erdal, and J. J. Seo, β€œA study on the fermionic p-adic q-integral representation on p associated with weighted q-Bernstein and q-Genocchi polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 649248, 10 pages, 2011. View at Publisher Β· View at Google Scholar
  2. A. Bayad, β€œModular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010. View at Google Scholar
  3. A. Bayad and T. Kim, β€œIdentities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials,” Russian Journal of Mathematical Physics, vol. 18, no. 2, pp. 133–143, 2011. View at Publisher Β· View at Google Scholar
  4. L. Carlitz, β€œNote on the integral of the product of several Bernoulli polynomials,” Journal of the London Mathematical Society, vol. 34, pp. 361–363, 1959. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. L. Carlitz, β€œMultiplication formulas for products of Bernoulli and Euler polynomials,” Pacific Journal of Mathematics, vol. 9, pp. 661–666, 1959. View at Google Scholar
  6. L. Carlitz, β€œArithmetic properties of generalized Bernoulli numbers,” Journal für die Reine und Angewandte Mathematik, vol. 202, pp. 174–182, 1959. View at Google Scholar Β· View at Zentralblatt MATH
  7. N. S. Jung, H. Y. Lee, and C. S. Ryoo, β€œSome relations between twisted (h,q)-Euler numbers with weight α and q-Bernstein polynomials with weight α,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 176296, 11 pages, 2011. View at Publisher Β· View at Google Scholar
  8. D. S. Kim, β€œIdentities of symmetry for q-Euler polynomials,” Open Journal of Discrete Mathematics, vol. 1, no. 1, pp. 22–31, 2011. View at Publisher Β· View at Google Scholar
  9. D. S. Kim, β€œIdentities of symmetry for generalized Euler polynomials,” International Journal of Combinatorics, vol. 2011, Article ID 432738, 12 pages, 2011. View at Publisher Β· View at Google Scholar
  10. T. Kim, β€œOn the weighted q-Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 207–215, 2011. View at Google Scholar
  11. T. Kim, β€œSymmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on p,” Russian Journal of Mathematical Physics, vol. 16, no. 1, pp. 93–96, 2009. View at Publisher Β· View at Google Scholar
  12. T. Kim, β€œSome identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on p,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491, 2009. View at Publisher Β· View at Google Scholar
  13. B. Kurt and Y. Simsek, β€œNotes on generalization of the Bernoulli type polynomials,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 906–911, 2011. View at Publisher Β· View at Google Scholar
  14. H. Y. Lee, N. S. Jung, and C. S. Ryoo, β€œA note on the q-Euler numbers and polynomials with weak weight α,” Journal of Applied Mathematics, vol. 2011, Article ID 497409, 14 pages, 2011. View at Publisher Β· View at Google Scholar
  15. H. Ozden, β€œp-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials,” Applied Mathematics and Computation, vol. 218, no. 3, pp. 970–973, 2011. View at Publisher Β· View at Google Scholar
  16. H. Ozden, I. N. Cangul, and Y. Simsek, β€œOn the behavior of two variable twisted p-adic Euler q-l-functions,” Nonlinear Analysis, vol. 71, no. 12, pp. e942–e951, 2009. View at Publisher Β· View at Google Scholar Β· View at Scopus
  17. S.-H. Rim, A. Bayad, E.-J. Moon, J.-H. Jin, and S.-J. Lee, β€œA new construction on the q-Bernoulli polynomials,” Advances in Difference Equations, vol. 2011, article 34, 2011. View at Publisher Β· View at Google Scholar
  18. C. S. Ryoo, β€œSome relations between twisted q-Euler numbers and Bernstein polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, no. 2, pp. 217–223, 2011. View at Google Scholar
  19. Y. Simsek, β€œComplete sum of products of (h,q)-extension of Euler polynomials and numbers,” Journal of Difference Equations and Applications, vol. 16, no. 11, pp. 1331–1348, 2010. View at Publisher Β· View at Google Scholar
  20. Y. Simsek, β€œGenerating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–278, 2008. View at Google Scholar Β· View at Zentralblatt MATH