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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 674210, 10 pages
http://dx.doi.org/10.1155/2012/674210
Research Article

Identities Involving -Bernoulli and -Euler Numbers

1Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
2Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
3Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Received 18 January 2012; Revised 24 February 2012; Accepted 24 February 2012

Academic Editor: Toka Diagana

Copyright © 2012 D. S. Kim et al. 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. A. Bayad and T. Kim, “Identities 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 · View at Google Scholar
  2. A. Bayad, “Modular 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, T. Kim, B. Lee, and S.-H. Rim, “Some identities on Bernstein polynomials associated with q-Euler polynomials,” Abstract and Applied Analysis, vol. 2011, Article ID 294715, 10 pages, 2011. View at Publisher · View at Google Scholar
  4. L. Carlitz, “The product of two Eulerian polynomials,” Mathematics Magazine, vol. 36, no. 1, pp. 37–41, 1963. View at Google Scholar
  5. L. C. Jang, “A note on Nörlund-type twisted q-Euler polynomials and numbers of higher order associated with fermionic invariant q-integrals,” Journal of Inequalities and Applications, vol. 2010, Article ID 417452, 12 pages, 2010. View at Publisher · View at Google Scholar
  6. T. Kim, “Some 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 · View at Google Scholar
  7. T. Kim, “An analogue of Bernoulli numbers and their congruences,” Reports of the Faculty of Science and Engineering. Saga University. Mathematics, vol. 22, no. 2, pp. 21–26, 1994. View at Google Scholar
  8. T. Kim, “New approach to q-Euler polynomials of higher order,” Russian Journal of Mathematical Physics, vol. 17, no. 2, pp. 218–225, 2010. View at Publisher · View at Google Scholar
  9. Y.-H. Kim, K.-W. Hwang, and T. Kim, “Interpolation functions of the q-Genocchi and the q-Euler polynomials of higher order,” Journal of Computational Analysis and Applications, vol. 12, no. 1-B, pp. 228–238, 2010. View at Google Scholar
  10. H. Ozden and Y. Simsek, “A new extension of q-Euler numbers and polynomials related to their interpolation functions,” Applied Mathematics Letters, vol. 21, no. 9, pp. 934–939, 2008. View at Publisher · View at Google Scholar
  11. H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-order q-Euler numbers and their applications,” Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008. View at Google Scholar
  12. K.-H. Park, Y.-H. Kim, and T. Kim, “A note on the generalized q-Euler numbers (2),” Journal of Computational Analysis and Applications, vol. 12, no. 3, pp. 630–636, 2010. View at Google Scholar
  13. C. S. Ryoo, “Some identities of the twisted q-Euler numbers and polynomials associated with q-Bernstein polynomials,” Proceedings of the Jangjeon Mathematical Society, vol. 14, no. 2, pp. 239–248, 2011. View at Google Scholar
  14. Y. Simsek, “On p-adic twisted p-L-functions related to generalized twisted Bernoulli numbers,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006. View at Publisher · View at Google Scholar
  15. T. Kim, “A note on q-Volkenborn integration,” Proceedings of the Jangjeon Mathematical Society, vol. 8, no. 1, pp. 13–17, 2005. View at Google Scholar
  16. T. Kim, “q-Euler numbers and polynomials associated with p-adic q-integrals,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007. View at Publisher · View at Google Scholar