Discrete Dynamics in Nature and Society

Volume 2012, Article ID 486158, 12 pages

http://dx.doi.org/10.1155/2012/486158

Research Article

## Some Identities on Bernoulli and Euler Numbers

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

Received 15 November 2011; Accepted 23 December 2011

Academic Editor: Delfim F. M. Torres

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

- T. Kim, “Euler numbers and polynomials associated with zeta functions,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 581582, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - T. Kim, “Note on the Euler numbers and polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 17, no. 2, pp. 131–136, 2008. View at Google Scholar · View at Zentralblatt MATH - T. Kim, “Symmetry of power sum polynomials and multivariate fermionic
*p*-adic invariant integral on ${\mathbb{Z}}_{p}$,”*Russian Journal of Mathematical Physics*, vol. 16, no. 1, pp. 93–96, 2009. View at Publisher · View at Google Scholar - T. Kim, “Symmetry
*p*-adic invariant integral on ${\mathbb{Z}}_{p}$ for Bernoulli and Euler polynomials,”*Journal of Difference Equations and Applications*, vol. 14, no. 12, pp. 1267–1277, 2008. View at Publisher · View at Google Scholar - 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 - D. Ding and J. Yang, “Some identities related to the Apostol-Euler and Apostol-Bernoulli polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 20, no. 1, pp. 7–21, 2010. View at Google Scholar · View at Zentralblatt MATH - T. Kim, “
*q*-Volkenborn integration,”*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288–299, 2002. View at Google Scholar · View at Zentralblatt MATH - T. Kim, “A note on
*q*-Bernstein polynomials,”*Russian Journal of Mathematical Physics*, vol. 18, no. 1, pp. 73–82, 2011. View at Publisher · View at Google Scholar - A. Kudo, “A congruence of generalized Bernoulli number for the character of the first kind,”
*Advanced Studies in Contemporary Mathematics*, vol. 2, pp. 1–8, 2000. View at Google Scholar · View at Zentralblatt MATH - Q.-M. Luo and F. Qi, “Relationships between generalized Bernoulli numbers and polynomials and generalized Euler numbers and polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 7, no. 1, pp. 11–18, 2003. View at Google Scholar · View at Zentralblatt MATH - Q.-M. Luo, “Some recursion formulae and relations for Bernoulli numbers and Euler numbers of higher order,”
*Advanced Studies in Contemporary Mathematics*, vol. 10, no. 1, pp. 63–70, 2005. View at Google Scholar · View at Zentralblatt MATH - H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on
*q*-Bernoulli numbers associated with Daehee numbers,”*Advanced Studies in Contemporary Mathematics*, vol. 18, no. 1, pp. 41–48, 2009. View at Google Scholar - Y.-H. Kim and K.-W. Hwang, “Symmetry of power sum and twisted Bernoulli polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 18, no. 2, pp. 127–133, 2009. View at Google Scholar · View at Zentralblatt MATH - G. Kim, B. Kim, and J. Choi, “The DC algorithm for computing sums of powers of consecutive integers and Bernoulli numbers,”
*Advanced Studies in Contemporary Mathematics*, vol. 17, no. 2, pp. 137–145, 2008. View at Google Scholar · View at Zentralblatt MATH - L. C. Jang, “A note on Kummer congruence for the Bernoulli numbers of higher order,”
*Proceedings of the Jangjeon Mathematical Society*, vol. 5, no. 2, pp. 141–146, 2002. View at Google Scholar · View at Zentralblatt MATH - L. C. Jang and H. K. Pak, “Non-Archimedean integration associated with
*q*-Bernoulli numbers,”*Proceedings of the Jangjeon Mathematical Society*, vol. 5, no. 2, pp. 125–129, 2002. View at Google Scholar - S.-H. Rim, J.-H. Jin, E.-J. Moon, and S.-J. Lee, “Some identities on the
*q*-Genocchi polynomials of higher-order and*q*-Stirling numbers by the fermionic*p*-adic integral on ${\mathbb{Z}}_{p}$,”*International Journal of Mathematics and Mathematical Sciences*, vol. 2010, Article ID 860280, 14 pages, 2010. View at Google Scholar - C. S. Ryoo, “On the generalized Barnes type multiple
*q*-Euler polynomials twisted by ramified roots of unity,”*Proceedings of the Jangjeon Mathematical Society*, vol. 13, no. 2, pp. 255–263, 2010. View at Google Scholar - C. S. Ryoo, “Some 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 - Y. Simsek, “Generating 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 · View at Zentralblatt MATH - I. Buyukyazici, “On generalized q-Bernstein polynomials,”
*The Global Journal of Pure and Applied Mathematics*, vol. 6, pp. 1331–1348, 2010. View at Google Scholar - L.-C. Jang, W.-J. Kim, and Y. Simsek, “A study on the
*p*-adic integral representation on ${\mathbb{Z}}_{p}$ associated with Bernstein and Bernoulli polynomials,”*Advances in Difference Equations*, vol. 2010, Article ID 163217, 6 pages, 2010. View at Google Scholar