About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 459763, 6 pages
http://dx.doi.org/10.1155/2013/459763
Research Article

Some Identities on the High-Order -Euler Numbers and Polynomials with Weight 0

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2Department of General Education-Mathematics, Kookmin University, Seoul 136-702, Republic of Korea

Received 7 February 2013; Accepted 2 April 2013

Academic Editor: Chun-Gang Zhu

Copyright © 2013 Jongsung Choi 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. L. Carlitz, “Eulerian numbers and polynomials,” Mathematics Magazine, vol. 32, pp. 247–260, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. L. Carlitz, “The product of two Eulerian polynomials,” Mathematics Magazine, vol. 36, no. 1, pp. 37–41, 1963. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. Choi, “A note on Eulerian polynomials of higher order,” Journal of the Chungcheong Mathematical Society, vol. 26, no. 1, pp. 191–196, 2013.
  4. J. Choi, T. Kim, and Y. H. Kim, “A recurrence formula for q-Euler numbers of higher order,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 3, pp. 321–326, 2010. View at Zentralblatt MATH · View at Scopus
  5. J. Choi, T. Kim, and Y.-H. Kim, “A note on the q-analogues of Euler numbers and polynomials,” Honam Mathematical Journal, vol. 33, no. 4, pp. 529–534, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  6. D. S. Kim, “Identities of symmetry for generalized Euler polynomials,” International Journal of Combinatorics, vol. 2011, Article ID 432738, 12 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  7. D. S. Kim, T. Kim, J. Choi, and Y. H. Kim, “Identities involving q-Bernoulli and q-Euler numbers,” Abstract and Applied Analysis, vol. 2012, Article ID 674210, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H.-M. Kim, J. Choi, and T. Kim, “On the extended q-Euler numbers and polynomials of higher-order with weight,” Honam Mathematical Journal, vol. 34, no. 1, pp. 1–9, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. T. Kim, “Identities involving Frobenius-Euler polynomials arising from non-linear differential equations,” Journal of Number Theory, vol. 132, no. 12, pp. 2854–2865, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  10. T. Kim and J. Choi, “A note on the product of Frobenius-Euler polynomials arising from the p-adic integral on p,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 2, pp. 215–223, 2012. View at MathSciNet
  11. T. Kim and J. Choi, “On the q-Euler numbers and polynomials with weight 0,” Abstract and Applied Analysis, vol. 2012, Article ID 795304, 7 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  12. 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 Publisher · View at Google Scholar · View at MathSciNet
  13. 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 · View at MathSciNet
  14. Y. Simsek, “Complete 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 · View at Google Scholar · View at MathSciNet
  15. Y. Simsek, “Generating functions for q-Apostol type Frobenius-Euler numbers and polynomials,” Axioms, vol. 1, no. 3, pp. 395–403, 2012. View at Publisher · View at Google Scholar
  16. Y. Simsek, “Generating functions for generalized Stirling type numbers, Array type polynomials, Eulerian type polynomials and their applications,” Fixed Point Theory and Applications, vol. 2013, article 87, 2013. View at Publisher · View at Google Scholar
  17. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differntial Equations with Applications, The Williams and Wilkins company, Baltimore, Md, USA, 1976.