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Abstract and Applied Analysis
Volume 2014, Article ID 286239, 6 pages
http://dx.doi.org/10.1155/2014/286239
Research Article

Identities of Symmetry for Higher-Order Generalized -Euler Polynomials

1Institute of Mathematics and Computer Sciences, Far Eastern Federal University, Vladivostok 690060, Russia
2Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
3Jangjeon Research Institute for Mathematics and Physics, 252-5 Hapcheon-Dong, Hapcheon-Gun Kyungshang Nam-Do 678-800, Republic of Korea
4Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
5Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea

Received 17 December 2013; Accepted 18 January 2014; Published 25 February 2014

Academic Editor: Alberto Fiorenza

Copyright © 2014 D. V. Dolgy 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. M. Cenkci, “The p-adic generalized twisted (h,q)-Euler-l-function and its applications,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 1, pp. 37–47, 2007. View at Google Scholar · View at MathSciNet
  2. T. Kim, “An identity of symmetry for the generalized Euler polynomials,” Journal of Computational Analysis and Applications, vol. 13, no. 7, pp. 1292–1296, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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 · View at Scopus
  4. 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 · View at MathSciNet · View at Scopus
  5. D. S. Kim, “Symmetry identities for generalized twisted Euler polynomials twisted by unramified roots of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 15, no. 3, pp. 303–316, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. T. Kim, “Symmetry p-adic invariant integral on 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 · View at MathSciNet · View at Scopus
  7. T. Kim, “On p-adic interpolating function for q-Euler numbers and its derivatives,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 598–608, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. 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 · View at MathSciNet · View at Scopus
  9. V. Kurt, “Some symmetry identities for the Apostol-type polynomials related to multiple alternating sums,” Advances in Difference Equations, vol. 2013, article 32, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  10. B. Kurt, “Some formulas for the multiple twisted (h,q)-Euler polynomials and numbers,” Applied Mathematical Sciences, vol. 5, no. 25–28, pp. 1263–1270, 2011. View at Google Scholar · View at MathSciNet · View at Scopus
  11. 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 · View at Scopus
  12. H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on sum of products of (h,q)-twisted Euler polynomials and numbers,” Journal of Inequalities and Applications, vol. 2008, Article ID 816129, 8 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S.-H. Rim and J. Jeong, “On the modified q-Euler numbers of higher order with weight,” Advanced Studies in Contemporary Mathematics, vol. 22, no. 1, pp. 93–98, 2012. View at Google Scholar · View at MathSciNet · View at Scopus
  14. Y. Simsek, “Twisted p-adic (h,q)-L-functions,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 2097–2110, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. Y. Simsek, “Interpolation functions of the Eulerian type polynomials and numbers,” Advanced Studies in Contemporary Mathematics, vol. 23, no. 2, pp. 301–307, 2013. View at Google Scholar · View at MathSciNet
  16. S. Araci, M. Acikgoz, and E. Şen, “On the extended Kim's p-adic q-deformed fermionic integrals in the p-adic integer ring,” Journal of Number Theory, vol. 133, no. 10, pp. 3348–3361, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Araci, J. J. Seo, and D. Erdal, “New construction weighted (h,q)-Genocchi numbers and polynomials related to zeta type functions,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 487490, 7 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. I. N. Cangul, H. Ozden, and Y. Simsek, “Generating functions of the (h,q) extension of twisted Euler polynomials and numbers,” Acta Mathematica Hungarica, vol. 120, no. 3, pp. 281–299, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. K.-W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, and S. H. Lee, “Some theorems on Bernoulli and Euler numbers,” Ars Combinatoria, vol. 109, pp. 285–297, 2013. View at Google Scholar · View at MathSciNet
  20. 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 Zentralblatt MATH · View at MathSciNet
  21. D. S. Kim, N. Lee, J. Na, and K. H. Park, “Identities of symmetry for higher-order Euler polynomials in three variables (II),” Journal of Mathematical Analysis and Applications, vol. 379, no. 1, pp. 388–400, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus