Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2008 (2008), Article ID 296159, 10 pages
http://dx.doi.org/10.1155/2008/296159
Research Article

On the -Extension of Apostol-Euler Numbers and Polynomials

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
2Natural Science Institute, KonKuk University, Chungju 380-701, South Korea
3Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea

Received 4 October 2008; Accepted 21 November 2008

Academic Editor: Lance Littlejohn

Copyright © 2008 Young-Hee 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. 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 · View at MathSciNet
  2. T. Kim, “On p-adic q-L-functions and sums of powers,” Discrete Mathematics, vol. 252, no. 1–3, pp. 179–187, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. T. Kim, “On Euler-Barnes multiple zeta functions,” Russian Journal of Mathematical Physics, vol. 10, no. 3, pp. 261–267, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. T. Kim, “Sums of powers of consecutive q-integers,” Advanced Studies in Contemporary Mathematics, vol. 9, no. 1, pp. 15–18, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. T. Kim, “Analytic continuation of multiple q-zeta functions and their values at negative integers,” Russian Journal of Mathematical Physics, vol. 11, no. 1, pp. 71–76, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. T. Kim, “q-Riemann zeta function,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 12, pp. 599–605, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. T. Kim, “Power series and asymptotic series associated with the q-analog of the two-variable p-adic L-function,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186–196, 2005. View at Google Scholar · View at MathSciNet
  8. T. Kim, “q-generalized Euler numbers and polynomials,” Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 293–298, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  9. T. Kim, “Multiple p-adic L-function,” Russian Journal of Mathematical Physics, vol. 13, no. 2, pp. 151–157, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. Kim, “On the analogs of Euler numbers and polynomials associated with p-adic q-integral on p at q=1,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 779–792, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, no. 4, pp. 987–1000, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. 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 MathSciNet
  13. L. Carlitz, “q-Bernoulli and Eulerian numbers,” Transactions of the American Mathematical Society, vol. 76, no. 2, pp. 332–350, 1954. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. 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
  15. M. Cenkci and M. Can, “Some results on q-analogue of the Lerch zeta function,” Advanced Studies in Contemporary Mathematics, vol. 12, no. 2, pp. 213–223, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. J. Choi, P. J. Anderson, and H. M. Srivastava, “Some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n, and the multiple Hurwitz zeta function,” Applied Mathematics and Computation, vol. 199, no. 2, pp. 723–737, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  17. M. Garg, K. Jain, and H. M. Srivastava, “Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions,” Integral Transforms and Special Functions, vol. 17, no. 11, pp. 803–815, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  18. L.-C. Jang, “Multiple twisted q-Euler numbers and polynomials associated with p-adic q-integrals,” Advances in Difference Equations, vol. 2008, Article ID 738603, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. T. Kim, “On the q-extension of Euler and Genocchi numbers,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1458–1465, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. T. Kim, “q-extension of the Euler formula and trigonometric functions,” Russian Journal of Mathematical Physics, vol. 14, no. 3, pp. 275–278, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  21. T. Kim, “On the symmetries of the q-Bernoulli polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 914367, 7 pages, 2008. View at Publisher · View at Google Scholar
  22. T. Kim, “The modified q-Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161–170, 2008. View at Google Scholar · View at MathSciNet
  23. T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,” Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008. View at Google Scholar · View at MathSciNet
  24. T. Kim, “On the multiple q-Genocchi and Euler numbers,” Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 481–486, 2008. View at Google Scholar
  25. T. Kim, J. Y. Choi, and J. Y. Sug, “Extended q-Euler numbers and polynomials associated with fermionic p-adic q-integral on p,” Russian Journal of Mathematical Physics, vol. 14, no. 2, pp. 160–163, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  26. T. Kim, M.-S. Kim, L. Jang, and S.-H. Rim, “New q-Euler numbers and polynomials associated with p-adic q-integrals,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 243–252, 2007. View at Google Scholar · View at MathSciNet
  27. Y. H. Kim, W. J. Kim, and C. S. Ryoo, “On the twisted q-Euler zeta function associated with twisted q-Euler numbers,” communicated.
  28. T. Kim, S.-H. Rim, and Y. Simsek, “A note on the alternating sums of powers of consecutive q-integers,” Advanced Studies in Contemporary Mathematics, vol. 13, no. 2, pp. 159–164, 2006. View at Google Scholar · View at MathSciNet
  29. T. Kim and Y. Simsek, “Analytic continuation of the multiple Daehee q-l-functions associated with Daehee numbers,” Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 58–65, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  30. S.-D. Lin and H. M. Srivastava, “Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations,” Applied Mathematics and Computation, vol. 154, no. 3, pp. 725–733, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. S.-D. Lin, H. M. Srivastava, and P.-Y. Wang, “Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions,” Integral Transforms and Special Functions, vol. 17, no. 11, pp. 817–827, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  32. Q.-M. Luo, “Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions,” Taiwanese Journal of Mathematics, vol. 10, no. 4, pp. 917–925, 2006. View at Google Scholar · View at MathSciNet
  33. Q.-M. Luo and H. M. Srivastava, “Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials,” Journal of Mathematical Analysis and Applications, vol. 308, no. 1, pp. 290–302, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. 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 Zentralblatt MATH · View at MathSciNet
  35. 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 Zentralblatt MATH · View at MathSciNet
  36. 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 Google Scholar · View at MathSciNet
  37. Y. Simsek, “On p-adic twisted q-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 · View at MathSciNet
  38. 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 MathSciNet
  39. H. M. Srivastava, T. Kim, and Y. Simsek, “q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 241–268, 2005. View at Google Scholar · View at MathSciNet
  40. T. M. Apostol, “On the Lerch zeta function,” Pacific Journal of Mathematics, vol. 1, pp. 161–167, 1951. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. W. Wang, C. Jia, and T. Wang, “Some results on the Apostol-Bernoulli and Apostol-Euler polynomials,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1322–1332, 2008. View at Google Scholar · View at MathSciNet