Honam Mathematical Journal (호남수학학술지)

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A NOTE ON GENOCCHI-ZETA FUNCTIONS

  • Received : 2008.09.13
  • Accepted : 2009.07.13
  • Published : 2009.09.25
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Abstract

In this paper, we study the Genoochi-zeta functions which are entire functions in whole complex s-plane these zeta functions have the values of the Genocchi numbers and the Genoochi polynomials at negative integers respectively. That is ${\zeta}_G(1-k)={\frac{G_k}{k}}$ and ${\zeta}_G(1-k,x)={\frac{G_k(x)}{k}}$.

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References

  1. H. M. Srivastava, T. Kim and Y. Simsek, q-Bernoulli numbers and polynomials associated with multiple q-zeta functions and basic L-series, Russ. J. Math. Phys. 12 (2005), no. 2, 241-268.
  2. H. Ozden, Y. Simsek, I. N. Cangul and S.-H. Rim, A note on p-adic q-Euler measure, Adv. Stud. Contemp. Math. 14 (2007), 233-239.
  3. K. Shiratani, S. Yamamoto, On a p-adic interpolating function for the Euler numbers and its derivatives, Mem. Fac. Sci. Kyushu Univ. Ser. A 39 (1985), 113-125. https://doi.org/10.2206/kyushumfs.39.113
  4. L.-C. Jang, T. Kim, D-H. Lee, and AD-W. Park An application of polylogarithms in analogs of Genocchi numbers, NNTDM: Notes Number Theory and Discrete Math. 7 (2001), no. 3, 65-70.
  5. L.-C. Jang and T. Kim, q-Genocchi Numbers and Polynomials Associated with. Fermionic p-Adic Invariant Integrals on Zp, Abstract and Applied Analysis, vol. 2008, Article ID 232187, 8 pages, 2008. doi:10.1155/2008/232187.
  6. M. Cenkci, V.Kurt, p-udic interpolation function and Kummer type congruence for q-twisted Euler numbers, Adv. Stud. Contemp. Math. 9 (2004), 203-216.
  7. M. Schork, Ward's "calculus of sequence", q-calculus and the limit q ${\rightarrow}$ -1, Adv. St ud. Contemp. Math. 13 (2006), 131-141.
  8. M. Schork, Combinatorial aspects of normal ordering and its connection to q-calculus, Adv. Stud. Contemp. Math. 15 (2007), 49-57.
  9. S.-H. Rim and T. Kim, A note on p-adic Euler measure on ${\mathbb{Z}}_p$, Russ. J. Math. Phys. 13 (2006), no. 3, 358-361. https://doi.org/10.1134/S1061920806030113
  10. T. Kim, L.-C. Jang and H. K. Pak, A note on q-Euler and Genocchi numbers, Proc. Japan Acad. 77, Ser. A (2001), 139-141. https://doi.org/10.3792/pjaa.77.139
  11. T. Kim, q-Volkenbom integration, Russ. J. Math. Phys. 9 (2002), no. 3, 288-299.
  12. T. Kim, Non-Archimedean q-integrals associated with multiple Changhee q-Bernoulli polynomials, Russ. J. Math. Phys. 10 (2003), no. 1, 91-98.
  13. T. Kim, On Euler-Barnes multiple zeta functions, Russ. J. Math. Phys. 10 (2003), no. 3, 261-267.
  14. T. Kim, Analytic continuation of multiple q-zeta functions and their values at negative integers, Russ. J. Math. Phys. 11 (2004), no. 1, 71-76.
  15. T. Kim et al, Introduction to Non-Archimedian Analysis, Kyo Woo Sa (Korea), 2004. (http://www.kyowoo.co.kr).
  16. T. Kim, A note on q-Volkenborn integration, Proc. Jangjeon Math. Soc. 8 (2005), 13-17.
  17. T. Kim, Power series and asymptotic series associated with the q-analog of the two-variable p-adic L-function, Russ. J. Math. Phys. 12 (2005), no. 2, 186-196.
  18. T. Kim, Multiple p-adic L-function, Russ. J. Math. Phys. 13 (2006), no. 2, 151-157. https://doi.org/10.1134/S1061920806020038
  19. T. Kim, q-generalized Euler numbers and polynomials, Russ. J. Math. Phys. 13 (2006). no. 3, 293-298. https://doi.org/10.1134/S1061920806030058
  20. T. Kim, Euler numbers and polynomials associated with zeta functions, Abstract and Applied Analysis, 2008 (2008), Article ID 581582, 11 pages.
  21. T. Kim, The modified q-Euler numbers and polynomials, Adv. Stud. Contemp. Math. 16 (2008), 161-170.
  22. T. Kim, A note on p-adic q-integml on ${\mathbb{Z}}_p$ Associated with q-Euler numbers, Adv. Stud. Contemp. Math. 15 (2007), 133-138.
  23. T. Kim, J. Y. Choi and J. Y. Sug, Ext ended q-Euier numbers and polynomials associat ed with fermionic p-adic q-integral on ${\mathbb{Z}}_p$, Russ. J . Math. Phys. 14 (2007), no. 2, 160-163. https://doi.org/10.1134/S1061920807020045
  24. T. Kim. On the analogs of Euler numbers and polynomials associated with p-adic q-integral on ${\mathbb{Z}}_p$ at q = -1, Journal of Mathematical Analysis and Applications. vol. 331 (2007), no. 2, 779-792. https://doi.org/10.1016/j.jmaa.2006.09.027
  25. T. Kim, On the q-extension of Euler and Genocchi numbers. J. Math. Anal. Appl. 326 (2007), no. 2. 1458-1465. https://doi.org/10.1016/j.jmaa.2006.03.037
  26. T. Kim, q-extension of the Euler formula and trigonometric functions, Russ. J. Math. Phys. 14 (2007). no. 3, 275-278. https://doi.org/10.1134/S1061920807030041
  27. T. Kim, A note on the q-Genocchi numbers an d polynomials, J. Inequal. Appl. 2007, Art. ID 71452, 8 pp. doi: 10.1155/2007/71452.
  28. T. Kim, q-Bernoulli numbers and polynomials associated with Gaussian. binomial coefficients, Russ. J. Math. Phys. 15 (2008), no. 1, 51-57. https://doi.org/10.1134/S1061920808010068
  29. T. Kim and Y. Simsek, Analytic continuation of the multiple Daehee q-l-functions associated with Daehee number, Russ. J. Math. Phys. 15 (2008). no. 1, 58-65. https://doi.org/10.1134/S106192080801007X
  30. Y. Simsek, On p-adic twist ed q-L-functions related to generalized twisted Bernoulli numbers, Russ. J. Math. Phys. 13 (2006), no.3, 340-348. https://doi.org/10.1134/S1061920806030095
  31. Y. Simsek, I. Naci Cangul, V. Kurt, and D. Kim q-Genocchi Numbers and Polynomials Associated with q-Genocchi- Type l-Functions, Advances in Difference Equations, vol. 2008, Article ID 815750, 12 pages, 2008. doi:10.1155/2008/815750.