Honam Mathematical Journal (호남수학학술지)

The Honam Mathematical Society (호남수학회)
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A NOTE ON THE q-EULER NUMBERS AND POLYNOMIALS WITH WEIGHT (α,ω)

  • Rim, Seog-Hoon (Department of Mathematics Education, Kyungpook National University) ;
  • Jeong, Joo-Hee (Department of Mathematics Education, Kyungpook National University)
  • Received : 2011.12.29
  • Accepted : 2012.03.27
  • Published : 2012.06.25
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Abstract

The main purpose of this paper is to introduce a new type of $q$-Euler numbers and polynomials with weak weight (${\alpha}$,${\omega}$): $\tilde{E}^{({\alpha},{\omega})}_{n,q}$ and $\tilde{E}^{({\alpha},{\omega})}_{n,q}(x)$, respectively. By using the fermionic $p$-adic $q$-integral on $\mathbb{Z}_p$, we can obtain some results and derive some recurrence identities for the $q$-Euler numbers and polynomials with weak weight (${\alpha}$,${\omega}$).

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References

  1. M. Cenkci, M. Can, V. Kurt, q-adic interpolation functions and kummer type congruence for q-twisted and q-generalized twisted Euler numbers, Adv. Stud. Contemp. Math. 2004 (9), 203-216.
  2. M. Can, M. Cenkci, V. Kurt, Y. Simsek, Twisted Dedekind type sums associated with Barnes' type multiple Frobenius-Euler l-functions, Adv. Stud. Contemp. Math. 18 (2009), no. 2, 135-160.
  3. A. Bayad, Modular properties of elliptic Bernoulli and Euler functions, Adv. Stud. Contemp. Math. 20 (2010), no. 3, 389-401.
  4. A. Bayad, T. Kim, Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomials, Russ. J. Math. Phys., 18(2011), no. 2, 133-143. https://doi.org/10.1134/S1061920811020014
  5. T. Kim, An analogus of Bernoulli numbers and their applications, Rep. Fac. Sci. Engrg. Saga Univ. Math., 22(1994 ), 21-26.
  6. T. Kim, On q-extention of Euler and Genocchi numbers, Journal of Mathematical Analysis and Applications, 326(2007), 1458-1465. https://doi.org/10.1016/j.jmaa.2006.03.037
  7. T. Kim, q-Volkenborn integration, Russ. J. Math. Phys., 9(2002), no.3, 288-299.
  8. T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys. 14 (2007), 15-27. https://doi.org/10.2991/jnmp.2007.14.1.3
  9. T. Kim, J. Choi, Y. H. Kim, C. S. Ryoo, A Note on the weighted p-adic q-Euler measure on $\mathbb{Z}_{p}$, Adv. Stud. Contemp. Math., 21 (2011), 35-40.
  10. T. Kim, A. Bayad, Y. H. Kim, A study on the p-adic q-integral representation on $\mathbb{Z}_{p}$ associated with the weighted q-Bernoulli polynomials, J. Inequal. Appl., 2011, Art. ID 513821, 8 pp.
  11. T. Kim, New approach to q-Euler polynomials of higher order, Russ. J. Math. Phys. 17 (2010), no. 2, 218-225. https://doi.org/10.1134/S1061920810020068
  12. T. Kim, Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on $\mathbb{Z}_p$, Russ. J. Math. Phys., 16 (2009), no. 4, 484-491. https://doi.org/10.1134/S1061920809040037
  13. T. Kim, Barnes-type multiple q-zeta functions and q-Euler polynomials, J. Phys. A, 43 (2010), no. 25, 255201, 11 pp.
  14. T. Kim, A note on q-Bernstein polynomials, Russ. J. Math. Phys., 18(2011), no.1, 73-82. https://doi.org/10.1134/S1061920811010080
  15. T. Kim, B. Lee, J. Choi, Y. H. Kim, A new approach of q-Euler numbers and polynomials, Proc. Janjeon Math. Soc., 14(2011), 7-14.
  16. T. Kim, On the multiple q-Genocchi and Euler numbers, Russ. J. Math. Phys, 15(2008), no.4, 481-486. https://doi.org/10.1134/S1061920808040055
  17. H. Ozden, Y. Simsek, S.-H. Rim, I. N. Cangul, A note on p-adic q-Euler measure, Adv. Stud. Contemp. Math. 14 (2007), no. 2, 233-239.
  18. S.-H. Rim, E.-J. Moon, S.-J. Lee, J.-H. Jin, Multivariate twisted p-adic q-integral on $\mathbb{Z}_p$ associated with twisted q-Bernoulli polynomials and numbers, J. Inequal. Appl., 2010, Art. ID 579509, 6 pp.
  19. S. H. Rim, 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
  20. H. M. Srivastava, T. Kim, 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.
  21. Y. Simsek, Theorem on twisted L-function and twisted Bernoulli numbers, Adv. Stud. Contemp. Math., 12(2006), 237-246.
  22. Y. Simsek, Special functions related to Dedekind-type DC-sums and their applications, Russ. J. Math. Phys., 17(2010), no.4, 495-508. https://doi.org/10.1134/S1061920810040114