DOI QR코드

DOI QR Code

SOME SYMMETRY IDENTITIES FOR GENERALIZED TWISTED BERNOULLI POLYNOMIALS TWISTED BY UNRAMIFIED ROOTS OF UNITY

  • Kim, Dae San (Department of Mathematics Sogang University)
  • Received : 2014.04.23
  • Published : 2015.03.31

Abstract

We derive three identities of symmetry in two variables and eight in three variables related to generalized twisted Bernoulli polynomials and generalized twisted power sums, both of which are twisted by unramified roots of unity. The case of ramified roots of unity was treated previously. The derivations of identities are based on the p-adic integral expression, with respect to a measure introduced by Koblitz, of the generating function for the generalized twisted Bernoulli polynomials and the quotient of p-adic integrals that can be expressed as the exponential generating function for the generalized twisted power sums.

Keywords

generalized twisted Bernoulli polynomial;generalized twisted power sum;Dirichlet character;unramified roots of unity;p-adic integral;identities of symmetry

References

  1. A. Bayad and T. Kim, Identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 2, 247-253.
  2. E. Deeba and D. Rodriguez, Stirling's and Bernoulli numbers, Amer. Math. Monthly 98 (1991), no. 5, 423-426. https://doi.org/10.2307/2323860
  3. F. T. Howard, Applications of a recurrence for the Bernoulli numbers, J. Number Theory 52 (1995), no. 1, 157-172. https://doi.org/10.1006/jnth.1995.1062
  4. D. S. Kim, Identities of symmetry for q-Bernoulli polynomials, Comput. Math. Appl. 60 (2010), no. 8, 2350-2359. https://doi.org/10.1016/j.camwa.2010.08.028
  5. D. S. Kim, Symmetry identities for generalized twisted Euler polynomials twisted by un-ramified roots of unity, Proc. Jangjeon Math. Soc. 15 (2012), no. 3, 303-316.
  6. D. S. Kim, Identities of symmetry for generalized twisted Bernoulli polynomials twisted by ramified roots of unity, An. Stiint. Univ. Al. I. Cuza Iasi, Ser. Noua, Mat. 60 (2014), no. 1, 19-36.
  7. D. S. Kim, N. Lee, J. Na, and K. H. Park, Identities of symmetry for higher-order Euler polynomials in three variables (I), Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 1, 51-74.
  8. D. S. Kim, N. Lee, J. Na, and K. H. Park, Abundant symmetry for higher-order Bernoulli polynomials (I), Adv. Stud. Contemp. Math. (Kyungshang) 23 (2013), no. 3, 461-482.
  9. D. S. Kim and K. H. Park, Identities of symmetry for Euler polynomials arising from quotients of fermionic integrals invariant under $S_3$, J. Inequal. Appl. 2010 (2010), Article ID 851521, 16 pages.
  10. D. S. Kim and K. H. Park, Identities of symmetry for Bernoulli polynomials arising from quotients of Volkenborn integrals invariant under $S_3$, Appl. Math. Comput. 219 (2013), no. 10, 5096-5104. https://doi.org/10.1016/j.amc.2012.11.061
  11. T. Kim, Symmetry p-adic invariant integral on $\mathbb{Z}_p$ for Bernoulli and Euler polynomials, J. Difference Equ. Appl. 14 (2008), no. 12, 1267-1277. https://doi.org/10.1080/10236190801943220
  12. T. Kim, On the symmetries of the q-Bernoulli polynomials, Abstr. Appl. Anal. 2008 (2008), Art. ID 914367, 7 pp.
  13. T. Kim, On the symmetric properties for the generalized twisted Bernoulli polynomials, J. Inequal. Appl. 2009 (2009), Article ID 164743, 8 pages.
  14. T. Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on $\mathbb{Z}_p$, Russ. J. Math. Phys. 16 (2009), no. 1, 93-96. https://doi.org/10.1134/S1061920809010063
  15. T. Kim, L.-C. Jang, Y.-H. Kim, and K.-W. Hwang, On the identities of symmetry for the generalized Bernoulli polynomials attached to of higher order, J. Inequal. Appl. 2009 (2009), Art. ID 640152, 7 pp.
  16. T. Kim, S.-H. Rim, and B. Lee, Some identities of symmetry for the generalized Bernoulli numbers and polynomials, Abstr. Appl. Anal. 2009 (2009), Art. ID 848943, 8 pp.
  17. Y.-H. Kim and K.-W. Hwang, Symmetry of power sum and twisted Bernoulli polyno-mials, Adv. Stud. Contemp. Math. (Kyungshang) 18 (2009), no. 2, 127-133.
  18. N. Koblitz, A new proof of certain formulas for p-adic L-functions, Duke Math. J. 46 (1979), no. 2, 455-468. https://doi.org/10.1215/S0012-7094-79-04621-0
  19. S.-H. Rim, Y.-H. Kim, B. Lee, and T. Kim, Some identities of the generalized twisted Bernoulli numbers and polynomials of higher order, J. Comput. Anal. Appl. 12 (2010), no. 3, 695-702.
  20. H. Tuenter, A symmetry of power sum polynomials and Bernoulli numbers, Amer.Math. Monthly 108 (2001), no. 3, 258-261. https://doi.org/10.2307/2695389
  21. S. Yang, An identity of symmetry for the Bernoulli polynomials, Discrete Math. 308 (2008), no. 4, 550-554. https://doi.org/10.1016/j.disc.2007.03.030