DOI QR코드

DOI QR Code

SOME IDENTITIES OF DEGENERATE GENOCCHI POLYNOMIALS

  • Lim, Dongkyu (School of Mathematical Sciences, Nankai University)
  • Received : 2015.03.20
  • Published : 2016.03.31

Abstract

L. Carlitz introduced higher order degenerate Euler polynomials in [4, 5] and studied a degenerate Staudt-Clausen theorem in [4]. D. S. Kim and T. Kim gave some formulas and identities of degenerate Euler polynomials which are derived from the fermionic p-adic integrals on ${\mathbb{Z}}_p$ (see [9]). In this paper, we introduce higher order degenerate Genocchi polynomials. And we give some formulas and identities of degenerate Genocchi polynomials which are derived from the fermionic p-adic integrals on ${\mathbb{Z}}_p$.

Keywords

References

  1. S. Araci, M. Acikgoz, H. Jolany, and J. J. Seo, A unified generating function of the q-Genocchi polynomials with their interpolation functions, Proc. Jangjeon Math. Soc. 15 (2012), no. 2, 227-233.
  2. 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.
  3. I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, On the higher-order w-q-Genocchi numbers, Adv. Stud. Contemp. Math. (Kyungshang) 19 (2009), no. 1, 39-57.
  4. L. Carlitz, A degenerate Staudt-Clausen theorem, Arch. Math. (Basel) 7 (1956), 28-33. https://doi.org/10.1007/BF01900520
  5. L. Carlitz, Degenerate Stirling, Bernoulli and Eulerian numbers, Utilitas Math. 15 (1979), 51-88.
  6. S. Gaboury, R. Tremblay, and B.-J. Fugere, Some explicit formulas for certain new classes of Bernoulli, Euler and Genocchi polynomials, Proc. Jangjeon Math. Soc. 17 (2014), no. 1, 115-123.
  7. D. Kang, J.-H. Jeong, B. J. Lee, S.-H. Rim, and S. H. Choi, Some identities of higher order Genocchi polynomials arising from higher order Genocchi basis, J. Comput. Anal. Appl. 17 (2014), no. 1, 141-146.
  8. D. S. Kim and T. Kim, A note on Boole polynomials, Integral Transforms Spec. Funct. 25 (2014), no. 8, 627-633. https://doi.org/10.1080/10652469.2014.891586
  9. D. S. Kim and T. Kim, Some identities of degenerate Euler polynomials arising from p-adic fermionic integrals on ${\mathbb{Z}}_p$, Integral Transforms Spec. Funct. 26 (2015), no. 4, 295-302. https://doi.org/10.1080/10652469.2014.1002497
  10. D. S. Kim and T. Kim, Higher-order Degenerate Euler Polynomials, Applied Mathematical Sciences, 9 (2015) no. 2, 57-73. https://doi.org/10.1007/s40096-015-0149-6
  11. D. S. Kim, T. Kim, D. V. Dolgy, and T. Komatsu, Barnes-type degenerate Bernoulli polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 25 (2015), no. 1, 121-146.
  12. D. S. Kim, T. Kim, S. H. Lee, and J.-J. Seo, A note on the lambda-Daehee polynomials, Int. J. Math. Anal. (Ruse) 7 (2013), no. 61-64, 3069-3080. https://doi.org/10.12988/ijma.2013.311264
  13. T. Kim, A note on q-Volkenborn integration, Proc. Jangjeon Math. Soc. 8 (2005), no. 1, 13-17.
  14. T. Kim, A note on the q-Genocchi numbers and polynomials, J. Inequal. Appl. 2007 (2007), Art. ID 71452, 8 pp.
  15. 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
  16. T. Kim, q-Euler numbers and polynomials associated with p-adic q-integrals, J. Nonlinear Math. Phys. 14 (2007), no. 1, 15-27. https://doi.org/10.2991/jnmp.2007.14.1.3
  17. T. Kim, Note on q-Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 17 (2008), no. 1, 9-15.
  18. 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
  19. T. Kim New approach to q-Euler, Genocchi numbers and their interpolation functions, Adv. Stud. Contemp. Math. (Kyungshang) 18 (2009), no. 2, 105-112.
  20. T. Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invari-ant integral on ${\mathbb{Z}}_p$, Russ. J. Math. Phys. 16 (2009), no. 1, 93-96. https://doi.org/10.1134/S1061920809010063
  21. T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 1, 23-28.
  22. T. Kim, D. V. Dolgy, D. S. Kim, and S.-H. Rim, A note on the identities of special polynomials, Ars Combin. 113A (2014), 97-106.
  23. T. Kim, L.-C. Jang, and Y.-H. Kim, Some properties on the p-adic invariant integral on ${\mathbb{Z}}_p$ associated with Genocchi and Bernoulli polynomials, J. Comput. Anal. Appl. 13 (2011), no. 7, 1201-1207.
  24. T. Kim, S.-H. Rim, D. V. Dolgy, and S.-H. Lee, Some identities of Genocchi polynomials arising from Genocchi basis, J. Inequal. Appl. 2013 (2013), 43, 6 pp. https://doi.org/10.1186/1029-242X-2013-6
  25. Y.-H. Kim, K.-W. Hwang, and T. Kim, Interpolation functions of the q-Genocchi and the q-Euler polynomials of higher order, J. Comput. Anal. Appl. 12 (2010), no. 1-B, 228-238.
  26. B. Kurt, The multiplication formulae for the Genocchi polynomials, Proc. Jangjeon Math. Soc. 13 (2010), no. 1, 89-96.
  27. H. Ozden, p-adic distribution of the unification of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Comput. 218 (2011), no. 3, 970-973. https://doi.org/10.1016/j.amc.2011.01.075
  28. S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, On the q-Genocchi numbers and polynomials associated with q-zeta function, Proc. Jangjeon Math. Soc. 12 (2009), no. 3, 261-267.
  29. C. S. Ryoo, T. Kim, J. Choi, and B. Lee, On the generalized q-Genocchi numbers and polynomials of higher-order, Adv. Difference Equ. 2011 (2011), Art. ID 424809, 8 pp. https://doi.org/10.1186/1687-1847-2011-8
  30. H. M. Srivastava, B. Kurt, and Y. Simsek, Some families of Genocchi type polynomials and their interpolation functions, Integral Transforms Spec. Funct. 23 (2012), no. 12, 919-938. https://doi.org/10.1080/10652469.2011.643627

Cited by

  1. Degenerate Changhee-Genocchi numbers and polynomials vol.2017, pp.1, 2017, https://doi.org/10.1186/s13660-017-1572-z
  2. On degenerate Apostol-type polynomials and applications pp.2296-4495, 2019, https://doi.org/10.1007/s40590-018-0220-z