Taiwanese Journal of Mathematics


Serkan Araci, Erdoğan Şen, and Mehmet Acikgoz

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Recently, Kim et al. [8] constructed a new method to obtain interesting identities related to Euler polynomials of higher order arising from Euler basis. In the present paper, we study to Genocchi polynomials of higher order arising from Genocchi basis by using the method of Kim et al. We also derive many interesting properties related to Genocchi polynomials of higher order.

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Taiwanese J. Math., Volume 18, Number 2 (2014), 473-482.

First available in Project Euclid: 10 July 2017

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Primary: 11S80: Other analytic theory (analogues of beta and gamma functions, $p$-adic integration, etc.) 11B68: Bernoulli and Euler numbers and polynomials

Genocchi polynomials Genocchi polynomials of higher order Bernoulli polynomials Bernoulli polynomials of higher order Hermite polynomials Euler polynomials Euler polynomials of higher order


Araci, Serkan; Şen, Erdoğan; Acikgoz, Mehmet. THEOREMS ON GENOCCHI POLYNOMIALS OF HIGHER ORDER ARISING FROM GENOCCHI BASIS. Taiwanese J. Math. 18 (2014), no. 2, 473--482. doi:10.11650/tjm.18.2014.3006. https://projecteuclid.org/euclid.twjm/1499706398

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  • T. Kim, Identities involving Frobenius-Euler polynomials arising from non-linear differential equations, J. Number Theory, 132(12) (2012), 2854-2865.
  • T. Kim, An identity of the symmetry for the Frobenius-Euler polynomials associated with the fermionic $p$-adic invariant $q$-integrals on $ \mathbb{Z}_{p}$, Rocky Mountain J. Math., 41(1) (2011), 239-247.
  • T. Kim, S. H. Rim, D. V. Dolgy and S. H. Lee, Some identities of Genocchi polynomials arising from Genocchi basis, J. Ineq. Appl., 2013 (2013), Article ID. 1433166741876263.
  • T. Kim, Symmetry $p$-adic invariant integral on $ \mathbb{Z}_{p}$ for Bernoulli and Euler polynomials, J. Difference Equ. Appl., 14(12) (2008), 1267-1277.
  • T. Kim, Symmetry of power sum polynomials and multivariate fermionic $p$-adic invariant integral on $\mathbb{Z} _{p}$, Russ. J. Math. Phys., 16(1) (2009), 93-96.
  • T. Kim, Some identities for the Bernoulli, the Euler and the Genocchi numbers and polynomials, Adv. Stud. Contemp. Math., 20(1) (2010), 23-28.
  • T. Kim, J. Choi and Y. H. Kim, On $q$-Bernstein and $q$-Hermite polynomials, Proc. Jangjeon Math. Soc., 14(2) (2011), 215-221.
  • D. S. Kim and T. Kim, Some identities of higher-order Euler polynomials arising from Euler basis, Integral Transforms and Special Functions, 2013, http://dx.doi.org/10.1080/ 10652469.2012.754756.
  • 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., 22(1) (2012), 51-74.
  • S. Araci, Novel identities for $q$-Genocchi numbers and polynomials, Journal of Function Spaces and Applications, Vol. 2012, Article ID 214961, 13 pages.
  • S. Araci, M. Acikgoz and J. J. Seo, Explicit formulas involving $q$-Euler numbers and polynomials, Abstract and Applied Analysis, Vol. 2012, Article ID 298531, 11 pages.
  • S. Araci, D. Erdal and J. J. Seo, A study on the fermionic $p$-adic $q$-integral representation on $\mathbb{Z}_{p}$ associated with weighted $q$-Bernstein and $q$-Genocchi polynomials, Abstract and Applied Analysis, Vol. 2011, Article ID 649248, 10 pages.
  • 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(2) (2012), 227-233.
  • S. Araci, M. Acikgoz and E. Şen, A note on the $p$-adic interpolation function for multiple Generalized Genocchi numbers, Turkish Journal of Analysis and Number Theory, 1(1) (2013), 17-22. doi: 10.12691/tjant-1-1-5.
  • S. Araci and M. Acikgoz, A note on the Frobenius-Euler numbers and polynomials associated with Bernstein polynomials, Adv. Stud. in Contemp. Math., 22(3) (2012), 399-406.
  • M. Acikgoz and Y. Simsek, On multiple interpolation function of the Nörlund-type $q$-Euler polynomials, Abst. Appl. Anal. 2009 (2009), Article ID 382574, 14 pages.
  • I. N. Cangul, V. Kurt, H. Ozden and Y. Simsek, On the higher-order $w$-$q$-Genocchi numbers, Adv. Stud. Contemp. Math., 19(1) (2009), 39-57.
  • K. Dilcher, Sums of products of Bernoulli numbers, J. Number Theory, 60 (1996), 23-41.
  • E. R. Hansen, A table of series and products, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1975.
  • Y. He and W. Zhang, A convolution formula for Bernoulli Polynomials, Ars Combinatoria, 2013, pp. 97-104.
  • K. Shiratani, On the Euler numbers, Mem. Fac. Sci., Kyushu Univ., Ser. A, 27 (1973), 1-5.