# A study on a class of q-Euler polynomials under the symmetric group of degree n

Volume 9, Issue 8, pp 5196--5201 Publication Date: August 23, 2016
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### Authors

Serkan Araci - Department of Economics, Faculty of Economics, Administrative and Social Science, Hasan Kalyoncu University, TR-27410 Gaziantep, Turkey. Ugur Duran - Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, TR-27310 Gaziantep, Turkey. Mehmet Acikgoz - Department of Mathematics, Faculty of Arts and Science, University of Gaziantep, TR-27310 Gaziantep, Turkey.

### Abstract

Motivated by the paper of Kim et al. [T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, D. V. Dolgy, J. Nonlinear Sci. Appl., 9 (2016), 1077-1082], we study a class of q-Euler polynomials earlier given by Kim et al. in [T. Kim, Y. H. Kim, K. W. Hwang, Proc. Jangjeon Math. Soc., 12 (2009), 77-92]. We derive some new symmetric identities for q-extension of $\lambda$-Euler polynomials, using fermionic p-adic invariant integral over the p-adic number field originally introduced by Kim in [T. Kim, Russ. J. Math. Phys., 16 (2009), 484-491], under symmetric group of degree n denoted by $S_n$.

### Keywords

• Symmetric identities
• q-extension of $\lambda$-Euler polynomials
• fermionic p-adic invariant integral on $\mathbb{Z}_p$
• invariant under $S_n$.

•  11B68
•  05A19
•  11S80
•  05A30

### References

• [1] E. Ağyüz, M. Acikgoz, S. Araci, A symmetric identity on the q-Genocchi polynomials of higher-order under third dihedral group $D_3$, Proc. Jangjeon Math. Soc., 18 (2015), 177--187

• [2] U. Duran, M. Acikgoz, S. Araci, Symmetric identities involving weighted q-Genocchi polynomials under $S_4$, Proc. Jangjeon Math. Soc., 18 (2015), 455--465

• [3] Y. He, S. Araci, Sums of products of Apostol-Bernoulli and Apostol-Euler polynomials, Adv. Difference Equ., 2014 (2014), 13 pages

• [4] Y. He, S. Araci, H. M. Srivastava, M. Acikgoz, Some new identities for the Apostol-Bernoulli polynomials and the Apostol-Genocchi polynomials, Appl. Math. Comput., 262 (2015), 31--41

• [5] T. Kim, q-Volkenborn integration, Russ. J. Math. Phys.,, 9 (2002), 288--299

• [6] 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), 484--491

• [7] T. Kim, Symmetry of power sum polynomials and multivariate fermionic p-adic invariant integral on $\mathbb{Z}_p$, Russ. J. Math. Phys., 16 (2009), 93--96

• [8] D. S. Kim, T. Kim, Some identities of symmetry for q-Bernoulli polynomials under symmetric group of degree n, Ars Combin., 126 (2016), 435--441

• [9] T. Kim, Y. H. Kim, K. W. Hwang, On the q-extensions of the Bernoulli and Euler numbers, related identities and Lerch zeta function, Proc. Jangjeon Math. Soc., 12 (2009), 77--92

• [10] T. Kim, D. S. Kim, H. I. Kwon, J. J. Seo, D. V. Dolgy, Some identities of q-Euler polynomials under the symmetric group of degree n, J. Nonlinear Sci. Appl., 9 (2016), 1077--1082

• [11] D. Q. Lu, H. M. Srivastava, Some series identities involving the generalized Apostol type and related polynomials, Comput. Math. Appl., 62 (2011), 3591--3602