Research Article | Open Access

# Identities of Symmetry for Higher-Order Generalized -Euler Polynomials

**Academic Editor:**Alberto Fiorenza

#### Abstract

We investigate the properties of symmetry in two variables related to multiple Euler --function which interpolates higher-order -Euler polynomials at negative integers. From our investigation, we can derive many interesting identities of symmetry in two variables related to generalized higher-order -Euler polynomials and alternating generalized -power sums.

#### 1. Introduction

Throughout this paper, the notations , , , and denote the sets of positive integers, integers, real numbers, and complex numbers, respectively, and . Let be a Dirichlet character with with conductor . Then the generalized Euler polynomials attached to are defined by the following generating function (see [1–3]): The generalized Euler polynomials of order attached to are also defined by the generating function: When , are called the generalized Euler numbers attached to (see [2, 4]).

Assume that with and define -numbers by (see [2–15]) Note that .

In [4, 8], Kim initiated to consider various -extensions (or -extensions) of Euler numbers and polynomials and constructed analytic continuations which interpolate his -numbers and polynomials. Until recently, many authors have studied -Euler or -Euler polynomials due to him (see [1–21]). In [4], Kim defined the -extension of generalized higher-order Euler polynomials attached to which is given by the generating function: where and .

Note that When , are called the -extension of generalized higher-order Euler numbers attached to .

We find from (4) that with the usual convention about replacing by .

In [4], Dirichlet-type multiple -function is defined by Kim to be where and , with .

By using Cauchy residue theorem, we get In this paper, we investigate certain properties of symmetry in two variables related to Dirichlet-type multiple -function which interpolates the -extension of generalized higher-order Euler polynomials attached to at negative integers. From our investigation, we can derive many interesting identities of symmetry in two variables related to -extension of generalized higher-order Euler polynomials and alternating generalized -power sums.

#### 2. Identities for the -Extension of Generalized Higher-Order Euler Polynomials

In this section, we assume that is a Dirichlet character with conductor with .

Let with and and . First, we observe that Thus, by (9), we get By using the same method as (10), we get

Therefore, by (10) and (11), we obtain the following theorem.

Theorem 1. *For with and , one has
*

By (8) and Theorem 1, we obtain the following theorem.

Theorem 2. *For and with and , one has
*

From (6), we note that By (14), we get where From (15), we have By using the same method as in (17), we get Therefore, by (17) and (18), we obtain the following theorem.

Theorem 3. *For and , with and , one has
*

Now, we observe that The left hand side of (20) multiplied by is given by The right hand side of (20) multiplied by is given by Therefore, by (21) and (22), we obtain the following theorem.

Theorem 4. *For , , one has
*

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors would like to thank the referee for his/her valuable and detailed comments on the paper. This work was supported by the National Research Foundation of Korea (NRF) Grant funded by the Korean Government (MOE) (no. 2012R1A1A2003786).

#### References

- M. Cenkci, “The $p$-adic generalized twisted $(h,q)$-Euler-$l$-function and its applications,”
*Advanced Studies in Contemporary Mathematics*, vol. 15, no. 1, pp. 37–47, 2007. View at: Google Scholar | MathSciNet - T. Kim, “An identity of symmetry for the generalized Euler polynomials,”
*Journal of Computational Analysis and Applications*, vol. 13, no. 7, pp. 1292–1296, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Simsek, “Complete sum of products of $(h,q)$-extension of Euler polynomials and numbers,”
*Journal of Difference Equations and Applications*, vol. 16, no. 11, pp. 1331–1348, 2010. View at: Publisher Site | Google Scholar | MathSciNet - T. Kim, “New approach to $q$-Euler polynomials of higher order,”
*Russian Journal of Mathematical Physics*, vol. 17, no. 2, pp. 218–225, 2010. View at: Publisher Site | Google Scholar | MathSciNet - D. S. Kim, “Symmetry identities for generalized twisted Euler polynomials twisted by unramified roots of unity,”
*Proceedings of the Jangjeon Mathematical Society*, vol. 15, no. 3, pp. 303–316, 2012. View at: Google Scholar | Zentralblatt MATH | MathSciNet - T. Kim, “Symmetry $p$-adic invariant integral on ${\mathbb{Z}}_{p}$ for Bernoulli and Euler polynomials,”
*Journal of Difference Equations and Applications*, vol. 14, no. 12, pp. 1267–1277, 2008. View at: Publisher Site | Google Scholar | MathSciNet - T. Kim, “On $p$-adic interpolating function for $q$-Euler numbers and its derivatives,”
*Journal of Mathematical Analysis and Applications*, vol. 339, no. 1, pp. 598–608, 2008. View at: Publisher Site | Google Scholar | MathSciNet - T. Kim, “$q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals,”
*Journal of Nonlinear Mathematical Physics*, vol. 14, no. 1, pp. 15–27, 2007. View at: Publisher Site | Google Scholar | MathSciNet - V. Kurt, “Some symmetry identities for the Apostol-type polynomials related to multiple alternating sums,”
*Advances in Difference Equations*, vol. 2013, article 32, 8 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet - B. Kurt, “Some formulas for the multiple twisted $(h,q)$-Euler polynomials and numbers,”
*Applied Mathematical Sciences*, vol. 5, no. 25–28, pp. 1263–1270, 2011. View at: Google Scholar | MathSciNet - H. Ozden and Y. Simsek, “A new extension of $q$-Euler numbers and polynomials related to their interpolation functions,”
*Applied Mathematics Letters*, vol. 21, no. 9, pp. 934–939, 2008. View at: Publisher Site | Google Scholar | MathSciNet - H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on sum of products of $(h,q)$-twisted Euler polynomials and numbers,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 816129, 8 pages, 2008. View at: Publisher Site | Google Scholar | MathSciNet - S.-H. Rim and J. Jeong, “On the modified $q$-Euler numbers of higher order with weight,”
*Advanced Studies in Contemporary Mathematics*, vol. 22, no. 1, pp. 93–98, 2012. View at: Google Scholar | MathSciNet - Y. Simsek, “Twisted $p$-adic $(h,q)\text{-}L$-functions,”
*Computers & Mathematics with Applications*, vol. 59, no. 6, pp. 2097–2110, 2010. View at: Publisher Site | Google Scholar | MathSciNet - Y. Simsek, “Interpolation functions of the Eulerian type polynomials and numbers,”
*Advanced Studies in Contemporary Mathematics*, vol. 23, no. 2, pp. 301–307, 2013. View at: Google Scholar | MathSciNet - S. Araci, M. Acikgoz, and E. Şen, “On the extended Kim's $p$-adic $q$-deformed fermionic integrals in the $p$-adic integer ring,”
*Journal of Number Theory*, vol. 133, no. 10, pp. 3348–3361, 2013. View at: Publisher Site | Google Scholar | MathSciNet - S. Araci, J. J. Seo, and D. Erdal, “New construction weighted $(h,q)$-Genocchi numbers and polynomials related to zeta type functions,”
*Discrete Dynamics in Nature and Society*, vol. 2011, Article ID 487490, 7 pages, 2011. View at: Publisher Site | Google Scholar | MathSciNet - I. N. Cangul, H. Ozden, and Y. Simsek, “Generating functions of the $(h,q)$ extension of twisted Euler polynomials and numbers,”
*Acta Mathematica Hungarica*, vol. 120, no. 3, pp. 281–299, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K.-W. Hwang, D. V. Dolgy, D. S. Kim, T. Kim, and S. H. Lee, “Some theorems on Bernoulli and Euler numbers,”
*Ars Combinatoria*, vol. 109, pp. 285–297, 2013. View at: Google Scholar | MathSciNet - D. S. Kim, “Identities of symmetry for generalized Euler polynomials,”
*International Journal of Combinatorics*, vol. 2011, Article ID 432738, 12 pages, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. S. Kim, N. Lee, J. Na, and K. H. Park, “Identities of symmetry for higher-order Euler polynomials in three variables (II),”
*Journal of Mathematical Analysis and Applications*, vol. 379, no. 1, pp. 388–400, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 D. V. Dolgy et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.