Abstract and Applied Analysis

Volume 2014 (2014), Article ID 286239, 6 pages

http://dx.doi.org/10.1155/2014/286239

Research Article

Identities of Symmetry for Higher-Order Generalized -Euler Polynomials

D. V. Dolgy,¹ D. S. Kim,² T. G. Kim,3,4 and J. J. Seo⁵

¹Institute of Mathematics and Computer Sciences, Far Eastern Federal University, Vladivostok 690060, Russia

²Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea

³Jangjeon Research Institute for Mathematics and Physics, 252-5 Hapcheon-Dong, Hapcheon-Gun Kyungshang Nam-Do 678-800, Republic of Korea

⁴Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

⁵Department of Applied Mathematics, Pukyong National University, Busan 608-737, Republic of Korea

Received 17 December 2013; Accepted 18 January 2014; Published 25 February 2014

Academic Editor: Alberto Fiorenza

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.

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).

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