- Research Article
- Open Access
- Published:

# Note on the -Extension of Barnes' Type Multiple Euler Polynomials

*Journal of Inequalities and Applications*
**volume 2009**, Article number: 136532 (2009)

## Abstract

We construct the -Euler numbers and polynomials of higher order, which are related to Barnes' type multiple Euler polynomials. We also derive many properties and formulae for our -Euler polynomials of higher order by using the multiple integral equations on .

## 1. Introduction

Let be a fixed odd prime number. Throughout this paper, symbols , , , and will denote the ring of rational integers, the ring of -adic integers, the field of -adic rational numbers, and the completion of algebraic closure of , respectively. Let be the set of natural numbers and . Let be the normalized exponential valuation of with . When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number . If one normally assumes If then one normally assumes We use the following notations:

Let a fixed positive odd integer with . For , we set

where lies in . The fermionic -adic -measures on are defined as

(see [5]).

We say that is a uniformly differentiable function at a point and write , if the difference quotients have a limit as . For , let us begin with expression

which represents a -analogue of Riemann sums for in the fermionic sense (see [4, 5]). The integral of on is defined by the limit of these sums (as ) if this limit exists. The fermionic invariant -adic -integral of function is defined as

Note that if in , then

The Barnes' type Euler polynomials are considered as follows:

where (cf. [7]).

From (1.5), we can derive the fermionic invariant integral on as follows:

For , let , one has

By (1.9), we see that

From (1.10), we note that

In the view point of (1.11), we try to study the -extension of Barnes' type Euler polynomials by using the -extension of fermionic -adic invariant integral on .

The purpose of this paper is to construct the -Euler numbers and polynomials of higher order, which are related to Barnes' type multiple Euler numbers and polynomials. Also, we give many properties and formulae for our -Euler polynomials of higher order. Finally, we give the generating function for these -Euler polynomials of higher order.

## 2. Barnes' Type Multiple -Euler Polynomials

Let . For and with , we define the Barnes' type multiple -Euler polynomials as follows:

where

In the special case , are called the Barnes' type multiple -Euler numbers. From (2.1), one has

Therefore, we obtain the following theorem.

Theorem 2.1.

Let and . For , one has

By (1.7), we easily see that

From (1.7), we can derive

By (2.6), one has

Hence we obtain the following theorem.

Theorem 2.2.

For and , one has

It is not difficult to show that the following integral equation is satisfied:

where with . By (2.9), we obtain the following theorem.

Theorem 2.3.

Let and . For with , one has

For the special case in Theorem 2.3, one has

By (2.1), (2.3), and (2.9), we obtain the following corollary.

Corollary 2.4.

For and , one has

From (2.3), we note that

where is an odd positive integer. By (2.13), we obtain the following theorem.

Theorem 2.5.

For with (mod), one has

Remark 2.6.

Let

From (2.4), we can easily derive the following equation:

By differentiating both sides of (2.16) with respect to and comparing coefficients on both sides, one has

The inversion formula of Equation (2.4) at is given by

Thus, one has

## References

Acikgoz M, Simsek Y:

**On multiple interpolation functions of the Nörlund-type**-**Euler polynomials.***Abstract and Applied Analysis*2009, Article ID 382574,**vol. 2009:**-14 Pages.Cangul IN, Kurt V, Ozden H, Simsek Y:

**On the higher-order**--**genocchi numbers.***Advanced Studies in Contemporary Mathematics*2009,**vol. 19**no.(1):pp.39–57.Govil NK, Gupta V:

**Convergence of**-**Meyer-König-Zeller-Durrmeyer operators.***Advanced Studies in Contemporary Mathematics*2009,**vol 19:**pp. 97–108.Kim T: -

**Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**vol. 9**no.(3):pp. 288–299.Kim T:

**Some identities on the**-**Euler polynomials of higher order and**-**stirling numbers by the fermionic**-**adic integral on**.*Russian Journal of Mathematical Physics*2010.,**vol 17:**Kim Y-H, Kim W, Ryoo CS:

**On the twisted**-**Euler zeta function associated with twisted**-**Euler numbers.***Proceedings of the Jangjeon Mathematical Society*2009,**vol. 12**no. (1):pp. 93–100.Simsek Y, Kim T, Pyung IS:

**Barnes' type multiple Changhee**-**zeta functions.***Advanced Studies in Contemporary Mathematics*2005,**vol. 10**no. (2):pp. 121–129.

## Acknowledgment

This paper was supported by Konkuk University (2009).

## Author information

### Authors and Affiliations

### Corresponding authors

## Rights and permissions

**Open Access**
This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (
https://creativecommons.org/licenses/by/2.0
), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

Jang, L., Kim, T., Kim, YH. *et al.* Note on the -Extension of Barnes' Type Multiple Euler Polynomials.
*J Inequal Appl* **2009**, 136532 (2009). https://doi.org/10.1155/2009/136532

Received:

Accepted:

Published:

DOI: https://doi.org/10.1155/2009/136532

### Keywords

- Integral Equation
- Inversion Formula
- Euler Polynomial