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

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:

(1.1)

for all (see [1–6]).

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

(1.2)

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

(1.3)

(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

(1.4)

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

(1.5)

Note that if in , then

(1.6)

The Barnes' type Euler polynomials are considered as follows:

(1.7)

where (cf. [7]).

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

(1.8)

For , let , one has

(1.9)

By (1.9), we see that

(1.10)

From (1.10), we note that

(1.11)

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:

(2.1)

where

(2.2)

(see [1, 5]).

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

(2.3)

Therefore, we obtain the following theorem.

Theorem 2.1.

Let and . For , one has

(2.4)

By (1.7), we easily see that

(2.5)

From (1.7), we can derive

(2.6)

By (2.6), one has

(2.7)

Hence we obtain the following theorem.

Theorem 2.2.

For and , one has

(2.8)

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

(2.9)

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

Theorem 2.3.

Let and . For with , one has

(2.10)

For the special case in Theorem 2.3, one has

(2.11)

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

Corollary 2.4.

For and , one has

(2.12)

From (2.3), we note that

(2.13)

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

Theorem 2.5.

For with (mod), one has

(2.14)

Remark 2.6.

Let

(2.15)

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

(2.16)

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

(2.17)

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

(2.18)

Thus, one has

(2.19)