Open Access

Multiple Twisted -Euler Numbers and Polynomials Associated with -Adic -Integrals

Advances in Difference Equations20082008:738603

DOI: 10.1155/2008/738603

Received: 14 January 2008

Accepted: 26 February 2008

Published: 28 February 2008

Abstract

By using -adic -integrals on , we define multiple twisted -Euler numbers and polynomials. We also find Witt's type formula for multiple twisted -Euler numbers and discuss some characterizations of multiple twisted -Euler Zeta functions. In particular, we construct multiple twisted Barnes' type -Euler polynomials and multiple twisted Barnes' type -Euler Zeta functions. Finally, we define multiple twisted Dirichlet's type -Euler numbers and polynomials, and give Witt's type formula for them.

1. Introduction

Let be a fixed odd prime number. Throughout this paper, , , and are, respectively, the ring of -adic rational integers, the field of -adic rational numbers, and the -adic completion of the algebraic closure of . The -adic absolute value in is normalized so that . When one talks about -extension, is variously considered as an indeterminate, a complex number, or a -adic number . If , one normally assumes that . If , one normally assumes that so that for each . We use the notations
(1.1)
  (cf. [114]), for all . For a fixed odd positive integer with , set
(1.2)
where lies in . For any ,
(1.3)

is known to be a distribution on (cf. [128]).

We say that is uniformly differentiable function at a point and denote this property by if the difference quotients
(1.4)

have a limit as (cf. [25]).

The -adic -integral of a function was defined as
(1.5)
(1.6)
(cf. [4, 24, 25, 28]), from (1.6), we derive
(1.7)
where . If we take , then we have . From (1.7), we obtain that
(1.8)

In Section 2, we define the multiple twisted -Euler numbers and polynomials on and find Witt's type formula for multiple twisted -Euler numbers. We also have sums of consecutive multiple twisted -Euler numbers. In Section 3, we consider multiple twisted -Euler Zeta functions which interpolate new multiple twisted -Euler polynomials at negative integers and investigate some characterizations of them. In Section 4, we construct the multiple twisted Barnes' type -Euler polynomials and multiple twisted Barnes' type -Euler Zeta functions which interpolate new multiple twisted Barnes' type -Euler polynomials at negative integers. In Section 5, we define multiple twisted Dirichlet's type -Euler numbers and polynomials and give Witt's type formula for them.

2. Multiple Twisted -Euler Numbers and Polynomials

In this section, we assume that with . For , by the definition of -adic -integral on , we have
(2.1)
where . If is odd positive integer, we have
(2.2)
Let be the locally constant space, where is the cyclic group of order . For , we denote the locally constant function by
(2.3)
(cf. [5, 714, 16, 18]). If we take , then we have
(2.4)
Now we define the twisted -Euler numbers as follows:
(2.5)
We note that by substituting , are the familiar Euler numbers. Over five decades ago, Carlitz defined -extension of Euler numbers (cf. [15]). From (2.4) and (2.5), we note that Witt's type formula for a twisted -Euler number is given by
(2.6)

for each and .

Twisted -Euler polynomials are defined by means of the generating function
(2.7)
where . By using the th iterative fermionic -adic -integral on , we define multiple twisted -Euler number as follows:
(2.8)

Thus we give Witt's type formula for multiple twisted -Euler numbers as follows.

Theorem 2.1.

For each and ,
(2.9)
where
(2.10)

From (2.8) and (2.9), we obtain the following theorem.

Theorem 2.2.

For and ,
(2.11)
From these formulas, we consider multivariate fermionic -adic -integral on as follows:
(2.12)
Then we can define the multiple twisted -Euler polynomials as follows:
(2.13)
From (2.12) and (2.13), we note that
(2.14)

Then by the th differentiation on both sides of (2.14), we obtain the following.

Theorem 2.3.

For each and ,
(2.15)
Note that
(2.16)
Then we see that
(2.17)

From (2.15) and (2.17), we obtain the sums of powers of consecutive -Euler numbers as follows.

Theorem 2.4.

For each and ,
(2.18)

3. Multiple Twisted -Euler Zeta Functions

For with and , the multiple twisted -Euler numbers can be considered as follows:

(31)

From (3.1), we note that

(32)
By the th differentiation on both sides of (3.2) at , we obtain that
(33)
From (3.3), we derive multiple twisted -Euler Zeta function as follows:
(34)

for all . We also obtain the following theorem in which multiple twisted -Euler Zeta functions interpolate multiple twisted -Euler polynomials.

Theorem 3.1.

For and ,
(35)

4. Multiple Twisted Barnes' Type -Euler Polynomials

In this section, we consider the generating function of multiple twisted -Euler polynomials:
(41)
We note that
(42)
By the th differentiation on both sides of (4.2) at , we obtain that
(43)
Thus we can consider multiple twisted Hurwitz's type -Euler Zeta function as follows:
(44)

for all and . We note that is analytic function in the whole complex -plane and . We also remark that if and , then is Hurwitz's type -Euler Zeta function (see [7, 27]). The following theorem means that multiple twisted -Euler Zeta functions interpolate multiple twisted -Euler polynomials at negative integers.

Theorem 4.1.

For , , , and ,
(45)
Let us consider
(46)
where and . Then will be called multiple twisted Barnes' type -Euler polynomials. We note that
(47)

By the th differentiation of both sides of (4.6), we obtain the following theorem.

Theorem 4.2.

For each , , , and ,
(48)
where
(49)
From (4.8), we consider multiple twisted Barnes' type -Euler Zeta function defined as follows: for each , , , and ,
(410)

We note that is analytic function in the whole complex -plane. We also see that multiple twisted Barnes' type -Euler Zeta functions interpolate multiple twisted Barnes' type -Euler polynomials at negative integers as follows.

Theorem 4.3.

For each , , , and ,
(411)

5. Multiple Twisted Dirichlet's Type -Euler Numbers and Polynomials

Let be a Dirichlet's character with conductor and . If we take , then we have . From (2.2), we derive
(51)
In view of (5.1), we can define twisted Dirichlet's type -Euler numbers as follows:
(52)

(cf. [17, 19, 21, 22]). From (5.1) and (5.2), we can give Witt's type formula for twisted Dirichlet's type -Euler numbers as follows.

Theorem 5.1.

Let be a Dirichlet's character with conductor . For each , , we have
(53)
We note that if , then is the generalized -Euler numbers attached to (see [18, 26]). From (5.2), we also see that
(54)
By (5.2) and (5.4), we obtain that
(55)
From (5.5), we can define the -function as follows:
(56)

for all . We note that is analytic function in the whole complex -plane. From (5.5) and (5.6), we can derive the following result.

Theorem 5.2.

Let be a Dirichlet's character with conductor . For each , , we have
(57)

Now, in view of (5.1), we can define multiple twisted Dirichlet's type -Euler numbers by means of the generating function as follows:

(58)

where . We note that if , then is a multiple generalized -Euler number (see [22]).

By using the same method used in (2.8) and (2.9),
(59)

From (5.9), we can give Witt's type formula for multiple twisted Dirichlet's type -Euler numbers.

Theorem 5.3.

Let be a Dirichlet's character with conductor . For each , , and , we have
(510)
where and
(511)

From (5.10), we also obtain the sums of powers of consecutive multiple twisted Dirichlet's type -Euler numbers as follows.

Theorem 5.4.

Let be a Dirichlet's character with conductor . For each , , and , we have
(512)
Finally, we consider multiple twisted Dirichlet's type -Euler polynomials defined by means of the generating functions as follows:
(513)
where and . From (5.13), we note that
(514)

Clearly, we obtain the following two theorems.

Theorem 5.5.

Let be a Dirichlet's character with conductor . For each , , , and , we have
(515)
where
(516)

Theorem 5.6.

Let be a Dirichlet's character with conductor . For each , , , and , we have
(517)

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Konkuk University

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Copyright

© Lee-Chae Jang. 2008

This article is published under license to BioMed Central Ltd. 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.