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Abstract and Applied Analysis
Volume 2008 (2008), Article ID 793297, 14 pages
http://dx.doi.org/10.1155/2008/793297
Research Article

On Multiple Twisted -adic -Euler -Functions and -Functions

1National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu, Daejeon 305-340, South Korea
2Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
3Department of Mathematics, Kyungnam University, Masan 631-701, South Korea

Received 6 May 2008; Accepted 13 August 2008

Academic Editor: Agacik Zafer

Copyright © 2008 Min-Soo Kim 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 give the existence of multiple twisted -adic -Euler -functions and -functions, which are generalization of the twisted -adic -zeta functions and twisted -adic -Euler -functions in the work of Ozden and Simsek (2008).

1. Introduction, Definitions, and Notations

The constants in the Taylor series expansion are known as the first kind Euler numbers (cf. [1]). From the generating function of the first kind Euler numbers, we note that and for The first few are and for Those numbers play an important role in number theory. For example, the Euler zeta-function essentially equals an Euler numbers at nonpositive integer: where (see [110]).

Throughout this paper , and will denote the ring of integers, the ring of -adic rational integers, the field of -adic rational numbers, and the completion of the algebraic closure of respectively. 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 then we normally assume so that for If then we assume that Also we use the following notations: cf. [24]. For the -adic -integral on was defined by Kim (cf. [24]) as follows: Furthermore, we can consider the fermionic integral in contrast to the conventional bosonic integral. That is, (cf. [5]). From this, we derive where Substitute into (1.7). The twisted -extension of Euler numbers is defined by [8] For we consider fermionic -adic -integral on which is the -extension of as follows: (cf. [5]). From (1.9), we can derive the following formula [5]: where is translation with If we take then we have From (1.10), we derive Hence, we obtain By (1.11), we define the twisted -Euler numbers, by means of the following generating function (cf. [5]): These numbers are interpolated by the twisted Euler -zeta function which is defined as follows: Note that is analytic function in the whole complex plane

In view of the functional equation for the twisted Euler -zeta function at nonpositive integers, we have (cf. [5]).

Twisted -Bernoulli and Euler numbers and polynomials are very important not only in practically every field of mathematics, in particular in combinatorial theory, finite difference calculus, numerical analysis, numbers theory, but also probability theory. Recently the -extensions of those Euler numbers (polynomials) and the multiple of -extensions of those Euler numbers (polynomials) have been studied by many authors, (cf. [115]). In [8], Ozden and Simsek have studied -extensions of twisted Euler numbers and polynomials by using -adic -integral on the ring of -adic integers From their -extensions of twisted Euler numbers and polynomials, they have derived -adic -extensions of Euler zeta function and -adic -extensions of Euler -functions. They also gave some interesting relations between their -Euler numbers and -Euler zeta functions, and found the -adic twisted interpolation function of the generalized twisted -Euler numbers. In [11], Jang defined twisted -Euler numbers and polynomials of higher order, and studied multiple twisted -Euler zeta functions. He also derived sums of products of -Euler numbers and polynomials by using fermionic -adic -integral. In [7, 9], Ozden et al. defined multivariate Barnes-type Hurwitz -Euler zeta functions and -functions. They also gave relation between multivariate Barnes-type Hurwitz -Euler zeta functions and multivariate -Euler -functions. In [16], Kim constructed multiple -adic -functions, which interpolate the Bernoulli numbers of higher order. He also derived that the values of the partial derivative of this multiple -adic -function at are given.

In this paper, we consider twisted -Euler numbers and polynomials of higher order, and study multiple twisted -adic, -Euler, -functions, and -functions, which are generalization of the twisted -adic -zeta functions and twisted -adic -Euler -functions in [8].

2. Preliminaries

We assume that with Let be a primitive th root of unity.

For an integer the twisted -Euler polynomials of higher order (the index may be negative), are defined by means of the following generating function (cf. [11, 14]): where Note that so Of course the explicit formulas in (2.1) depend on which is a positive integer. If in the above, we obtain the generating function of the twisted -extension of Euler polynomials in [8, cf. Section 1, (1.3)]. In fact, if then Therefore, the generating function is the form The twisted -Euler numbers of higher order are Then, it is immediate that

From now on, we assume and in general whenever is actually an index then Jang [11] defined the two-variable multiple twisted -Euler zeta functions as follows. Definition 2.1. For and one defines

is an analytical function in the whole complex plane.

The value of at nonpositive integers, is given explicitly as follows. Theorem 2.2 (see [11]). Let Then,

Let be a Dirichlet character with odd conductor We define a twisted Dirichlet's type -Euler polynomials of higher order by means of the following generating function (cf. [11, 14]):

We now see that the twisted Dirichlet's type -Euler polynomials of higher order are easily expressed by the twisted -Euler polynomials of higher order as follows. Proposition 2.3. Let be an odd multiple of the conductor Then, Proof. Let By (2.1) and (2.5), we note that Then, we have On the other hand, if then we get This completes the proof.

The two-variable multiple twisted -Euler -functions are defined by the following definition. Definition 2.4 (see [14]). Let be a Dirichlet character. For and one has

The value of at nonpositive integers is given explicitly by the following theorem. Theorem 2.5 (see [14]). Let Then Proof. Let be a Dirichlet character with odd conductor and let be an odd number of multiple Set and Beside the multiple twisted -Euler -function we consider the multiple twisted -Euler zeta function in Definition 2.1. Then (cf. [14]).
In the integral for we make the change of variable where to obtain or Summing over all we find This gives us If we divide the infinite integral into two parts: it is easily seen that the second term is an entire function on
Consider By the definition of we have Therefore, This has an analytic continuation to a meromorphic function in the entire complex plane. It is holomorphic except at where it has a pole of order 1. Note that is holomorphic except at where it has a pole of order 1. does not have a zero. Therefore, has an analytic continuation to the whole complex plane. For an integer , we have If we have and thus we obtain Consequently, by using Propositions 2.3 and (2.6) and the above equation, we have Therefore, we obtain another proof of Theorem 2.5.
Remark 2.6 (see [11, 14]). We put Let and let From (2.1) and (2.4), we obtain the following: Similarly, by (2.5) and (2.11), we have

3. Partial Multiple Twisted -Euler -Functions

Let and with as an odd integer and where Then, partial multiple twisted -Euler -functions are as follows (cf. [14, 16, 1820]): We give a relationship between and as follows. For substituting with as an odd into (3.1), we have

By using (2.3) and Theorem 2.2 and substituting in the above, we arrive at the following theorem.

Theorem 3.1. Let be an odd integer, and let Then In particular, if then

By using Theorem 3.1 and (2.11), we arrive at the following theorem.

Theorem 3.2. Let be a Dirichlet character with conductor and as an odd multiple of Then, where and

4. Multiple Twisted -Adic -Euler -Functions

Let be an odd prime. and will always denote, respectively, the ring of -adic integers, the field of -adic numbers, and the completion of the algebraic closure of Let ( the field of rational numbers) denote the -adic valuation of normalized so that The absolute value on will be denoted as and for We let A -adic integer in is sometimes called a -adic unit. For each integer will denote the multiplicative group of the primitive -th roots of unity in Set The dual of in the sense of -adic Pontrjagin duality, is the direct limit (under inclusion) of cyclic groups of order with the discrete topology.

When one talks of -extension, is variously considered as an indeterminate, a complex number or a -adic number If then we normally assume

We will consider the -adic analogue of the -functions which are introduced in the previous section. In order to consider -adic and complex -functions simultaneously, we will use an isomorphism, between the algebraic closure of the rational numbers in and the algebraic closure of the rational numbers within the complex numbers Our purpose is to discuss the values of -functions, so we will consider as fixed throughout this section and use to identify -adic algebraic numbers with complex algebraic numbers. We will write when and

Let be denoted as the Teichmüller character having conductor For an arbitrary character let where in sense of the product of characters. We put whenever We then have for these values of Note that we extend this notation by defining for all with and such that Thus, so that (cf. [21, 22]).

The significance of Theorem 3.1 lies in the fact that the right-hand side is essentially a liner combination of terms of the form which makes sense when is replaced by a -adic variable and Set (cf. [8, 16, 1822]). Let be an odd integer, and let with for Suppose that and with We apply [18, Proposition 5.8, page 53] to the series Let where In [13], we see that since Observe that we have for odd so that we can take and in [18, Proposition 5.8]. This prove that (4.6) is analytic in Note that is analytic in for and such that Definition 4.1. Let be an odd integer, and let with for Suppose that and with One defines the partial multiple twisted -adic -Euler -functions for :

Theorem 4.2. Let be an odd integer with and let with for Suppose that and with Then is a -adic analytic function on such that for In particular, if then Proof. We have already remarked that is analytic in for and such that Also we see that (4.6) is analytic in It is clear that is a -adic analytic function on For one has This completes the proof.Definition 4.3. Let be a Dirichlet character with odd conductor and let be a positive multiple of and We can define the multiple twisted -adic -Euler -function:

Theorem 4.4. Let be a Dirichlet's character with an odd conductor and let be a positive multiple of and Then is a -adic analytic function on with where and and in one sums over as many times as is expressed in the form by various 's, and with the Teichmüller in the sense of the product of characters.Remark 4.5. Theorem 4.4 can be extended to obtain similar results for the multiple -adic -function in [16]. In the case we note that Observe that if then This function interpolates the twisted generalized -Euler polynomials at negative integers. For the twisted -adic -Euler -functions similar results were obtained (cf. see for detail [8, Theorem 9]). If in the above, then which is called the twisted -adic -function of two variables.Proof. The formula for is -adic analytic function in by the Theorem 4.2. On the other hand, by substituting into Definition 4.3, we have From Theorems 2.2, 3.1, and 4.2, we obtain Therefore, This completes the proof.

Acknowledgment

This work is supported by Kyungnam University Foundation Grant, 2008.

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