Open Access

On Interpolation Functions of the Generalized Twisted -Euler Polynomials

Journal of Inequalities and Applications20092009:946569

DOI: 10.1155/2009/946569

Received: 5 November 2008

Accepted: 14 January 2009

Published: 20 January 2009

Abstract

The aim of this paper is to construct -adic twisted two-variable Euler-( , )- -functions, which interpolate generalized twisted ( , )-Euler polynomials at negative integers. In this paper, we treat twisted ( , )-Euler numbers and polynomials associated with -adic invariant integral on . We will construct two-variable twisted ( , )-Euler-zeta function and two-variable ( , )- -function in Complex -plane.

1. Introduction

Tsumura and Young treated the interpolation functions of the Bernoulli and Euler polynomials in [1, 2]. Kim and Simsek studied on -adic interpolation functions of these numbers and polynomials [348]. In [49], Carlitz originally constructed -Bernoulli numbers and polynomials. Many authors studied these numbers and polynomials [4, 28, 38, 41, 50]. After that, twisted -Bernoulli and Euler numbers(polynomials) were studied by several authors [132, 3265]. In [62], Whashington constructed one-variable -adic- -function which interpolates generalized classical Bernoulli numbers at negative integers. Fox introduced the two-variable -adi -functions [53]. Young defined -adic integral representation for the two-variable -adic -functions [64]. Furthermore, Kim constructed the two-variable -adic - -function, which is interpolation function of the generalized -Bernoulli polynomials [8]. This function is the -extension of the two-variable -adic -function. Kim constructed -extension of the generalized formula for two-variable of Diamond and Ferrero and Greenberg formula for two-variable -adic -function in the terms of the -adic gamma and log-gamma functions [8]. Kim and Rim introduced twisted -Euler numbers and polynomials associated with basic twisted - -functions [28]. Also, Jang et al. investigated the -adic analogue twisted - -function, which interpolates generalized twisted -Euler numbers attached to Dirichlet's character [55]. Kim et al. have studied two-variable -adic -functions, which interpolate the generalized Bernoulli polynomials at negative integers. In this paper, we will construct two-variale -adic twisted Euler - -functions. This functions interpolation functions of the generalized twisted -Euler polynomials.

Let be a fixed odd prime number. Throughout this paper and will respectively denote the ring of rational integers, the ring of -adic rational integers, the field of -adic rational numbers and the completion of the algebraic closure of . Let be the normalized exponential valuation of such that . If , then . If , we normally assume , so that for . Throughout this paper we use the following notations (cf. [132, 3248, 50, 51, 5465]):
(1.1)

Hence, , for any with in the present -adic case.

For a fixed positive integer with , set
(1.2)
where satisfies the condition . The distribution is defined by
(1.3)

We say that is uniformly differential function at a point , and we write , if the difference quotients, have a limit as .

For , the -adic invariant -integral on is defined as [4, 18]
(1.4)
The fermionic -adic -measures on is defined as (cf. [1416, 18, 22, 28])
(1.5)
for . For , the ferminoic -adic invariant -integral on is defined as
(1.6)
which has a sense as we see readily that the limit is convergent. For , we note that (cf. [14, 16, 18, 22, 28])
(1.7)
From the fermionic invariant integral on , we derive the following integral equation (cf. [14, 35]):
(1.8)

where .

2. Twisted -Euler Numbers and Polynomials

In this section, we will treat some properties of twisted -Euler numbers and polynomials associated with -adic invariant integral on . From now on, we take and with . Let be the space of primitive th root of unity,
(2.1)
Then, we denote
(2.2)
Hence is a -adic locally constant space. For , we denote by defined by , the locally constant function. If we take , then we have (cf. [35])
(2.3)
By induction in (1.8), Kim constructed the following useful identity (cf. [14, 28]):
(2.4)
where . From (2.4), if is odd, then we have
(2.5)
If we replace by into (2.5), we obtain
(2.6)
Let . Let be a Dirichlet's character of conductor , which is any multiple of with . By substituting into (2.6), we have
(2.7)

Remark 2.1.

In complex case, the generating function of the Euler numbers is given by (cf. [28])
(2.8)
By using Taylor series of , then we can define the generalized twisted Euler numbers attached to as follows (cf. [55]):
(2.9)
In [8], -Euler numbers were defined by
(2.10)

where and . In particular, if we take , then . These numbers are called -Euler numbers.

By using iterative method of -adic invariant integral on in the sense of fermionic, we define twisted -Euler numbers as follows (cf. [55]):
(2.11)
For and , we have that (cf. [55])
(2.12)
(2.13)

where with .

Let be the generating function of in complex plane as follows (cf. [55]):
(2.14)

Let be the Dirichlet's character with conductor with . Then the generalized twisted -Euler polynomials attached to is given by as follows:

For ,
(2.15)

where is any multiple of with and .

Then the distribution relation of the generalized twisted -Euler polynomials is given by as follows (cf. [14]):
(2.16)

3. Two-Variable Twisted -Euler-Zeta Function and - -Function

In this section, we will construct two-variable twisted -Euler-zeta function and two-variable - -function in Complex -plane. We assume with .

Firstly, we consider twisted -Euler numbers and polynomials in as follows (cf. [55]):
(3.1)

where and is th root of unity. In particular, if we take , then we have . These numbers are called twisted Euler numbers. By using derivative operator, we have .

From (3.1), we can define Hurwitz-type twisted -Euler-zeta function as follows (cf. [55]):
(3.2)
where and . Note that if in (3.2), then we see that the twisted -Euler-zeta function is defined by (cf. [28, 55])
(3.3)
For , we know (cf. [28])
(3.4)

From now on, we will define the two-variable - -functions which interpolates the generalized -Euler polynomials.

Definition 3.1.

Let be the Dirichlet's character with conductor with . For and , we define
(3.5)
By substituting and into (3.5), then using (3.2), we have
(3.6)

Thus, we see the function which interpolates the generalized -Euler polynomials as follows.

Theorem 3.2.

For , let be the Dirichlet's character with conductor with . Then one has
(3.7)
By substituting with , into (3.7), we obtain
(3.8)

where .

Thus, we have the following theorem.

Theorem 3.3.

For , let be the Dirichlet's character with conductor with . Then one has
(3.9)

Remark 3.4.

If we take in (3.5), then we have (cf. [28, 55])
(3.10)

From (3.9) and (3.10), we have the following corollary.

Corollary 3.5.

Let be the Dirichlet's character with conductor with . Then one has
(3.11)

Secondly, we will define two-variable twisted Euler - -function as follows.

Definition 3.6.

Let be the Dirichlet's character with conductor with . For and with , we define
(3.12)
We consider the well-known identity (cf. [44, 65])
(3.13)
By using (3.12), we define two-variable twisted Euler - -function as follows:
(3.14)

We will investigate the relations between and as follows.

Substituting with into (3.12), we have
(3.15)

Thus we obtain the following theorem.

Theorem 3.7.

For with , let be the Dirichlet character with conductor with and with . Then one has
(3.16)
By substituting with into (3.16) and using (3.4), we can obtain
(3.17)

Thus, we see that the function interpolates generalized -Euler polynomials attached to at negative integer values of as followings.

Theorem 3.8.

For , let be the Dirichlet's character with odd conductor . Then one has
(3.18)

Note that if we take , then Theorem 3.8 reduces to Theorem 3.3.

Let and be integers with and . For , we define partial -Hurwitz type zeta function as follows:
(3.19)
By substituting , we have
(3.20)
By substituting (3.2), for , we get
(3.21)

Equation (3.20) means that the function interpolates polynomials at negative integers.

From (3.16) and (3.20), we have the following theorem.

Theorem 3.9.

For with , let be the Dirichlet's character with conductor with and , is any multiple of . Then one has
(3.22)

Remark 3.10.

If we take in (3.22), then we have
(3.23)

From (2.12), if we take , then we have the following corollary.

Corollary 3.11.

For with , let be the Dirichlet's character with conductor with and , is any multiple of . Then one has
(3.24)

4. -Adic Twisted Two-Variable Euler -L-Functions

In [62], Washington constructed one-variable -adic- -function which interpolates generalized classical Bernoulli numbers negative integers. Kim [22] investigated the -adic analogues of two-variables Euler - -function. In this section, we will construct -adic twisted two-variable Euler- - -functions, which interpolate generalized twisted -Euler polynomials at negative integers. Our notations and methods are essentially due to Kim and Washington (cf. [22, 62]).

We assume that with , so that . Let be an odd prime number. Let denote the Teichmüller character having conductor . For an arbitrary character , we define , where , in the sense of the product of characters. Let . Then . Hence we see that
(4.1)

where with .

We denote the subset of by (cf. [62])
(4.2)
Let
(4.3)

be a sequence of power series, each of which converges in a fixed subset such that

(1) as and

(2)for each and , there exists such that
(4.4)

Then for all (cf. [2, 22, 50, 51, 60, 62]).

Let be the Dirichlet's character with conductor with and let be a positive multiple of and .

Now we set
(4.5)
Then is analytic for with , when . For with , we have
(4.6)
is analytic for . It readily follows that
(4.7)
is analytic for with when . Thus we see that
(4.8)
Let and fixed with . Then we have that
(4.9)
If , then , so is a multiple of . Therefore, we have
(4.10)
Then we note that
(4.11)
The difference of these equations yields
(4.12)
Using distribution for -Euler polynomials, we easily see that
(4.13)
Since , for , and , with , we have
(4.14)
From (4.5)–(4.14), we can derive that
(4.15)

Therefore we obtain the following theorem.

Theorem 4.1.

Let be a positive integral multiple of and with , and let
(4.16)
Then is analytic for , provides when . Furthermore, for each , we have
(4.17)

Thus we note that for all , where is twisted -adic Euler - -function, (cf. [15, 22]).

We now generalized to two-variable -adic Euler - -function, which is first defined by the interpolation function
(4.18)

for .

From (4.18), we have that
(4.19)
By using the definition of , we can express for all and with as follows:
(4.20)
We know that is analytic for , when . The value of is the coefficients of in the expansion of at . Using the Taylor expansion at , we see that
(4.21)
The -adic logarithmic function, , is the unique function that satisfies
(4.22)
By employing these expansion and some algebraic manipulations, we evaluate the derivative . It follows from the definition of that
(4.23)
Thus, we have
(4.24)
Since is a root of unity for , we have
(4.25)

Thus we have the following theorem.

Theorem 4.2.

Let be a primitive Dirichlet's character with odd conductor and let be a odd positive integral multiple of and . Then for any with , one has
(4.26)

Authors’ Affiliations

(1)
Department of Mathematics, Sogang University

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© Kyoung Ho Park. 2009

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