Abstract and Applied Analysis

Volume 2008 (2008), Article ID 793297, 14 pages

http://dx.doi.org/10.1155/2008/793297

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

^{1}National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu, Daejeon 305-340, South Korea^{2}Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea^{3}Department 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 expansionare 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 [1–10]).

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. [2–4]. Forthe -adic -integral on was defined by Kim (cf. [2–4]) as follows:Furthermore, we can consider the fermionic integral in contrast to the conventional bosonic integral. That is, (cf. [5]). From this, we derivewhere 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 obtainBy (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. [1–15]). 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 formThe 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 thatThen, we haveOn the other hand, if then we getThis 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 obtainorSumming over all we findThis gives usIf 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 haveTherefore,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 haveIf we have and thus we obtainConsequently, by using
Propositions 2.3 and (2.6) and the above equation, we haveTherefore, we obtain another
proof of Theorem 2.5.*Remark 2.6 (see [11, 14]). *We putLet 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, 18–20]):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 SetThe 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 putwhenever We then have for these values of Note that we extend this notation by definingfor 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 formwhich makes sense when is replaced by a -adic variable and Set(cf. [8, 16, 18–22]). Let be an odd integer, and let with for Suppose that and with We apply [18, Proposition 5.8, page 53] to the
seriesLet where In [13], we see thatsince 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 hasThis 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 thatObserve that if thenThis 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, thenwhich 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 haveFrom Theorems 2.2, 3.1,
and 4.2, we obtainTherefore,This completes the proof.

#### Acknowledgment

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

#### References

- T. Kim, “Euler numbers and polynomials associated with zeta functions,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 581582, 11 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kim, “On the $q$-extension of Euler and Genocchi numbers,”
*Journal of Mathematical Analysis and Applications*, vol. 326, no. 2, pp. 1458–1465, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “On $p$-adic interpolating function for $q$-Euler numbers and its derivatives,”
*Journal of Mathematical Analysis and Applications*, vol. 339, no. 1, pp. 598–608, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kim, “On the multiple $q$-Genocchi and Euler numbers,” to appear in
*Russian Journal of Mathematical Physics*. - T. Kim, M.-S. Kim, L. Jang, and S.-H. Rim, “New $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals,”
*Advanced Studies in Contemporary Mathematics*, vol. 15, no. 2, pp. 243–252, 2007. View at Google Scholar · View at MathSciNet - T. Kim, L.-C. Jang, and C.-S. Ryoo, “Note on $q$-extensions of Euler numbers and polynomials of higher order,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 371295, 9 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-order $q$-Euler numbers and their applications,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 390857, 16 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - H. Ozden and Y. Simsek, “Interpolation function of the $(h,q)$-extension of twisted Euler numbers,”
*Computers & Mathematics with Applications*, vol. 56, no. 4, pp. 898–908, 2008. View at Publisher · View at Google Scholar - H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on sum of products of $(h,q)$-twisted Euler polynomials and numbers,”
*Journal of Inequalities and Applications*, vol. 2008, Article ID 816129, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Simsek, “$q$-analogue of twisted $l$-series and $q$-twisted Euler numbers,”
*Journal of Number Theory*, vol. 110, no. 2, pp. 267–278, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L.-C. Jang, “Multiple twisted $q$-Euler numbers and polynomials associated with $p$-adic $q$-integrals,”
*Advances in Difference Equations*, vol. 2008, Article ID 738603, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L.-C. Jang and C.-S. Ryoo, “A note on the multiple twisted Carlitz's type $q$-Bernoulli polynomials,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 498173, 7 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kim, “A note on $q$-Euler numbers and polynomials,” preprint 2006, http://arxiv.org/abs/math.NT/0608649.
- M.-S. Kim, T. Kim, and J.-W. Son, “Multivariate $p$-adic fermionic $q$-integral on ${\mathbb{Z}}_{p}$ and related multiple zeta-type functions,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 304539, 13 pages, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kim, “$q$-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,”
*Russian Journal of Mathematical Physics*, vol. 15, no. 1, pp. 51–57, 2008. View at Google Scholar · View at MathSciNet - T. Kim, “Multiple $p$-adic $L$-function,”
*Russian Journal of Mathematical Physics*, vol. 13, no. 2, pp. 151–157, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. Kato, N. Kurokawa, and T. Saito,
*Number Theory 1: Fermat's Dream*, vol. 186 of*Translations of Mathematical Monographs*, American Mathematical Society, Providence, RI, USA, 2000. View at Zentralblatt MATH · View at MathSciNet - L. C. Washington,
*Introduction to Cyclotomic Fields*, vol. 83 of*Graduate Texts in Mathematics*, Springer, New York, NY, USA, 2nd edition, 1997. View at Zentralblatt MATH · View at MathSciNet - L. C. Washington, “A note on $p$-adic $L$-functions,”
*Journal of Number Theory*, vol. 8, no. 2, pp. 245–250, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. C. Washington, “$p$-adic $L$-functions and sums of powers,”
*Journal of Number Theory*, vol. 69, no. 1, pp. 50–61, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. J. Fox, “A method of Washington applied to the derivation of a two-variable $p$-adic $L$-function,”
*Pacific Journal of Mathematics*, vol. 209, no. 1, pp. 31–40, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. T. Young, “On the behavior of some two-variable $p$-adic $L$-functions,”
*Journal of Number Theory*, vol. 98, no. 1, pp. 67–88, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet