Multiple Twisted -Euler Numbers and Polynomials Associated with -Adic -Integrals
- Lee-Chae Jang^{1}Email author
DOI: 10.1155/2008/738603
© Lee-Chae Jang. 2008
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
is known to be a distribution on (cf. [1–28]).
have a limit as (cf. [25]).
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
for each and .
Thus we give Witt's type formula for multiple twisted -Euler numbers as follows.
Theorem 2.1.
From (2.8) and (2.9), we obtain the following theorem.
Theorem 2.2.
Then by the th differentiation on both sides of (2.14), we obtain the following.
Theorem 2.3.
From (2.15) and (2.17), we obtain the sums of powers of consecutive -Euler numbers as follows.
Theorem 2.4.
3. Multiple Twisted -Euler Zeta Functions
For with and , the multiple twisted -Euler numbers can be considered as follows:
From (3.1), we note that
for all . We also obtain the following theorem in which multiple twisted -Euler Zeta functions interpolate multiple twisted -Euler polynomials.
Theorem 3.1.
4. Multiple Twisted Barnes' Type -Euler Polynomials
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.
By the th differentiation of both sides of (4.6), we obtain the following theorem.
Theorem 4.2.
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.
5. Multiple Twisted Dirichlet's Type -Euler Numbers and Polynomials
(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.
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.
Now, in view of (5.1), we can define multiple twisted Dirichlet's type -Euler numbers by means of the generating function as follows:
where . We note that if , then is a multiple generalized -Euler number (see [22]).
From (5.9), we can give Witt's type formula for multiple twisted Dirichlet's type -Euler numbers.
Theorem 5.3.
From (5.10), we also obtain the sums of powers of consecutive multiple twisted Dirichlet's type -Euler numbers as follows.
Theorem 5.4.
Clearly, we obtain the following two theorems.
Theorem 5.5.
Theorem 5.6.
Authors’ Affiliations
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