## Advances in Difference Equations

Impact Factor 0.335

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

## References

1. Kim T:Analytic continuation of multiple -zeta functions and their values at negative integers. Russian Journal of Mathematical Physics 2004, 11(1):71-76.
2. Kim T:On the analogs of Euler numbers and polynomials associated with -adic -integral on at . Journal of Mathematical Analysis and Applications 2007, 331(2):779-792. 10.1016/j.jmaa.2006.09.027
3. Kim T:A note on some formulae for the -Euler numbers and polynomials. Proceedings of the Jangjeon Mathematical Society 2006, 9(2):227-232.
4. Kim T, Kim M-S, Jang L-C, Rim S-H:New -Euler numbers and polynomials associated with -adic -integrals. Advanced Studies in Contemporary Mathematics 2007, 15(2):243-252.
5. Jang L-C:On a -analogue of the -adic generalized twisted -functions and -adic -integrals. Journal of the Korean Mathematical Society 2007, 44(1):1-10.
6. Jang L-C, Kim S-D, Park D-W, Ro Y-S: A note on Euler number and polynomials. Journal of Inequalities and Applications 2006, 2006:-5.Google Scholar
7. Ozden H, Simsek Y:A new extension of -Euler numbers and polynomials related to their interpolation functions. Applied mathematics Letters. In pressGoogle Scholar
8. Ozden H, Simsek Y, Cangul IN:Remarks on sum of products of v-twisted Euler polynomials and numbers. Journal of Inequalities and Applications 2008, 2008:-8.Google Scholar
9. Ozden H, Simsek Y, Cangul IN:Multivariate interpolation functions of higher order -Euler numbers and their applications. Abstract and Applied Analysis. In pressGoogle Scholar
10. Simsek Y:Twisted -Bernoulli numbers and polynomials related to twisted -zeta function and -function. Journal of Mathematical Analysis and Applications 2006, 324(2):790-804. 10.1016/j.jmaa.2005.12.057
11. Simsek Y:On -adic twisted - -functions related to generalized twisted Bernoulli numbers. Russian Journal of Mathematical Physics 2006, 13(3):340-348. 10.1134/S1061920806030095
12. Simsek Y:On twisted -Hurwitz zeta function and -two-variable -function. Applied Mathematics and Computation 2007, 187(1):466-473. 10.1016/j.amc.2006.08.146
13. Simsek Y:The behavior of the twisted -adic - -functions at . Journal of the Korean Mathematical Society 2007, 44(4):915-929. 10.4134/JKMS.2007.44.4.915
14. Simsek Y, Kurt V, Kim D:New approach to the complete sum of products of the twisted -Bernoulli numbers and polynomials. Journal of Nonlinear Mathematical Physics 2007, 14(1):44-56. 10.2991/jnmp.2007.14.1.5
15. Carlitz L: -Bernoulli numbers and polynomials. Duke Mathematical Journal 1948, 15(4):987-1000. 10.1215/S0012-7094-48-01588-9
16. Cenkci M:The -adic generalized twisted -Euler- -function and its applications. Advanced Studies in Contemporary Mathematics 2007, 15(1):37-47.
17. Cenkci M, Can M:Some results on -analogue of the Lerch zeta function. Advanced Studies in Contemporary Mathematics 2006, 12(2):213-223.
18. Cenkci M, Can M, Kurt V: -adic interpolation functions and Kummer-type congruences for -twisted and -generalized twisted Euler numbers. Advanced Studies in Contemporary Mathematics 2004, 9(2):203-216.
19. Cenkci M, Simsek Y, Kurt V:Further remarks on multiple -adic - -function of two variables. Advanced Studies in Contemporary Mathematics 2007, 14(1):49-68.
20. Kim T: On Euler-Barnes multiple zeta functions. Russian Journal of Mathematical Physics 2003, 10(3):261-267.
21. Kim T:Power series and asymptotic series associated with the -analog of the two-variable -adic -function. Russian Journal of Mathematical Physics 2005, 12(2):186-196.
22. Kim T:Multiple -adic -function. Russian Journal of Mathematical Physics 2006, 13(2):151-157. 10.1134/S1061920806020038
23. Kim T:A note on -adic -integral on associated with -Euler numbers. Advanced Studies in Contemporary Mathematics 2007, 15(2):133-137.
24. Kim T: -Euler numbers and polynomials associated with -adic -integrals. Journal of Nonlinear Mathematical Physics 2007, 14(1):15-27. 10.2991/jnmp.2007.14.1.3
25. Kim T:An invariant -adic -integral on . Applied Mathematics Letters 2008, 21(2):105-108. 10.1016/j.aml.2006.11.011
26. Kim T:On -adic interpolating function for -Euler numbers and its derivatives. Journal of Mathematical Analysis and Applications 2008, 339(1):598-608. 10.1016/j.jmaa.2007.07.027
27. Kim T, Rim S-H:New Changhee -Euler numbers and polynomials associated with -adic -integrals. Computers & Mathematics with Applications 2007, 54(4):484-489. 10.1016/j.camwa.2006.12.028
28. Kim T, Choi JY, Sug JY:Extended -Euler numbers and polynomials associated with fermionic -adic -integral on . Russian Journal of Mathematical Physics 2007, 14(2):160-163. 10.1134/S1061920807020045