- Lee-Chae Jang^{1}Email author
https://doi.org/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
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
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
References
- Kim T:Analytic continuation of multiple -zeta functions and their values at negative integers. Russian Journal of Mathematical Physics 2004, 11(1):71-76.MATHMathSciNetGoogle Scholar
- 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.2006.09.027MATHMathSciNetView ArticleGoogle Scholar
- Kim T:A note on some formulae for the -Euler numbers and polynomials. Proceedings of the Jangjeon Mathematical Society 2006, 9(2):227-232.MATHMathSciNetGoogle Scholar
- 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.MATHMathSciNetGoogle Scholar
- 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.MATHMathSciNetView ArticleGoogle Scholar
- 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
- Ozden H, Simsek Y:A new extension of -Euler numbers and polynomials related to their interpolation functions. Applied mathematics Letters. In pressGoogle Scholar
- 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
- Ozden H, Simsek Y, Cangul IN:Multivariate interpolation functions of higher order -Euler numbers and their applications. Abstract and Applied Analysis. In pressGoogle Scholar
- 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.2005.12.057MATHMathSciNetView ArticleGoogle Scholar
- Simsek Y:On -adic twisted - -functions related to generalized twisted Bernoulli numbers. Russian Journal of Mathematical Physics 2006, 13(3):340-348. 10.1134/S1061920806030095MATHMathSciNetView ArticleGoogle Scholar
- Simsek Y:On twisted -Hurwitz zeta function and -two-variable -function. Applied Mathematics and Computation 2007, 187(1):466-473. 10.1016/j.amc.2006.08.146MATHMathSciNetView ArticleGoogle Scholar
- Simsek Y:The behavior of the twisted -adic - -functions at . Journal of the Korean Mathematical Society 2007, 44(4):915-929. 10.4134/JKMS.2007.44.4.915MATHMathSciNetView ArticleGoogle Scholar
- 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.2007.14.1.5MATHMathSciNetView ArticleGoogle Scholar
- Carlitz L: -Bernoulli numbers and polynomials. Duke Mathematical Journal 1948, 15(4):987-1000. 10.1215/S0012-7094-48-01588-9MATHMathSciNetView ArticleGoogle Scholar
- Cenkci M:The -adic generalized twisted -Euler- -function and its applications. Advanced Studies in Contemporary Mathematics 2007, 15(1):37-47.MATHMathSciNetGoogle Scholar
- Cenkci M, Can M:Some results on -analogue of the Lerch zeta function. Advanced Studies in Contemporary Mathematics 2006, 12(2):213-223.MATHMathSciNetGoogle Scholar
- 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.MATHMathSciNetGoogle Scholar
- 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.MATHMathSciNetGoogle Scholar
- Kim T: On Euler-Barnes multiple zeta functions. Russian Journal of Mathematical Physics 2003, 10(3):261-267.MATHMathSciNetGoogle Scholar
- 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.MATHMathSciNetGoogle Scholar
- Kim T:Multiple -adic -function. Russian Journal of Mathematical Physics 2006, 13(2):151-157. 10.1134/S1061920806020038MATHMathSciNetView ArticleGoogle Scholar
- Kim T:A note on -adic -integral on associated with -Euler numbers. Advanced Studies in Contemporary Mathematics 2007, 15(2):133-137.MATHMathSciNetGoogle Scholar
- Kim T: -Euler numbers and polynomials associated with -adic -integrals. Journal of Nonlinear Mathematical Physics 2007, 14(1):15-27. 10.2991/jnmp.2007.14.1.3MATHMathSciNetView ArticleGoogle Scholar
- Kim T:An invariant -adic -integral on . Applied Mathematics Letters 2008, 21(2):105-108. 10.1016/j.aml.2006.11.011MATHMathSciNetView ArticleGoogle Scholar
- 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.2007.07.027MATHMathSciNetView ArticleGoogle Scholar
- 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.2006.12.028MATHMathSciNetView ArticleGoogle Scholar
- 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/S1061920807020045MATHMathSciNetView ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. 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.