Abstract and Applied Analysis

Volume 2008 (2008), Article ID 296159, 10 pages

http://dx.doi.org/10.1155/2008/296159

## On the -Extension of Apostol-Euler Numbers and Polynomials

^{1}Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea^{2}Natural Science Institute, KonKuk University, Chungju 380-701, South Korea^{3}Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea

Received 4 October 2008; Accepted 21 November 2008

Academic Editor: Lance Littlejohn

Copyright © 2008 Young-Hee 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

Recently, Choi et al. (2008) have studied the -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order and multiple Hurwitz zeta function. In this paper, we define Apostol's type -Euler numbers and -Euler polynomials . We obtain the generating functions of and , respectively. We also have the distribution relation for Apostol's type -Euler polynomials. Finally, we obtain -zeta function associated with Apostol's type -Euler numbers and Hurwitz_s type -zeta function associated with Apostol's type -Euler polynomials for negative integers.

#### 1. Introduction

Let be a fixed odd prime. Throughout this paper, and and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the complex number field, and the completion of algebraic closure of . Let be the set of natural numbers and . 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 one normally assumes If then one assumes We also use the notations

For a fixed odd positive integer with , letwhere lies in . The distribution is defined by

We say that is a uniformly differentiable function at a point and denote this property by , if the difference quotients have a limit as . For , the -adic invariant -integral is defined asThe fermionic -adic -measures on are defined asand the fermionic -adic invariant -integral on is defined asfor . For details see [1–10].

Classical Euler numbers are defined by the generating functionand these numbers are interpolated by the Euler zeta function which is defined as

After Carlitz [11] gave -extensions of the classical Bernoulli numbers and polynomials, the -extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors (cf. [1–16, 18–26, 34–39]).

By using -adic -integral, the -Euler numbers are defined asThe -Euler numbers are defined by means of the generating function(cf. [8, 26]). Kim [22] gave a new construction of the -Euler numbers which can be uniquely determined bywith the usual convention of replacing by .

The twisted -Euler numbers and -Euler polynomials are very important in several fields of mathematics and physics, and so they have been studied by many authors. Simsek [37, 38] constructed generating functions of -generalized Euler numbers and polynomials and twisted -generalized Euler numbers and polynomials. Recently, Y. H. Kim et al. [27] gave the twisted -Euler zeta function associated with twisted -Euler numbers and obtained -Euler's identity. They also have a -extension of the Euler zeta function for negative integers and the -analog of twisted Euler zeta function. Kim [24] defined twisted -Euler numbers and polynomials of higher order and studied multiple twisted -Euler zeta functions.

The Apostol-Bernoulli and the Apostol-Euler polynomials and numbers have been studied by several authors (cf. [15, 17, 32, 33, 40, 41]). Recently, -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials and numbers have been studied by many authors with great interest. In [15], Cenkci and Can introduced and investigated -extensions of the Bernoulli polynomials. Choi et al. [16] have studied some -extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order and multiple Hurwitz zeta function.

In this paper, we define Apostol's type -Euler numbers and -Euler polynomials. Then, we have the generating functions of Apostol's type -Euler numbers and -Euler polynomials and the distribution relation for Apostol's type -Euler polynomials. In Section 2, we define Apostol's type -Euler numbers and -Euler polynomials . Then, we obtain the generating functions of and , respectively. We also have the distribution relation for Apostol's type -Euler polynomials. In Section 3, we obtain -zeta function associated with Apostol's type -Euler numbers and Hurwitz's type -zeta function associated with Apostol's type -Euler polynomials for negative integers.

#### 2. on The -Extensions of The Apostol-Euler Numbers and Polynomials

In this section, we will assume with For , let be the cyclic group of order , and let be the space of locally constant space, that is,

Let . We define Apostol's type -Euler numbers byThen, we havewhere are the binomial coefficients.

Apostol's type -Euler polynomials are defined asSincewe have from (2.4) thatBy (2.2) and (2.6), we haveSincewe haveTherefore, we also have

Note that (2.7) and (2.10) are two representations for . Hence, we have the following result.

Theorem 2.1. *For and , one
has*

Now, we will find the generating function of and , respectively. Let be the generating function of . Then, we haveTherefore, the generating function of equalsNote thatFor the generating function of , we haveHence, we obtain the following theorem.

Theorem 2.2. *For ,
one has*

Since (2.16) equals to the generating functions (2.17) equals to the generating functions , we have the following result.

Corollary 2.3. *For and , one
has*

Now, we will find the distribution relation for . By (2.4), we haveNote that for odd numbers and ,By (2.19), we haveTherefore, we obtain the distribution relation for as follows.

Theorem 2.4. *For ,
and with ,
one has
*

#### 3. Further Remark on The Basic -Zeta Functions Associated with Apostol's Type -Euler Numbers and Polynomials

In this section, we assume that with . Let . For , -zeta function associated with Apostol's type -Euler numbers is defined aswhich is analytic in whole complex -plane. Substituting with into and using Corollary 2.3, then we arrive at

Now, we also consider Hurwitz's type -zeta function associated with the Apostol's type -Euler polynomials as follows:Substituting with into and using Corollary 2.3, then we arrive atHence, we obtain -zeta function associated with Apostol's type -Euler numbers and Hurwitz's type -zeta function associated with Apostol's type -Euler polynomials for negative integers.

#### References

- T. Kim, “$q$-Volkenborn integration,”
*Russian Journal of Mathematical Physics*, vol. 9, no. 3, pp. 288–299, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “On $p$-adic $q$-$L$-functions and sums of powers,”
*Discrete Mathematics*, vol. 252, no. 1–3, pp. 179–187, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “On Euler-Barnes multiple zeta functions,”
*Russian Journal of Mathematical Physics*, vol. 10, no. 3, pp. 261–267, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “Sums of powers of consecutive $q$-integers,”
*Advanced Studies in Contemporary Mathematics*, vol. 9, no. 1, pp. 15–18, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “Analytic continuation of multiple $q$-zeta functions and their values at negative integers,”
*Russian Journal of Mathematical Physics*, vol. 11, no. 1, pp. 71–76, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “$q$-Riemann zeta function,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2004, no. 12, pp. 599–605, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Kim, “Power series and asymptotic series associated with the $q$-analog of the two-variable $p$-adic $L$-function,”
*Russian Journal of Mathematical Physics*, vol. 12, no. 2, pp. 186–196, 2005. View at Google Scholar · View at MathSciNet - T. Kim, “$q$-generalized Euler numbers and polynomials,”
*Russian Journal of Mathematical Physics*, vol. 13, no. 3, pp. 293–298, 2006. View at Publisher · 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 - T. Kim, “On the analogs of Euler numbers and polynomials associated with $p$-adic $q$-integral on ${\mathbb{Z}}_{p}$ at $q=-1$,”
*Journal of Mathematical Analysis and Applications*, vol. 331, no. 2, pp. 779–792, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Carlitz, “$q$-Bernoulli numbers and polynomials,”
*Duke Mathematical Journal*, vol. 15, no. 4, pp. 987–1000, 1948. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. N. Cangul, H. Ozden, and Y. Simsek, “Generating functions of the $(h,q)$ extension of twisted Euler polynomials and numbers,”
*Acta Mathematica Hungarica*, vol. 120, no. 3, pp. 281–299, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - L. Carlitz, “$q$-Bernoulli and Eulerian numbers,”
*Transactions of the American Mathematical Society*, vol. 76, no. 2, pp. 332–350, 1954. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Cenkci, “The $p$-adic generalized twisted $(h,q)$-Euler-$l$-function and its applications,”
*Advanced Studies in Contemporary Mathematics*, vol. 15, no. 1, pp. 37–47, 2007. View at Google Scholar · View at MathSciNet - M. Cenkci and M. Can, “Some results on $q$-analogue of the Lerch zeta function,”
*Advanced Studies in Contemporary Mathematics*, vol. 12, no. 2, pp. 213–223, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Choi, P. J. Anderson, and H. M. Srivastava, “Some $q$-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order $n$, and the multiple Hurwitz zeta function,”
*Applied Mathematics and Computation*, vol. 199, no. 2, pp. 723–737, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - M. Garg, K. Jain, and H. M. Srivastava, “Some relationships between the generalized Apostol-Bernoulli polynomials and Hurwitz-Lerch zeta functions,”
*Integral Transforms and Special Functions*, vol. 17, no. 11, pp. 803–815, 2006. View at Publisher · View at Google Scholar · 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 - 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, “$q$-extension of the Euler formula and trigonometric functions,”
*Russian Journal of Mathematical Physics*, vol. 14, no. 3, pp. 275–278, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - T. Kim, “On the symmetries of the $q$-Bernoulli polynomials,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 914367, 7 pages, 2008. View at Publisher · View at Google Scholar - T. Kim, “The modified $q$-Euler numbers and polynomials,”
*Advanced Studies in Contemporary Mathematics*, vol. 16, no. 2, pp. 161–170, 2008. 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, “On the multiple $q$-Genocchi and Euler numbers,”
*Russian Journal of Mathematical Physics*, vol. 15, no. 4, pp. 481–486, 2008. View at Google Scholar - T. Kim, J. Y. Choi, and J. Y. Sug, “Extended $q$-Euler numbers and polynomials associated with fermionic $p$-adic $q$-integral on ${\mathbb{Z}}_{p}$,”
*Russian Journal of Mathematical Physics*, vol. 14, no. 2, pp. 160–163, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - 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 - Y. H. Kim, W. J. Kim, and C. S. Ryoo, “On the twisted $q$-Euler zeta function associated with twisted $q$-Euler numbers,” communicated.
- T. Kim, S.-H. Rim, and Y. Simsek, “A note on the alternating sums of powers of consecutive $q$-integers,”
*Advanced Studies in Contemporary Mathematics*, vol. 13, no. 2, pp. 159–164, 2006. View at Google Scholar · View at MathSciNet - T. Kim and Y. Simsek, “Analytic continuation of the multiple Daehee $q$-$l$-functions associated with Daehee numbers,”
*Russian Journal of Mathematical Physics*, vol. 15, no. 1, pp. 58–65, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - S.-D. Lin and H. M. Srivastava, “Some families of the Hurwitz-Lerch zeta functions and associated fractional derivative and other integral representations,”
*Applied Mathematics and Computation*, vol. 154, no. 3, pp. 725–733, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-D. Lin, H. M. Srivastava, and P.-Y. Wang, “Some expansion formulas for a class of generalized Hurwitz-Lerch zeta functions,”
*Integral Transforms and Special Functions*, vol. 17, no. 11, pp. 817–827, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - Q.-M. Luo, “Apostol-Euler polynomials of higher order and Gaussian hypergeometric functions,”
*Taiwanese Journal of Mathematics*, vol. 10, no. 4, pp. 917–925, 2006. View at Google Scholar · View at MathSciNet - Q.-M. Luo and H. M. Srivastava, “Some generalizations of the Apostol-Bernoulli and Apostol-Euler polynomials,”
*Journal of Mathematical Analysis and Applications*, vol. 308, no. 1, pp. 290–302, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · 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 Zentralblatt MATH · View at MathSciNet - 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 - H. Ozden and Y. Simsek, “A new extension of $q$-Euler numbers and polynomials related to their interpolation functions,”
*Applied Mathematics Letters*, vol. 21, no. 9, pp. 934–939, 2008. View at Google Scholar · View at MathSciNet - Y. Simsek, “On $p$-adic twisted $q$-$L$-functions related to generalized twisted Bernoulli numbers,”
*Russian Journal of Mathematical Physics*, vol. 13, no. 3, pp. 340–348, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their
interpolation functions,”
*Advanced Studies in Contemporary Mathematics*, vol. 16, no. 2, pp. 251–278, 2008. View at Google Scholar · View at MathSciNet - H. M. Srivastava, T. Kim, and Y. Simsek, “$q$-Bernoulli numbers and polynomials associated with multiple $q$-zeta functions and basic $L$-series,”
*Russian Journal of Mathematical Physics*, vol. 12, no. 2, pp. 241–268, 2005. View at Google Scholar · View at MathSciNet - T. M. Apostol, “On the Lerch zeta function,”
*Pacific Journal of Mathematics*, vol. 1, pp. 161–167, 1951. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Wang, C. Jia, and T. Wang, “Some results on the Apostol-Bernoulli and Apostol-Euler polynomials,”
*Computers & Mathematics with Applications*, vol. 55, no. 6, pp. 1322–1332, 2008. View at Google Scholar · View at MathSciNet