AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation29615910.1155/2008/296159296159Research ArticleOn the q-Extension of Apostol-Euler Numbers and PolynomialsKimYoung-Hee1KimWonjoo2JangLee-Chae3LittlejohnLance1Division of General Education-MathematicsKwangwoon UniversitySeoul 139-701South Koreakwangwoon.ac.kr2Natural Science InstituteKonKuk UniversityChungju 380-701South Koreakonkuk.ac.kr3Department of Mathematics and Computer ScienceKonKuk UniversityChungju 380-701South Koreakonkuk.ac.kr200810122008200804102008211120082008Copyright © 2008This 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.

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

1. Introduction

Let p be a fixed odd prime. Throughout this paper, p,p and and p will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of algebraic closure of p. Let be the set of natural numbers and +={0}. Let vp be the normalized exponential valuation of p with |p|p=pvp(p)=p1. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q, or a p-adic number qp. If q, one normally assumes |q|<1. If qp, then one assumes |q1|p<1. We also use the notations[x]q=1qx1q,[x]q=1(q)x1+qxp

For a fixed odd positive integer d with (p,d)=1, letX=Xd=limNdpN,X1=p,X=0<a<dp(a,p)=1(a+dpp),a+dpNp={xXxa(moddpN)},where a lies in 0a<dpN. The distribution is defined byμq(a+dpNp)=qa[dpN]q.

We say that f is a uniformly differentiable function at a point ap and denote this property by fUD(p), if the difference quotients Ff(x,y)=(f(x)f(y))/(xy) have a limit l=f(a) as (x,y)(a,a). For fUD(p), the p-adic invariant q-integral is defined asIq(f)=pf(x)dμq(x)=limN1[pN]qx=0pN1f(x)qx.The fermionic p-adic q-measures on p are defined asμq(a+dpNp)=(q)a[dpN]q,and the fermionic p-adic invariant q-integral on p is defined asIq(f)=pf(x)dμq(x)=limN1[pN]qx=0pN1f(x)(q)xfor fUD(p). For details see .

Classical Euler numbers are defined by the generating function2et+1=n=0Entnn!,and these numbers are interpolated by the Euler zeta function which is defined asζE(s)=n=0(1)nns,s.

After Carlitz  gave q-extensions of the classical Bernoulli numbers and polynomials, the q-extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors (cf. [116, 1826, 3439]).

By using p-adic q-integral, the q-Euler numbers En,q are defined asEn,q=p[t]qndμq(t),forn.The q-Euler numbers En,q are defined by means of the generating functionFq(t)=qn=0(1)nqne[n]qt(cf. [8, 26]). Kim  gave a new construction of the q-Euler numbers En,q which can be uniquely determined byE0,q=q2,(qE+1)n+En,q={{q,ifn=0,0,ifn0,with the usual convention of replacing En by En,q.

The twisted q-Euler numbers and q-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 q-generalized Euler numbers and polynomials and twisted q-generalized Euler numbers and polynomials. Recently, Y. H. Kim et al.  gave the twisted q-Euler zeta function associated with twisted q-Euler numbers and obtained q-Euler's identity. They also have a q-extension of the Euler zeta function for negative integers and the q-analog of twisted Euler zeta function. Kim  defined twisted q-Euler numbers and polynomials of higher order and studied multiple twisted q-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, q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials and numbers have been studied by many authors with great interest. In , Cenkci and Can introduced and investigated q-extensions of the Bernoulli polynomials. Choi et al.  have studied some q-extensions of the Apostol-Bernoulli and the Apostol-Euler polynomials of order n and multiple Hurwitz zeta function.

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

2. on The <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M110"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Extensions of The Apostol-Euler Numbers and Polynomials

In this section, we will assume qp with |q1|p<1. For n+, let Cpn={ξξpn=1} be the cyclic group of order pn, and let Tp be the space of locally constant space, that is,Tp=limnCpn=n0Cpn.

Let ξTp. We define Apostol's type q-Euler numbers byEn,q,ξ=pqxξx[x]qndμq(x).Then, we haveEn,q,ξ=q(1q)nl=0n(nl)(1)l11+qlξ,where (nl) are the binomial coefficients.

Apostol's type q-Euler polynomials are defined asEn,q,ξ(x)=pqyξy[x+y]qndμq(y).Since[x+y]qn=([x]q+qx[y]q)n=l=0n(nl)[x]qnlqlx[y]ql,we have from (2.4) thatEn,q,ξ(x)=l=0n(nl)[x]qnlqlxpqyξy[y]qldμq(y).By (2.2) and (2.6), we haveEn,q,ξ(x)=l=0n(nl)[x]qnlqlxEl,q,ξ.Since[x+y]qn=1(1q)nl=0n(nl)(1)lq(x+y)l=1(1q)nl=0n(nl)(1)lqlxqly,we havepqyξy[x+y]qndμq(y)=1(1q)nl=0n(nl)(1)lqlxpq(l1)yξydμq(y).Therefore, we also haveEn,q,ξ(x)=q1(1q)nl=0n(nl)qlx(1)l11+qlξ.

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

Theorem 2.1.

For n+ and ξTp, one hasEn,q,ξ=q(1q)nl=0n(nl)(1)l11+qlξ,En,q,ξ(x)=q(1q)nl=0n(nl)(1)lqlx1+qlξ=l=0n(nl)[x]qnlqlxEl,q,ξ.

Now, we will find the generating function of En,q,ξ and En,q,ξ(x), respectively. Let F(t) be the generating function of En,q,ξ. Then, we haveF(t)=n=0En,q,ξtnn!=n=0(q(1q)nl=0n(nl)(1)l11+qlξ)tnn!=qn=01(1q)nl=0n(nl)(1)l(m=0qlmξm(1)m)tnn!=qm=0(1)mξmn=01(1q)n(l=0n(nl)(1)lqlm)tnn!=qm=0(1)mξmn=01(1q)n(1qm)ntnn!=qm=0(1)mξmn=0[m]qntnn!=qm=0(1)mξme[m]qt.Therefore, the generating function F(t) of En,q,ξ equalsF(t)=n=0En,q,ξtnn!=qm=0(1)mξme[m]qt.Note thatpqxξxe[x]qtdμq(x)=n=0pqxξx[x]qndμq(x)tnn!=n=0En,q,ξtnn!=F(t).For the generating function of En,q,ξ(x), we havepqyξye[x+y]qtdμq(y)=qm=0(1)mξme[m+x]qt.Hence, we obtain the following theorem.

Theorem 2.2.

For ξTp, one haspqxξxe[x]qtdμq(x)=qm=0(1)mξme[m]qt,pqyξye[x+y]qtdμq(y)=qm=0(1)mξme[m+x]qt.

Since (2.16) equals to the generating functions (2.17) equals to the generating functions n=0En,q,ξ(x)(tn/n!), we have the following result.

Corollary 2.3.

For n+ and ξTp, one hasEn,q,ξ=qm=0(1)mξm[m]qn,En,q,ξ(x)=qm=0(1)mξm[m+x]qn.

Now, we will find the distribution relation for En,q,ξ(x). By (2.4), we haveEn,q,ξ(x)=Xqyξy[x+y]qndμq(y)=limN1[dpN]qy=0dpN1ξy(1)y[x+y]qn=limN1[dpN]qa=0d1y=0pN1ξa+dy(1)a+dy[x+a+dy]qn.Note that for odd numbers d and p,[dpN]q=[d]q[pN]qd,[x+a+dy]q=[d]q[x+ad+y]qd.By (2.19), we haveEn,q,ξ(x)=1[d]qa=0d1ξa(1)alimN1[pN]qdy=0pN1(ξd)y(1)y[d]qn[x+ad+y]qdn=[d]qn[d]qa=0d1ξa(1)ap(ξd)y(qd)y[x+ad+y]qdndμqd(y).Therefore, we obtain the distribution relation for En,q,ξ(x) as follows.

Theorem 2.4.

For n+,ξTp, and d+ with d1(mod2), one has En,q,ξ(x)=[d]qn[d]qa=0d1ξa(1)aEn,qd,ξd(x+ad).

3. Further Remark on The Basic <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M164"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Zeta Functions Associated with Apostol's Type <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M165"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Euler Numbers and Polynomials

In this section, we assume that q with |q|<1. Let ξTp. For s, q-zeta function associated with Apostol's type q-Euler numbers is defined asζq,ξ(s)=qn=1ξn(1)n[n]qs,which is analytic in whole complex s-plane. Substituting s=k with k+ into ζq,ξ(s) and using Corollary 2.3, then we arrive atζq,ξ(k)=qn=1ξn(1)n[n]qk=Ek,q,ξ.

Now, we also consider Hurwitz's type q-zeta function associated with the Apostol's type q-Euler polynomials as follows:ζq,ξ(s,x)=qn=0ξn(1)n[n+x]qs.Substituting s=k with k+ into ζq,ξ(s,x) and using Corollary 2.3, then we arrive atζq,ξ(k,x)=qn=0ξn(1)n[n+x]qk=Ek,q,ξ(x).Hence, we obtain q-zeta function associated with Apostol's type q-Euler numbers and Hurwitz's type q-zeta function associated with Apostol's type q-Euler polynomials for negative integers.

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