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

Authors: Young-Hee Kim; Wonjoo Kim; Lee-Chae Jang
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

Recently, Choi et al. (2008) have studied theq-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.


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## Body

## 1. Introduction

Letp be  a fixed odd prime. Throughout  this paper, ℤp,ℚp and ℂ and ℂp 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 ℚp.
Let ℕ be the set of natural numbers and ℤ+=ℕ∪{0}.
Let vp be the normalized exponential valuation of ℂp with |p|p=p−vp(p)=p−1. When one talks of q-extension, q is variously considered as an indeterminate, a
complex number q∈ℂ, or a p-adic number q∈ℂp.
If q∈ℂ, one normally assumes |q|<1. If q∈ℂp, then one assumes |q−1|p<1. We also use the notations(1.1)[x]q=1−qx1−q,[x]−q=1−(−q)x1+q∀x∈ℤpFor a fixed odd
positive  integerd with
(p,d)=1,
let(1.2)X=Xd=limN←ℤdpNℤ,X1=ℤp,X∗=⋃0<a<dp(a,p)=1(a+dpℤp),a+dpNℤp={x∈X∣x≡a(moddpN)},where a∈ℤ lies in 0≤a<dpN.
The distribution is defined by(1.3)μq(a+dpNℤp)=qa[dpN]q.We say thatf is a uniformly differentiable function at a
point a∈ℤp and denote this property by f∈UD(ℤp),
if the difference quotients Ff(x,y)=(f(x)−f(y))/(x−y) have a limit l=f′(a) as (x,y)→(a,a).
For f∈UD(ℤp),
the p-adic invariant q-integral is defined as(1.4)Iq(f)=∫ℤpf(x)dμq(x)=limN→∞1[pN]q∑x=0pN−1f(x)qx.The fermionic p-adic q-measures on ℤp are defined as(1.5)μ−q(a+dpNℤp)=(−q)a[dpN]−q,and the fermionic p-adic invariant q-integral on ℤp is defined as(1.6)I−q(f)=∫ℤpf(x)dμ−q(x)=limN→∞1[pN]−q∑x=0pN−1f(x)(−q)xfor f∈UD(ℤp).
For details see [1–10].Classical Euler numbers are defined by the generating
function(1.7)2et+1=∑n=0∞Entnn!,and these numbers are
interpolated by the Euler zeta function which is defined as(1.8)ζE(s)=∑n=0∞(−1)nns,s∈ℂ.After Carlitz [11] 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.
[1–16, 18–26, 34–39]).By usingp-adic q-integral, the q-Euler numbers En,q are defined as(1.9)En,q=∫ℤp[t]qndμ−q(t),forn∈ℕ.The q-Euler numbers En,q are defined by means of the generating
function(1.10)Fq(t)=[2]q∑n=0∞(−1)nqne[n]qt(cf.
[8, 26]). Kim [22] gave a new construction of
the q-Euler numbers En,q which can be uniquely determined by(1.11)E0,q=[2]q2,(qE+1)n+En,q={{[2]q,ifn=0,0,ifn≠0,with the usual convention of replacing En by En,q.The twistedq-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. [27] 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
[24] 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 [15], Cenkci and Can
introduced and investigated q-extensions of the Bernoulli polynomials. Choi
et al. [16] 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 typeq-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 Theq-Extensions of The Apostol-Euler Numbers and Polynomials

In this
section, we will assumeq∈ℂp with |q−1|p<1. For n∈ℤ+,
let Cpn={ξ∣ξpn=1} be the cyclic group of order pn,
and let
Tp be the space of locally constant space, that
is,(2.1)Tp=limn→∞Cpn=⋃n≥0Cpn.Letξ∈Tp.
We define Apostol's type q-Euler numbers by(2.2)En,q,ξ=∫ℤpq−xξx[x]qndμ−q(x).Then, we have(2.3)En,q,ξ=[2]q(1−q)n∑l=0n(nl)(−1)l11+qlξ,where (nl) are the binomial coefficients.Apostol's typeq-Euler polynomials are defined
as(2.4)En,q,ξ(x)=∫ℤpq−yξy[x+y]qndμ−q(y).Since(2.5)[x+y]qn=([x]q+qx[y]q)n=∑l=0n(nl)[x]qn−lqlx[y]ql,we have from (2.4)
that(2.6)En,q,ξ(x)=∑l=0n(nl)[x]qn−lqlx∫ℤpq−yξy[y]qldμ−q(y).By (2.2) and (2.6), we
have(2.7)En,q,ξ(x)=∑l=0n(nl)[x]qn−lqlxEl,q,ξ.Since(2.8)[x+y]qn=1(1−q)n∑l=0n(nl)(−1)lq(x+y)l=1(1−q)n∑l=0n(nl)(−1)lqlxqly,we have(2.9)∫ℤpq−yξy[x+y]qndμ−q(y)=1(1−q)n∑l=0n(nl)(−1)lqlx∫ℤpq(l−1)yξydμ−q(y).Therefore, we also
have(2.10)En,q,ξ(x)=[2]q1(1−q)n∑l=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.
Forn∈ℤ+ and ξ∈Tp, one
has(2.11)En,q,ξ=[2]q(1−q)n∑l=0n(nl)(−1)l11+qlξ,En,q,ξ(x)=[2]q(1−q)n∑l=0n(nl)(−1)lqlx1+qlξ=∑l=0n(nl)[x]qn−lqlxEl,q,ξ.Now, we will find the generating function ofEn,q,ξ and En,q,ξ(x),
respectively. Let F(t) be the generating function of En,q,ξ.
Then, we have(2.12)F(t)=∑n=0∞En,q,ξtnn!=∑n=0∞([2]q(1−q)n∑l=0n(nl)(−1)l11+qlξ)tnn!=[2]q∑n=0∞1(1−q)n∑l=0n(nl)(−1)l(∑m=0∞qlmξm(−1)m)tnn!=[2]q∑m=0∞(−1)mξm∑n=0∞1(1−q)n(∑l=0n(nl)(−1)lqlm)tnn!=[2]q∑m=0∞(−1)mξm∑n=0∞1(1−q)n(1−qm)ntnn!=[2]q∑m=0∞(−1)mξm∑n=0∞[m]qntnn!=[2]q∑m=0∞(−1)mξme[m]qt.Therefore, the generating
function F(t) of En,q,ξ equals(2.13)F(t)=∑n=0∞En,q,ξtnn!=[2]q∑m=0∞(−1)mξme[m]qt.Note that(2.14)∫ℤpq−xξxe[x]qtdμ−q(x)=∑n=0∞∫ℤpq−xξx[x]qndμ−q(x)tnn!=∑n=0∞En,q,ξtnn!=F(t).For the generating function of En,q,ξ(x),
we have(2.15)∫ℤpq−yξye[x+y]qtdμ−q(y)=[2]q∑m=0∞(−1)mξme[m+x]qt.Hence, we obtain the following
theorem.Theorem 2.2.
Forξ∈Tp,
one has(2.16)∫ℤpq−xξxe[x]qtdμ−q(x)=[2]q∑m=0∞(−1)mξme[m]qt,(2.17)∫ℤpq−yξye[x+y]qtdμ−q(y)=[2]q∑m=0∞(−1)mξme[m+x]qt.Since (2.16) equals to the generating functions
(2.17)
equals to the generating functions ∑n=0∞En,q,ξ(x)(tn/n!),
we have the following result.Corollary 2.3.
Forn∈ℤ+ and ξ∈Tp, one
has(2.18)En,q,ξ=[2]q∑m=0∞(−1)mξm[m]qn,En,q,ξ(x)=[2]q∑m=0∞(−1)mξm[m+x]qn.Now, we will find the distribution relation forEn,q,ξ(x).
By (2.4), we have(2.19)En,q,ξ(x)=∫Xq−yξy[x+y]qndμ−q(y)=limN→∞1[dpN]−q∑y=0dpN−1ξy(−1)y[x+y]qn=limN→∞1[dpN]−q∑a=0d−1∑y=0pN−1ξa+dy(−1)a+dy[x+a+dy]qn.Note that for odd numbers d and p,(2.20)[dpN]−q=[d]−q[pN]−qd,[x+a+dy]q=[d]q[x+ad+y]qd.By (2.19), we
have(2.21)En,q,ξ(x)=1[d]−q∑a=0d−1ξa(−1)alimN→∞1[pN]−qd∑y=0pN−1(ξd)y(−1)y[d]qn[x+ad+y]qdn=[d]qn[d]−q∑a=0d−1ξa(−1)a∫ℤp(ξd)y(qd)−y[x+ad+y]qdndμ−qd(y).Therefore, we obtain the
distribution relation for En,q,ξ(x) as follows.Theorem 2.4.
Forn∈ℤ+,ξ∈Tp,
and d∈ℤ+ with d≡1(mod2),
one has
(2.22)En,q,ξ(x)=[d]qn[d]−q∑a=0d−1ξa(−1)aEn,qd,ξd(x+ad).


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

In this section, we assume thatq∈ℂ with |q|<1.
Let ξ∈Tp.
For s∈ℂ, q-zeta function associated with Apostol's type q-Euler numbers is defined as(3.1)ζq,ξ(s)=[2]q∑n=1∞ξn(−1)n[n]qs,which is analytic in whole
complex s-plane. Substituting s=−k with k∈ℤ+ into ζq,ξ(s) and using Corollary 2.3, then we arrive
at(3.2)ζq,ξ(−k)=[2]q∑n=1∞ξn(−1)n[n]qk=Ek,q,ξ.Now, we also consider Hurwitz's typeq-zeta function associated with the Apostol's
type q-Euler polynomials as follows:(3.3)ζq,ξ(s,x)=[2]q∑n=0∞ξn(−1)n[n+x]qs.Substituting s=−k with k∈ℤ+ into ζq,ξ(s,x) and using Corollary 2.3, then we arrive
at(3.4)ζq,ξ(−k,x)=[2]q∑n=0∞ξn(−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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*Source: 296159-2009-01-04.xml*