Journal of Number Theory 129 (2009) 1798–1804

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Journal of Number Theory

www.elsevier.com/locate/jnt

Note on the Euler q-zeta functions

Taekyun Kim

Division of General Education-Mathematics, Kwangwoon University, 447-1, Wolgye-Dong, Nowon-Gu, Seoul 139-701, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 August 2008Revised 3 October 2008Available online 13 December 2008Communicated by David Goss

MSC:11S8011B68

Keywords:q-Euler numberp-Adic q-integralsZeta functionp-Adic fermionic integrals

We consider the q-analogue of the Euler zeta function which isdefined by

ζq,E (s) = [2]q

∞∑n=1

(−1)nqns

[n]sq

, 0 < q < 1, �(s) > 1.

In this paper, we give the q-extensions of the Euler numbers whichcan be viewed as interpolating of the above q-analogue of Eulerzeta function at negative integers, in the same way that Riemannzeta function interpolates Bernoulli numbers at negative integers.Also, we will treat some identities of the q-extensions of the Eulernumbers by using fermionic p-adic q-integration on Zp .

© 2008 Elsevier Inc. All rights reserved.

1. Introduction

Throughout this paper Zp , Qp , C and Cp will respectively denote the ring of p-adic rational inte-gers, the field of p-adic rational numbers, the complex number field and the completion of algebraicclosure of Qp .

The p-adic absolute value in Cp is normalized so that |p|p = 1p . 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, then we normally assume |q| < 1, and when q ∈ Cp , then we normally assume |q − 1|p < 1.We use the notation:

[x]q = [x : q] = 1 − qx

1 − qand [x]−q = 1 − (−q)x

1 + q.

Note that limq→1[x]q = x for x ∈ Zp in presented p-adic case.

E-mail address: tkkim@kw.ac.kr.

0022-314X/$ – see front matter © 2008 Elsevier Inc. All rights reserved.doi:10.1016/j.jnt.2008.10.007

T. Kim / Journal of Number Theory 129 (2009) 1798–1804 1799

Let UD(Zp) be denoted by the set of uniformly differentiable functions on Zp .For f ∈ UD(Zp), let us start with the expression

1

[pN ]−q

∑0� j<pN

(−q) j f ( j) =∑

0� j<pN

f ( j)μ−q(

j + pNZp)

representing analogue of Riemann’s sums for f , cf. [1–22,24–31].The fermionic p-adic q-integral of f on Zp will be defined as the limit (N → ∞) of these sums,

which it exists. The fermionic p-adic q-integral of a function f ∈ UD(Zp) is defined as

∫Zp

f (x)dμ−q(x) = limN→∞

1

[pN ]−q

∑0� j<pN

f ( j)(−q) j (see [13,15,17]

).

For d a fixed positive integer with (p,d) = 1, let

X = Xd = lim←N

Z/dpNZ, X1 = Zp,

X∗ =⋃

0<a<dp(a,p)=1

a + dpZp,

a + dpNZp = {x ∈ X

∣∣ x ≡ a(mod dpN)}

,

where a ∈ Z lies in 0 � a < dpN (see [1–22,24–31]).Let N be the set of positive integers. For m,k ∈ N , the q-Euler polynomials E(−m,k)

m (x,q) of higherorder in the variables x in Cp by making use of the p-adic q-integral, cf. [13,15], are defined by

E(−m,k)m,q (x) =

∫Zp

∫Zp

· · ·∫Zp︸ ︷︷ ︸

k times

[x + x1 + x2 + · · · + xk]mq

· q−x1(m+1)−x2(m+2)−···−xk(m+k) dμ−q(x1)dμ−q(x2) · · · dμ−q(xk). (1)

Now, we define the q-Euler numbers of higher order as follows:

E(−m,k)m,q = E(−m,k)

m,q (0).

From (1), we can derive

E(−m,k)m,q = lim

N→∞1

[pN ]k−q

pN −1∑x1=0

· · ·pN −1∑xk=0

[x1 + · · · + xk]mq (−1)x1+···+xk q−x1m−···−xk(m+k−1)

= [2]kq

(1 − q)m

m∑i=0

(m

i

)(−1)i 1

(1 + qi−m)(1 + qi−m−1) · · · (1 + qi−m−k+1), (2)

where(m

i

)is binomial coefficient.

Note that limq→1 E(−m,k)m,q = E(k)

m where E(k)m are ordinary Euler numbers of order k, which are de-

fined as

1800 T. Kim / Journal of Number Theory 129 (2009) 1798–1804

(2

et + 1

)k

=∞∑

n=0

E(k)n

tn

n! .

By (1) and (2), it is easy to see that

E(−m,1)m,q (x) =

m∑i=0

(m

i

)qxi E(−m,1)

i,q [x]m−i = [2]q

(1 − q)m

m∑j=0

q jx(

m

j

)(−1) j 1

1 + q j−m. (3)

We define q-zeta functions as follows.

ζq,E (s) = [2]q

∞∑n=1

(−1)nqsn

[n]sq

, q ∈ R with 0 < q < 1 and s ∈ C with �(s) > 1. (4)

The numerator ensures the convergence. In (4), we can consider the following problem:“Are there q-Euler numbers which can be viewed as interpolating of ζq,E(s) at negative integers,

in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers?”In this paper, we give the value ζq,E(−m), for m ∈ N, which is an answer of the above problem

and construct a new complex q-analogue of Hurwitz’s type Euler zeta function and q–L-series relatedto q-Euler numbers. Also, we will treat some interesting identities of q-Euler numbers.

2. Some identities of q-Euler numbers E(−m,1)m,q

In this section, we assume q ∈ Cp with |1 − q|p < 1. By (1), we see that

E(−n,1)n,q (x) =

∫X

q−(n+1)t [x + t]nq dμ−q(t)

= [2]q

[2]qd[d]n

q

d−1∑i=0

(−1)iq−ni∫Zp

q−(n+1)dt[

x + i

d+ t

]n

qddμ−qd (t).

Thus we have

E(−n,1)n,q (x) = [2]q

[2]qd[d]n

q

d−1∑i=0

(−1)iq−ni E(−n,1)

n,qd

(x + i

d

), (5)

where d, n are positive integers with d ≡ 1 (mod 2).If we take x = 0, then we have

E(−m,1)m,q = [2]q

[2]qn

m∑k=0

(m

k

)[n]k

q E(−m,1)

k,qn

n−1∑j=0

(−1) jq−(m−k) j[ j]m−kq , where n ≡ 1 (mod 2). (6)

From (6), we can easily derive the following Eq. (7).

E(−m,1)m,q − [2]q

[2]qn[n]m

q E(−m,1)m,qn = [2]q

[2]qn

m−1∑k=0

(m

k

)[n]k

q E(−m,1)

k,qn

n−1∑j=1

q−(m−k) j(−1) j[ j]m−kq . (7)

T. Kim / Journal of Number Theory 129 (2009) 1798–1804 1801

It is easy to see that limq→1 E(−m,1)m,q = E(−m,1)

m,1 = Em , where Em are the mth ordinary Euler numbers,cf. [15,23]. From (7), we note that

(1 − nm)

Em =m−1∑k=0

(m

k

)nk Ek

n−1∑j=1

(−1) j jm−k.

Let Fq(t, x) be generating function of E(−n,1)n,q as follows:

Fq(t, x) =∞∑

k=0

E(−k,1)

k,q (x)tk

k! . (8)

By (3), (8), we easily see:

Fq(t, x) = [2]q

∞∑k=0

(1

(q − 1)k

k∑j=0

(−1)k− j(

k

j

)q jx

1 + q j−k

)tk

k!

=∞∑

k=0

([2]q

∞∑n=0

(−1)nq−kn[n + x]kq

)tk

k! . (9)

Differentiating both sides with respect to t in (5), (6) and comparing coefficients, we obtain thefollowing:

Theorem 1. For m � 0, we have

E(−m,1)m,q (x) = [2]q

∞∑n=0

q−nm[n + x]mq (−1)n. (10)

Corollary 2. Let m ∈ N. Then there exists

E(−m,1)m,q = [2]q

∞∑n=1

q−nm[n]mq (−1)n and E(0,1)

0,q = [2]q

2. (11)

Note that Corollary 2 is a q-analogue of ζE (m), for any positive integer m.Let χ be a primitive Dirichlet character with conductor d ∈ N with d ≡ 1 (mod 2).For m ∈ N, we define

E(−m,1)m,χ,q =

∫X

q−(m+1)xχ(x)[x]m dμ−q(x), for m � 0. (12)

Note that

E(−m,1)m,χ,q = [2]q

[2]qd[d]m

q

d−1∑i=0

q−mi(−1)iχ(i)

∫Zp

q−dx(m+1)

[i

d+ x

]m

qddμ−qd (x)

= [2]q

[2]qd[d]m

q

d−1∑i=0

χ(i)(−1)iq−mi E(−m,1)

m,qd

(i

d

). (13)

1802 T. Kim / Journal of Number Theory 129 (2009) 1798–1804

3. q-Analogs of zeta functions

In this section, we assume q ∈ R with 0 < q < 1. Now we consider the q-extension of the Eulerzeta function as follows:

ζq,E (s) = [2]q

∞∑n=1

(−1)n qns

[n]sq, where s ∈ C.

By (11), we obtain the following theorem.

Theorem 3. For m ∈ N, we have

ζq,E (−m) = E(−m,1)m,q .

From Theorem 1, we can also define the q-extension of Hurwitz’s type Euler ζ -function as follows:For s ∈ C, define

ζq,E (s, x) = [2]q

∞∑n=0

(−1)nqsn

[n + x]sq

. (14)

Note that ζq,E (s, x) is an analytic continuation in whole complex s-plane.By (14) and Theorem 1, we have the following theorem.

Theorem 4. For any positive integer k, we have

ζq,E(−k, x) = E(−k,1)

k,q (x,q). (15)

For d ∈ N with d ≡ 1 (mod 2), let χ be Dirichlet character with conductor d. By (13), the generalizedq-Euler numbers attached to χ can be defined as

E(−m,1)m,χ,q = [2]q

[2]qd[d]m

q

d−1∑i=0

χ(i)q−mi(−1)i E(−m,1)

m,qd

(i

d

). (16)

For s ∈ C, we define

Lq,E(s,χ) = [2]q

∞∑n=1

χ(n)(−1)nqsn

[n]sq

. (17)

It is easy to see that

Lq,E(s,χ) = [2]q

[2]qd[d]−s

q

d∑a=1

χ(a)(−1)aqsaζqd,E

(s,

a

d

). (18)

By (16)–(18), we obtain the following theorem.

Theorem 5. Let k ∈ N. Then there exists

Lq,E(−k,χ) = E(−k,1)

k,χ,q .

T. Kim / Journal of Number Theory 129 (2009) 1798–1804 1803

Let a and F be integers with 0 < a < F . For s ∈ C, we consider the functions Hq(s,a, F ) as follows:

Hq,E(s,a, F ) = [2]q

∑m≡a(F ),m>0

qms(−1)m

[m]sq

= [2]q

[2]qF[F ]−s

q (−1)aqaζqF

(s,

a

F

).

Then we have

Hq,E (−n,a, F ) = (−1)aqa [2]q

[2]qF[F ]n

q E(−n,1)

n,qF

(a

F

),

where n is any positive integer.In the recent paper, the q-analogue of Riemann zeta function related to twisted q-Bernoulli num-

bers was studied by Y. Simsek (see [4,22,26]). In [23], Y. Simsek have studied the twisted q-Eulernumbers which can be viewed as an interpolating of the q-analogue of l-function at negative inte-gers. In this paper, we have shown that the q-analogue of Euler zeta function interpolates the q-Eulernumbers at negative integers, in the same way that Riemann zeta function interpolates Bernoullinumbers at negative integers, cf. [13,15,23,26].

Acknowledgments

The author expresses his gratitude to the referees for their valuable suggestions and comments.

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