J. Math. Anal. Appl. 336 (2007) 738–744

www.elsevier.com/locate/jmaa

On the twisted q-Euler numbers and polynomialsassociated with basic q-l-functions

Taekyun Kim a, Seog-Hoon Rim b,∗

a EECS, Kyungpook National University, Daegu 702-701, South Koreab Department of Mathematics Education, Kyungpook National University, Daegu 702-701, South Korea

Received 27 November 2006

Available online 20 March 2007

Submitted by D. Waterman

Abstract

One of the purposes of this paper is to construct the twisted q-Euler numbers by using p-adic invariantintegral on Zp in the fermionic sense. Moreover, we consider the twisted Euler q-zeta functions and q-l-functions which interpolate the twisted q-Euler numbers and polynomials at a negative integer.© 2007 Elsevier Inc. All rights reserved.

Keywords: Twisted q-Bernoulli numbers; Twisted l-function; Zeta function; Twisted Euler numbers

1. Introduction

Let p be a fixed odd prime number. Throughout this paper Zp , Qp , C and Cp are, respectively,denoted as the ring of p-adic rational integers, the field of p-adic rational numbers, the complexnumber field and the completion of algebraic closure of Qp . The p-adic absolute value in Cp isnormalized so that |p|p = 1/p. When one talks of q-extension, q is variously considered as anindeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp . If q ∈ C, we usually assumethat |q| < 1. If q ∈ Cp , we usually assume that |q − 1|p < p−1/p−1 so that qx = exp(x logq) for|x|p � 1, cf. [1,2,15,20].

We use the notations as

�x�q = 1 − qx

1 − q= 1 + q + q2 + · · · + qx−1,

* Corresponding author.E-mail addresses: tkim64@hanmail.net, tkim@knu.ac.kr (T. Kim), shrim@knu.ac.kr (S.-H. Rim).

0022-247X/$ – see front matter © 2007 Elsevier Inc. All rights reserved.doi:10.1016/j.jmaa.2007.03.035

T. Kim, S.-H. Rim / J. Math. Anal. Appl. 336 (2007) 738–744 739

�x�−q = 1 − (−q)x

1 + q

= 1 − q + q2 − q3 + · · · + (−1)x−1qx−1, see [3,8,9,12].

Let UD(Zp) be the set of uniformly differentiable function on Zp . For f ∈ UD(Zp), thep-adic invariant q-integral on Zp was defined by Kim as follows:

Iq(f ) =∫Zp

f (x) dμq(x) = limv→∞

1

�pv�q

pv−1∑x=0

f (x)qx,

where v is a fixed natural number.

In [4,13,21], the bosonic integral was considered from a more physical point of view to thebosonic limit q → 1 as follows:

I1(f ) = limq→1

Iq(f ) =∫Zp

f (x) dμ1(x) = limv→∞

1

pv

pv−1∑x=0

f (x).

We now consider the fermionic integral in contrast to the conventional “bosonic.” That is,

I−1(f ) = limq→−1

Iq(f ) =∫Zp

f (x) dμ−1(x).

For a fixed positive integer d with (p, d) = 1, v ∈ N, set

X = Xd = lim←−vZ/dpvZ,

X1 = Zp,

X∗ =⋃

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

a + dpZp,

a + dpvZp = {x ∈ X

∣∣ x ≡ a(mod dpv

)},

where a ∈ Z satisfies the condition 0 � a < dpv , cf. [15].Recently Kim has defined the q-Euler numbers as follows:

Fq(x, t) = �2�q

∞∑n=0

(−1)nqne�n+x�q t =∞∑

n=0

En,q(x)tn

n! , see [5,12].

Let us consider the q-deformed fermionic integral on Zp as follows:

I−q(f ) = limq→−q

Iq(f ) =∫Zp

f (x) dμ−q(x).

For h ∈ Z, k ∈ N, the q-Euler polynomials of higher order are defined as

E(h,k)n,q (x) =

∫Zp

· · ·∫Zp︸ ︷︷ ︸

k-times

�x + x1 + · · · + xk�nqq

∑ki=1 xi (h−i)dμ−q(x1) · · ·dμ−q(xk),

see [8,15]. (1)

740 T. Kim, S.-H. Rim / J. Math. Anal. Appl. 336 (2007) 738–744

The q-Euler polynomials of higher order at x = 0 are called the q-Euler numbers, cf. [1,2,14].Hence the q-Euler numbers of higher order can be defined by E

(h,k)n,q = E

(h,k)n,q (0). Let χ be the

Dirichlet character with conductor f (= odd) ∈ N. Then the l-function is defined as

l(s,χ) = 2∞∑

n=1

χ(n)(−1)n

ns, for s ∈ C, see [4,13].

When χ = 1, this is the Euler zeta function, which is defined

ζE(s) = 2∞∑

n=1

(−1)n

ns, see [3,4,13].

In [8,12], q-analogue of Euler ζ -function is defined as

ζ(h)E,q(s, x) = �2�q

∞∑n=0

(−1)nqnh

�n + x�sq

, s ∈ C, h ∈ Z. (2)

Note that ζ(h)E,q(s, x) is an analytic function on C with

ζ(h)E,q(−n,x) = E(h,1)

n,q (x), for n ∈ N, h ∈ Z, see [5,8]. (3)

It was known that

E(h,1)n,q (x) = �2�q

∞∑k=0

(−1)kqkh�k + x�nq, see [8]. (4)

It follows from (2) that

limq→1

ζ(h)E,q(s, x) = ζE(s, x) = 2

∞∑n=0

(−1)n

(n + x)s, see [13].

With the meaning of q-analogue of the Dirichlet’s type l-function, we consider the followingq-l-function: for s ∈ C,

lq(s,χ) = �2�q

∞∑k=1

χ(k)(−1)kqk

�k�sq

, see [3,5,12]. (5)

It is easy to see that lq(s,χ) is an analytic function in the whole complex plane. The generalizedq-Euler numbers attached to χ are also defined as

�2�q

∞∑n=0

e�n�q tχ(n)(−1)nqn =∞∑

n=0

En,χ,q

tn

n! , see [5,6,12]. (6)

Note that∫X

χ(x)�x�nq dμ−q(x) = En,χ,q and lq(−n,x) = En,χ,q , see [3,5]. (7)

For m � 0, the extended generalized q-Euler numbers attached to χ are defined as

T. Kim, S.-H. Rim / J. Math. Anal. Appl. 336 (2007) 738–744 741

E(h,k)n,χ,q =

∫X

· · ·∫X︸ ︷︷ ︸

k-times

⌈k∑

l=1

xl

⌉n

q

q∑k

j=1(h−j)xj

(k∏

j=1

χ(xj )

)dμ−q(x1) · · ·dμ−q(xk)

= �2�kq

�2�kqf

�f �nq

f −1∑a1,...,ak=0

q∑k

l=1(h−l+1)al (−1)a1+···+ak

× E(h,k)

n,qf

(al + a2 + · · · + ak

f

) k∏i=1

χ(ai) (see [8,10]). (8)

From (8), we derive

E(h,1)n,χ,q = �2�q

�2�qf

�f �nq

f −1∑a=0

qha(−1)aE(h,1)

n,qf

(a

f

)χ(a)

= �2�q

∞∑k=1

χ(k)(−1)kqhk�k�nq, see [5, p. 5]. (9)

From (9), we can consider the following q-l-function:

l(h)q (s,χ) = �2�q

∞∑k=1

χ(k)(−1)kqhk

�k�sq

, see [5,8,10],

where s ∈ C. Note that l(h)q (−n,χ) = E

(h,1)n,χ,q , for n ∈ N, h ∈ Z.

In the present paper, we give the twisted q-Euler numbers by using p-adic invariant q-integralon Zp in the fermionic sense. Moreover, we construct the q-analogues of Euler zeta function andthe q-l-function which interpolate the twisted q-Euler numbers and polynomials at a negativeinteger.

2. Twisted q-extension of Euler numbers

In this section, we assume that q ∈ Cp with |1 − q|p < 1. From the definition of p-adicinvariant q-integral on Zp in the fermionic sense, we derive

limq→−1

Iq(f1) = I−1(f1) = −I−1(f ) + 2f (0), where f1(x) = f (x + 1), see [7]. (10)

Let Tp = ⋃n�1 Cpn = limn→∞ Cpn = Cp∞ be the locally constant space, where Cpn = {w |

wpn = 1} is the cyclic group of order pn. For w ∈ Tp , we denote the locally constant function byφw : Zp → Cp , x → wx . If we take f (x) = φw(x)etx , then we have∫

Zp

etxφw(x) dμ−1(x) = 2

wet + 1, see [4,11,13,16–19]. (11)

From (10), we derive

I−1(fn) = (−1)nI−1(f ) + 2n−1∑

(−1)n−1−lf (l), see [6,7], (12)

l=0

742 T. Kim, S.-H. Rim / J. Math. Anal. Appl. 336 (2007) 738–744

where fn(x) = f (x + n). By (12), we easily see that∫X

etxφw(x)χ(x) dμ−1(x) = 2∑f −1

i=0 χ(i)φw(i)eit

wf ef t + 1. (13)

Now we define the analogue of Euler numbers as follows:

2∑f −1

i=0 χ(i)φw(i)eit

wf ef t + 1=

∞∑n=0

En,χ,w

tn

n! . (14)

From (13) and (14), we note that∫X

xnφw(x)χ(x) dμ−1(x) = En,χ,w. (15)

In the viewpoint of (15), we consider the twisted q-Euler numbers using p-adic invariant q-integral on Zp in the fermionic sense as follows:

E(h,1)n,w,q(x) =

∫Zp

q(h−1)ywy�x + y�nq dμ−q(y). (16)

Observe that limq→1 E(h,1)n,q,w(x) = En,w(x). When x = 0, we write E

(h)n,w(0, q) = E

(h)n,w . Note that

E(h,1)n,w,q is the twisted form of E

(h,1)n,q , see [8].

From (16), we derive

E(h,1)n,w,q(x) = �2�q

(1 − q)n

n∑j=0

(n

j

)(−1)j qxj 1

1 + qh+jw. (17)

Equation (17) is equivalent to

E(h,1)n,w,q(x) = �2�q

∞∑k=0

(−1)kwkqhk�x + k�nq, h ∈ Z, n ∈ N. (18)

By (16), we see that

E(h,1)n,w,q(x) = �2�q

�2� qd

�d�nq

d−1∑a=0

qhawa(−1)aE(h,1)

n,qd

(x + a

d

),

where n,d(= odd) ∈ N.Let χ be the Dirichlet character with conductor f (= odd) ∈ N. Then we define the generalized

twisted q-Euler numbers as follows: For n � 0,

E(h,1)n,w,χ,q =

∫X

χ(x)q(h−1)xwx�x�nq dμ−q(x). (19)

Note that E(1,1) = En,χ,q , see [3,5,12].

n,1,χ,q

T. Kim, S.-H. Rim / J. Math. Anal. Appl. 336 (2007) 738–744 743

From (19), we can also derive

E(h,1)n,w,χ,q = �f �n

q

�2�q

�2�qf

f −1∑a=0

qhawaχ(a)(−1)aE(h,1)

n,wf ,qf

(a

f

)

= �2�q

∞∑k=1

χ(k)(−1)kqhkwk�k�nq . (20)

It is easy to check that

limq→1

E(h,1)n,w,χ,q = En,w,χ , see [4].

3. q-Euler zeta function and q-l-function

In this section, we assume that q ∈ C with |q| < 1. Here we construct the twisted q-Eulerzeta function and twisted q-l-function. Let w be the pnth root of unity. For s ∈ C and h ∈ Z, wedefine the twisted Euler q-zeta function as follows:

ζ(h,1)E,q,w(s) = �2�q

∞∑k=1

(−1)kwkqhk

�k�sq

.

Note that ζ(h,1)E,q,w is an analytic function on C. In [8], it is easy to see that

ζ(h,1)E,q,1(s) = ζ

(h)E,q(s).

By (18), we have

ζ(h,1)E,q,w(−n) = E(h,1)

n,w,q , for n ∈ N, h ∈ Z.

We now also consider the Hurwitz’s type twisted q-Euler zeta function as follows: For s ∈ C,

ζ(h,1)E,q,w(s, x) = �2�q

∞∑k=0

(−1)kqhkwk

�x + k�sq

. (21)

Note that ζ(h,1)E,q,w(s, x) is an analytic function on C. By (18), we obtain

ζ(h,1)E,q,w(−n,x) = E(h,1)

n,w,q(x), for n ∈ N, h ∈ Z.

From [8], we notice that

ζ(h,1)E,q,1(s, x) = ζ

(h)E,q(s, x).

Let χ be the Dirichlet’s character with conductor f (= odd) ∈ N. Then we define the twistedq-l-function which interpolates the twisted generalized q-Euler numbers attached to χ as fol-lows: For s ∈ C, h ∈ Z,

l(h,1)q,w (s,χ) = �2�q

∞∑k=1

χ(k)(−1)kqhkwk

�k�sq

. (22)

For any positive integer n, we obtain

l(h,1)q,w (−n,χ) = E(h,1)

n,w,χ,q , for n ∈ N, h ∈ Z.

744 T. Kim, S.-H. Rim / J. Math. Anal. Appl. 336 (2007) 738–744

From (22), we can also derive

l(h,1)q,w (s,χ) = �2�q

∞∑k=1

χ(k)(−1)kqhkwk

�k�sq

= �f �−sq

�2�q

�2�qf

f∑a=1

χ(a)(−1)aqhawaζE,qf ,wf

(s,

a

f

).

Question. Find a q-analogue of the p-adic twisted l-function which interpolates the generalizedtwisted q-Euler numbers attached to χ , E

(h,1)n,w,χ,q [4,5,10,13].

Acknowledgment

The authors wish to express their sincere gratitude to the referee for his/her valuable suggestions and comments.

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