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# On the twisted $q$-zeta functions and $q$-Bernoulli polynomials

ON THE TWISTED q-ZETA FUNCTIONS

arXiv:math/0502306v1 [math.NT] 15 Feb 2005

AND q-BERNOULLI POLYNOMIALS

Taekyun Kim, Lee Chae Jang, Seog-Hoon Rim and Hong-Kyung Pak To the 62nd birthday of Og-Yeon Choi
Abstract. One purpose of this paper is to de?ne the twisted q-Bernoulli numbers by using p-adic invariant integrals on Zp . Finally, we construct the twisted q-zeta function and q-L-series which interpolate the twisted q-Bernoulli numbers.

1. Introduction Throughout this paper Zp , Qp , C and Cp are respectively denoted as the ring of p-adic rational integers, the ?eld of p-adic rational numbers, the complex number ?eld and the completion of algebraic closure of Qp . The p-adic absolute value in Cp is normalized so that |p|p = 1 . When one talks of q-extension, q is considered in p many ways such as an indeterminate, a complex number q ¡Ê C, or a p-adic number q ¡Ê Cp . If q ¡Ê C one normally assumes that |q| < 1. If q ¡Ê Cp , we normally assume that |q ? 1|p < p?1/(p?1) so that q x = exp(x log q) for |x|p ¡Ü 1. We use the notation as [x] = [x : q] = 1 + q + ¡¤ ¡¤ ¡¤ + q x?1 , for x ¡Ê Zp . Let U D(Zp ) be the set of uniformly di?erentiable functions on Zp . For f ¡Ê U D(Zp ) the p-adic q-integral was de?ned as Iq (f ) =
Zp

f (x)d?q (x) = lim

1 N¡ú¡Þ [pN ]

f (x)q x ,
0¡Üx<pN

cf. [1, 2, 3].

Note that I1 (f ) = lim Iq (f ) =
q¡ú1 Zp

f (x)d?1 (x) = lim

1 N¡ú¡Þ pN

f (x).
0¡Üx<pN

Key words and phrases. q-Bernoulli numbers, Riemann zeta function. 2000 Mathematics Subject Classi?cation : 11B68, 11S40 .

2 TAEKYUN KIM, LEE CHAE JANG, SEOG-HOON RIM AND HONG-KYUNG PAK

For a ?xed positive integer d with (p, d) = 1, let X = Xd = lim Z/dpN Z, ¡û ?
N (a,p)=1

X1 = Zp ,

X ? = ¡È 0<a<dp (a + dpZp ), a + dpN Zp = {x ¡Ê X|x ¡Ô a(mod dpN )}, where a ¡Ê Z lies in 0 ¡Ü a < dpN , cf. [1, 2, 3, 4, 5, 6]. Let Z be the set of integers. For h ¡Ê Z, k¡ÊN, the q-Bernoulli polynomials were de?ned as
k

(h,k) (1) ¦Ân (x, q) =

[x + x1 + ¡¤ ¡¤ ¡¤ + xk ]n q i=1
Zk p

xi (h?i)

d?q (x1 ) ¡¤ ¡¤ ¡¤ d?q (xk ),

cf. [1, 2].

The q-Bernoulli polynomials at x = 0 are called q-Bernoulli numbers. In [1] it was shown that the q-Bernoulli numbers were written as
(h,k) (h,k) (h,k) ¦Ân (= ¦Ân (q)) = ¦Ân (0, q).

Indeed, limq¡ú1 ¦Ân (q) = Bn , where Bn are Bernoulli numbers of order k, see [1, 2, 3]. Let ¦Ö be a Dirichlet character with conductor f ¡Ê N. Then the Dirichlet L-series attached to ¦Ö is de?ned as L(s, ¦Ö) = ¦Ö(n) , for s ¡Ê C, cf. [7, 8]. ns n=1
¡Þ

(h,k)

(k)

(k)

When ¦Ö = 1, this is the Riemann zeta function. In [1], q-analogue of ¦Æ-function was de?ned as follows: For h ¡Ê Z, s ¡Ê C, (2)
(h) ¦Æq (s, x) =

1?s+h q (n+x)h q (n+x)h (q ? 1) + . 1?s [n + x]s?1 n=0 [n + x]s n=0

¡Þ

¡Þ

Note that ¦Æq (s, x) is an analytic continuation on C except for s = 1 with (3)
(h) ¦Æq (1 ? m, x) = ?

(h)

¦Âm

(h,1)

(x, q) , for m ¡Ê N. m

In [1], we easily see that
¡Þ (h,1) (4) ¦Âm (x, q) = ?m n=0 ¡Þ

q (n+x)h [n + x]m?1 ? (q ? 1)(m + h)
n=0

q (n+x)h [n + x]m .

It follows from (2) that
¡Þ

lim

(h) ¦Æq (s, x)

= ¦Æ(s, x) =

1
s

.

ON THE TWISTED q-ZETA FUNCTIONS AND q-BERNOULLI POLYNOMIALS

3

By the meaning of the q-analogue of Dirichlet L-series, we consider the following L-series: (5)
(h) Lq (s, ¦Ö)

q nh ¦Ö(n) q nh ¦Ö(n) 1?s+h (q ? 1) + , cf. [1], = 1?s [n]s?1 [n]s n=1 n=1
(h)

¡Þ

¡Þ

for h ¡Ê Z, s ¡Ê C. It is easy to see that Lq (s, ¦Ö) is an analytic continuation on C except for s = 1. For m ¡Ý 0, the generalized extended q-Bernoulli numbers with ¦Ö are de?ned as
k f ?1
k

(6)

(h,k) ¦Âm,¦Ö (q)

= [f ]

m?k i1 ,¡¤¡¤¡¤ ,ik =0

q l=1

(h?l+1)il

il
(h,k) ¦Âm ( l=1

f

, qf )

j=1

¦° ¦Ö(ij ) , see [1] .

k

By (5) and (6), we easily see that
(h) Lq (1 ? m, ¦Ö) = ?

¦Âm,¦Ö , for m ¡Ê N, cf. [1]. m

(h,1)

In the present paper we give twisted q-Bernoulli numbers by using p-adic invariant integrals on Zp . Moreover, we construct the analogs of q-zeta function and q-L-series which interpolate the twisted q-Bernoulli numbers at negative integers. 2. q-extension of Bernoulli numbers In this section we assume that q ¡Ê Cp with |1 ? q|p < 1. By the de?nition of p-adic invariant integrals, we see that (7) where f1 (x) = f (x + 1). Let Tp = ¡Èn¡Ý1 Cpn = lim Cpn ,
n¡ú¡Þ

I1 (f1 ) = I1 (f ) + f ¡ä (x),

where Cpn = {w|wp = 1} is the cyclic group of order pn . For w ¡Ê Tp , we denote by ¦Õw : Zp ?¡ú Cp the locally constant function x ¡ú wx . If we take f (x) = ¦Õw (x)etx , then we easily see that (8)
Zp

n

etx ¦Õw (x)d?1 (x) =

t , wet ? 1

cf. [5].

It is obvious from (7) that
f

¦Ö(i)¦Õw (i)eit , cf. [5] .

(9)

etx ¦Ö(x)¦Õ (x)d? (x) =

i=1

4 TAEKYUN KIM, LEE CHAE JANG, SEOG-HOON RIM AND HONG-KYUNG PAK

Now we de?ne the analogue of Bernoulli numbers as follows: t tn e = Bn,w (x) , wet ? 1 n=0 n!
xt ¡Þ

(10)

f i=1

¦Ö(i)¦Õw (i)eit wf ef t ? 1 =

¡Þ

Bn,w,¦Ö
n=0

tn , cf. [5] . n!

By (8), (9) and (10), it is not di?cult to see that (11)
Zp

xn ¦Õw (x)d?1 (x) = Bn,w

and
X

¦Ö(x)xn ¦Õw (x)d?1 (x) = Bn,w,¦Ö ,

where Bn,w = Bn,w (0). From (11) we consider twisted q-Bernoulli numbers using p-adic q-integral on Zp . For w ¡Ê Tp and h ¡Ê Z, we de?ne the twisted q-Bernoulli polynomials as (12) Observe that
q¡ú1 (h) (h) lim ¦Âm,w (x, q) = Bm,w (x). (h) (h) ¦Âm,w (x, q) = Zp

q (h?1)y wy [x + y]m d?q (y).

When x = 0, we write ¦Âm,w (0, q) = ¦Âm,w (q), which are called twisted q-Bernoulli numbers. It follows from (12) that (13)
(h) ¦Âm,w (x, q) =

1 (1 ? q)m?1

m

k=0

k+h m xk q (?1)k . k 1 ? q h+k w

The Eq.(13) is equivalent to
¡Þ ¡Þ

(14)

(h) ¦Âm,w (q) = ?m n=0

[n]m?1 q hn wn ? (q ? 1)(m + h)
n=0

[n]m q hn wn .

From (13), we obtain the below distribution relation for the twisted q-Bernoulli polynomials as follows: For n ¡Ý 0,
f ?1 (h) ¦Ân,w (x, q)

= [f ]

n?1 a=0

wa q ha ¦Ân,wf (

(h)

a+x f , q ). f

Let ¦Ö be the Dirichlet character with conductor f ¡Ê N. Then we de?ne the generalized twisted q-Bernoulli numbers as follows: For n ¡Ý 0, (15) ¦Â (h) (q) = ¦Ö(x)q (h?1)x wx [x]m d? (x).

ON THE TWISTED q-ZETA FUNCTIONS AND q-BERNOULLI POLYNOMIALS

5

By (15), it is easy to see that
f ?1

(16)

(h) ¦Âm,w,¦Ö (q)

= [f ]

m?1

a (h) ¦Ö(a)wa q ha ¦Âm,wf ( , q f ). f a=0

Remark. We note that limq¡ú1 ¦Âm,w,¦Ö (q) = Bm,w,¦Ö , ( see Eq. (10) ). 3. q-zeta functions In this section we assume that q ¡Ê C with |q| < 1. Here we construct the twisted q-zeta function and the twisted q-L-series (see Eq.(2) and Eq.(3)). Let R be the ?eld of real numbers and let w be the pr -th root of unity. For q ¡Ê R with 0 < q < 1, s ¡Ê C and h ¡Ê Z, we de?ne the twisted q-zeta function as (17)
(h) (h) ¦Æq,w (s)

(h)

1?s+h q nh wn q nh wn = (q ? 1) + . 1?s [n]s?1 n=1 [n]s n=1
(h) q¡ú1

¡Þ

¡Þ

Note that ¦Æq,w (s) is an analytic continuation on C except for s = 1 and lim ¦Æq,w (s) = ¦Æ(s, w) = (18)
¡Þ wn n=1 ns ,

cf. [4]. We see, by (17), that
(s?1) ¦Æq,w (s)

=

q n(s?1) wn . [n]s n=1

¡Þ

In what follows, the notation ¦Æq,w (s) will be replaced by ¦Æq,w (s), that is, ¦Æq,w (s)(=
(s?1) ¦Æq,w (s))

(s?1)

q n(s?1) wn = . [n]s n=1

¡Þ

We note that Eq.(18) is the q-extension of Riemann zeta function. By (14) and (h) (17) we give the values of ¦Æq,w (s) at negative integers as follows: For m ¡Ê N, we have (19) By (17), we also see that
¡Þ (h) ¦Æq,w (1

¦Âm,w (q) ? m) = ? . m

(h)

(20)

¦Æq,w (1 ? m) =
n=1

[n]m?1 q ?mn wn .

The Eq.(20) seems to be the q-analogue of Euler divergence theorem for Riemann zeta function. Now we also consider the twisted q-analogue of Hurwitz zeta function as follows: For s ¡Ê C, de?ne (21)
(h) ¦Æq,w (s, x)

=

1?s+h

¡Þ

(q ? 1)

q (n+x)h wn
s?1

¡Þ

+

q (n+x)h wn
s

.

6 TAEKYUN KIM, LEE CHAE JANG, SEOG-HOON RIM AND HONG-KYUNG PAK

Note that ¦Æq (s, x) has an analytic continuation on C with only one simple poles at s = 1. By Eq.(13), Eq.(14) and Eq.(21), we obtain the following: ¦Âm,w (x, q) , for m > 0. ? m, x) = ? m Now we consider the twisted q-L-series which interpolate twisted generalized q-Bernoulli numbers as follows: For s ¡Ê C, de?ne
(h) ¦Æq,w (1 (h)

(h)

(22)

(h) Lq,w (s, ¦Ö)

1?s+h q nh wn ¦Ö(n) q nh wn ¦Ö(n) = (q ? 1) + , 1?s [n]s?1 [n]s n=1 n=1

¡Þ

¡Þ

where w is the pr -th root of unity. For any positive integer m we have
(h) Lq,w (1 ? m, ¦Ö) = ?

¦Âm,w,¦Ö (q) . m

(h)

The Eq.(22) implies that L(s?1) (s, ¦Ö) = q,w q n(s?1) wn ¦Ö(n) [n]s n=1
?s ¡Þ

= [f ]

a ¦Ö(a)wa q (s?1)a ¦Æq f ,wf (s, ). f a=1

f

Question. Find a q-analogue of the p-adic twisted L-function which interpolates (h) q-Bernoulli numbers ¦Âm,w,¦Ö (q), cf. [3]. ACKNOWLEDGEMENTS: This paper was supported by Korea Research Foundation Grant( KRF-2002-050-C00001). References
1. T. Kim and S. H. Rim, Generalized Carlitz¡¯s q-Bernoulli numbers in the p-adic number ?eld, Adv. Studies Contemp. Math. 2 (2000), 9¨C19. 2. T. Kim, q-Riemann zeta function, to appear in Inter. J. Math. Math. Sci. 2003 (2003), 00-00. 3. T. Kim, q-Volkenborn integration, Russian J. Math. Phys. 9 (2002), 288-299. 4. T. Kim, Barnes-Euler multiple zeta functions, Russian J. Math. Phys. 10 (2003), 261¨C267. 5. T. Kim, An analogue of Bernoulli numbers and their applications, Rep. Fac. Sci. Engrg. Saga Univ. Math. 22 (1994), 21-26. 6. N. Koblitz, A new proof of certain formulas for p-adic L-functions, Duke Math. J. 40 (1979), 455-468. 7. K. Shiratani and S. Yamamoto, On a p-adic interpolating function for the Euler number and its derivative, Mem. Fac. Sci. Kyushu Univ. 39 (1985), 113¨C125. 8. L. C. Washington, Introduction to cyclotomic ?eld, vol. 83, Graduate Texts in Math., Springer, 1996.

Institute of Science Education, Kongju National University, Kongju 314-701, Korea, tkim@kongju.ac.kr Department of Mathematics and Computer Science, KonKuk University, Choongju, Chungbuk 380-701, Korea, leechae.jang@kku.ac.kr Department of Mathematics Education, Kyungpook University, Taegu, 702-701, Korea, shrim@kyungpook.ac.kr Faculty of Informatoin and Science, Daegu Haany University, Kyungsan, 712-240,

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