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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 ∈ C, or a p-adic number q ∈ Cp . If q ∈ C one normally assumes that |q| < 1. If q ∈ Cp , we normally assume that |q ? 1|p < p?1/(p?1) so that q x = exp(x log q) for |x|p ≤ 1. We use the notation as [x] = [x : q] = 1 + q + · · · + q x?1 , for x ∈ Zp . Let U D(Zp ) be the set of uniformly di?erentiable functions on Zp . For f ∈ U D(Zp ) the p-adic q-integral was de?ned as Iq (f ) =

Zp

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

1 N→∞ [pN ]

f (x)q x ,

0≤x<pN

cf. [1, 2, 3].

Note that I1 (f ) = lim Iq (f ) =

q→1 Zp

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

1 N→∞ pN

f (x).

0≤x<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, ← ?

N (a,p)=1

X1 = Zp ,

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

k

(h,k) (1) βn (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) βn (= βn (q)) = βn (0, q).

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

∞

(h,k)

(k)

(k)

When χ = 1, this is the Riemann zeta function. In [1], q-analogue of ζ-function was de?ned as follows: For h ∈ Z, s ∈ C, (2)

(h) ζq (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

∞

∞

Note that ζq (s, x) is an analytic continuation on C except for s = 1 with (3)

(h) ζq (1 ? m, x) = ?

(h)

βm

(h,1)

(x, q) , for m ∈ N. m

In [1], we easily see that

∞ (h,1) (4) βm (x, q) = ?m n=0 ∞

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

∞

lim

(h) ζq (s, x)

= ζ(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, χ)

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

(h)

∞

∞

for h ∈ Z, s ∈ C. It is easy to see that Lq (s, χ) is an analytic continuation on C except for s = 1. For m ≥ 0, the generalized extended q-Bernoulli numbers with χ are de?ned as

k f ?1

k

(6)

(h,k) βm,χ (q)

= [f ]

m?k i1 ,··· ,ik =0

q l=1

(h?l+1)il

il

(h,k) βm ( l=1

f

, qf )

j=1

Π χ(ij ) , see [1] .

k

By (5) and (6), we easily see that

(h) Lq (1 ? m, χ) = ?

βm,χ , for m ∈ 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 ∈ 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 = ∪n≥1 Cpn = lim Cpn ,

n→∞

I1 (f1 ) = I1 (f ) + f ′ (x),

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

Zp

n

etx φw (x)d?1 (x) =

t , wet ? 1

cf. [5].

It is obvious from (7) that

f

χ(i)φw (i)eit , cf. [5] .

(9)

etx χ(x)φ (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 ∞

(10)

f i=1

χ(i)φw (i)eit wf ef t ? 1 =

∞

Bn,w,χ

n=0

tn , cf. [5] . n!

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

Zp

xn φw (x)d?1 (x) = Bn,w

and

X

χ(x)xn φw (x)d?1 (x) = Bn,w,χ ,

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

q→1 (h) (h) lim βm,w (x, q) = Bm,w (x). (h) (h) βm,w (x, q) = Zp

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

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

(h) βm,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

∞ ∞

(14)

(h) βm,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 ≥ 0,

f ?1 (h) βn,w (x, q)

= [f ]

n?1 a=0

wa q ha βn,wf (

(h)

a+x f , q ). f

Let χ be the Dirichlet character with conductor f ∈ N. Then we de?ne the generalized twisted q-Bernoulli numbers as follows: For n ≥ 0, (15) β (h) (q) = χ(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) βm,w,χ (q)

= [f ]

m?1

a (h) χ(a)wa q ha βm,wf ( , q f ). f a=0

Remark. We note that limq→1 βm,w,χ (q) = Bm,w,χ , ( see Eq. (10) ). 3. q-zeta functions In this section we assume that q ∈ 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 ∈ R with 0 < q < 1, s ∈ C and h ∈ Z, we de?ne the twisted q-zeta function as (17)

(h) (h) ζq,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→1

∞

∞

Note that ζq,w (s) is an analytic continuation on C except for s = 1 and lim ζq,w (s) = ζ(s, w) = (18)

∞ wn n=1 ns ,

cf. [4]. We see, by (17), that

(s?1) ζq,w (s)

=

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

∞

In what follows, the notation ζq,w (s) will be replaced by ζq,w (s), that is, ζq,w (s)(=

(s?1) ζq,w (s))

(s?1)

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

∞

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

∞ (h) ζq,w (1

βm,w (q) ? m) = ? . m

(h)

(20)

ζq,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 ∈ C, de?ne (21)

(h) ζq,w (s, x)

=

1?s+h

∞

(q ? 1)

q (n+x)h wn

s?1

∞

+

q (n+x)h wn

s

.

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

Note that ζq (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: βm,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 ∈ C, de?ne

(h) ζq,w (1 (h)

(h)

(22)

(h) Lq,w (s, χ)

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

∞

∞

where w is the pr -th root of unity. For any positive integer m we have

(h) Lq,w (1 ? m, χ) = ?

βm,w,χ (q) . m

(h)

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

?s ∞

= [f ]

a χ(a)wa q (s?1)a ζq 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 βm,w,χ (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), 9–19. 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), 261–267. 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), 113–125. 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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a e 1. Introduction Let Sn denote

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by using

with q-Genocchi type

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