EXPLORING THE q-RIEMANN ZETA FUNCTION
AND q-BERNOULLI POLYNOMIALS
T. KIM, C. S. RYOO, L. C. JANG, AND S. H. RIM
Received 15 July 2004
We study that the q-Bernoulli polynomials, which were constructed by Kim, are analytic
continued to βs (z). A new formula for the q-Riemann zeta function ζq (s) due to Kim in
terms of nested series of ζq (n) is derived. The new concept of dynamics of the zeros of
analytic continued polynomials is introduced, and an interesting phenomenon of “scattering” of the zeros of βs (z) is observed. Following the idea of q-zeta function due to Kim,
we are going to use “Mathematica” to explore a formula for ζq (n).
1. Introduction
Throughout this paper, Z, R, and C will denote the ring of integers, the field of real numbers, and the complex numbers, respectively.
When one talks of q-extension, q is variously considered as an indeterminate, a complex number, or a p-adic number. In the complex number field, we will assume that
|q| < 1 or |q| > 1. The q-symbol [x]q denotes [x]q = (1 − qx )/(1 − q).
In this paper, we study that the q-Bernoulli polynomials due to Kim (see [2, 8]) are
analytic continued to βs (z). By those results, we give a new formula for the q-Riemann
zeta function due to Kim (cf. [4, 6, 8]) and investigate the new concept of dynamics of the
zeros of analytic continued polynomials. Also, we observe an interesting phenomenon of
“scattering” of the zeros of βs (z). Finally, we are going to use a software package called
“Mathematica” to explore dynamics of the zeros from analytic continuation for q-zeta
function due to Kim.
2. Generating q-Bernoulli polynomials and numbers
For h ∈ Z, the q-Bernoulli polynomials due to Kim were defined as
∞
βn x,h | q
n =0
n!
t n = −t
∞
ql(h+1)+x e[l+x]q t + (1 − q)h
l=0
for x, q ∈ C (cf. [6, 8]).
Copyright © 2005 Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society 2005:2 (2005) 171–181
DOI: 10.1155/DDNS.2005.171
∞
l =0
qlh e[l+x]q t ,
(2.1)
172
q-Riemann zeta function
In the special case x = 0, βn (0,h | q) = βn (h | q) are called q-Bernoulli numbers (cf.
[1, 5, 7, 8]).
By (2.1), we easily see that
1
βn x,h | q =
(1 − q)n
n n
j
j =0
(−1) j
j + h jx
q ,
[ j + h]q
(cf. [2, 6]),
(2.2)
where nj is a binomial coefficient.
In (2.1), it is easy to see that
h
q qβ h | q + 1
n
− βn h | q =


1
if n = 1,

0
if n > 1,
(2.3)
with the usual convention of replacing βn (h | q) by βn (h | q).
By differentiating both sides with respect to t in (2.1), we have
β m h | q = −m
∞
n =0
−1
qhn [n]m
− (q − 1)(m + h)
q
∞
n =0
qhn [n]m
q.
(2.4)
Expanding (2.1) as a series and matching the coefficients on both sides give
β0 2 | q =
2
,
[2]q
β1 2 | q =
2q + 1
,
[2]q [3]q
q2 (q − 1) 2[3]q + q
β3 2 | q = −
,...,
[3]q [4]q [5]q
h
,
[h]q
β2 2 | q =
β0 h | q =
2q2
,
[3]q [4]q
(2.5)
1 + q + · · · + q h −1 + q 1 + q + · · · + q h −2 + · · · + q h −1
β1 h | q = −
,....
[h]q [h + 1]q
By (2.1), the q-Bernoulli polynomials can be written as
βm x,h | q =
m m
j =0
j
n− j
[x]q q jx β j h | q .
(2.6)
T. Kim et al. 173
0.6
0.4
βm
0.2
0
−0.2
−0.4
−0.6
−1
−0.5
0
0.5
1
x
Figure 3.1. The curve of βm (x,1 | 1/2), 1 ≤ m ≤ 10, −1 ≤ x ≤ 1.
In the case h = 0, βm (x,0 | q) will be symbolically written as βm,q (x). Let Gq (x,t) be the
generating function of q-Bernoulli polynomials as follows:
Gq (x,t) =
∞
n =0
βn,q (x)
tn
.
n!
(2.7)
Then we easily see that
Gq (x,t) =
∞
q − 1 t/(1−q)
−t
qh+x e[n+x]q t ,
e
log q
n =0
|t | < 1, (cf. [2, 3, 4, 6]).
(2.8)
For x = 0, βn,q = βn,q (0) will be called q-Bernoulli numbers.
By (2.8), we easily see that
βm,q (n) − βm,q = m
n
−1
l =0
−1
ql [l]m
.
q
(2.9)
Thus, we have
n
−1
l=0
q
l
−1
[l]m
q
m−1
1
1 m nl
−l
=
q βl,q [n]m
+
1 − qmn βm,q .
q
m l=0 l
m
(2.10)
3. Beautiful shape of q-Bernoulli polynomials
In this section, we display the shapes of the q-Bernoulli polynomials βm (x,1|1/2). For
m = 1,2,...,10, we can draw a plot of βm (x,1|1/2), respectively. This shows the ten plots
combined into one. For m = 1,...,10, q, Figure 3.1 displays the shapes of the q-Bernoulli
q-Riemann zeta function
3
3
2
2
1
1
Im(x)
Im(x)
174
0
0
−1
−1
−2
−2
−3
−2
−1
0
1
2
3
−3
4
−2
−1
Re(x)
0
1
2
3
4
Re(x)
2
2
1.5
1.5
1
1
0.5
0.5
Im(x)
Im(x)
Figure 3.2. Zeros of q-Bernoulli polynomials βm (x,1 | 1/2), m = 40,60, and x ∈ C.
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−0.75 −0.5 −0.25
0
Re(x)
0.25 0.5 0.75
1
−0.75 −0.5 −0.25
0
0.25 0.5 0.75
1
Re(x)
Figure 3.3. Zeros of q-Bernoulli polynomials βm (x,1 | −1/2), m = 40,60, and x ∈ C.
polynomials βm (x,1|1/2). We plot the zeros of βm (x,1|1/2), m = 40, m = 60, and x ∈ C
(Figure 3.2). We plot the zeros of βm (x,1| − 1/2), m = 40, m = 60, and x ∈ C (Figure 3.3).
We plot the zeros of βm (x,1|11/10), m = 40, m = 60, and x ∈ C (Figure 3.4). We plot
the zeros of βm (x,1| − 11/10), m = 40, m = 60, and x ∈ C (Figure 3.5). Stacks of zeros
of βn (x,1|1/2), 1 ≤ n ≤ 60, from a 3D structure are presented in Figure 3.6. The curve
β(s) runs through the points β−n (n|1/2) (Figure 3.7). We draw the curve of β−n (n|q) and
limn→∞ = nζq (n + 1), q = 3/10, 5/10, 7/10, 9/10, 99/100, 999/1000 (Figures 3.8, 3.9, and
3.10).
20
20
15
15
10
10
5
5
Im(x)
Im(x)
T. Kim et al. 175
0
0
−5
−5
−10
−10
−15
−15
−15 −10 −5
0
5
10
15
−15 −10 −5
20
0
5
10
15
20
Re(x)
Re(x)
2
2
1.5
1.5
1
1
0.5
0.5
Im(x)
Im(x)
Figure 3.4. Zeros of q-Bernoulli polynomials βm (x,1 | 11/10), m = 40,60, and x ∈ C.
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−0.75 −0.5 −0.25 0
0.25 0.5 0.75
1
Re(x)
−0.75 −0.5 −0.25 0
0.25 0.5 0.75
1
Re(x)
Figure 3.5. Zeros of q-Bernoulli polynomials βm (x,1 | −11/10), m = 40,60, and x ∈ C.
4. q-Riemann zeta function
We display the plot of βq (s), 0.1 ≤ s ≤ 0.9, 1.1 ≤ q ≤ 2 (Figure 4.1). We display the plot
of βq (s), 1.03 ≤ s ≤ 2, 0.1 ≤ q ≤ 2 (Figure 4.2). We draw the curve of ζq (n), q = 7/10,
9/10 (Figure 4.3). We draw the curve of β−q (s,w), 2 ≤ s ≤ 3, −0.5 ≤ w ≤ 0.5, q = 11/10
(Figure 4.4).
The q-Riemann zeta function due to Kim was defined as
ζq(h) (s) =
∞
∞
qnh
qnh
1−s+h
+
,
(q − 1)
s
−
1
1−s
[n]q
[n]sq
n =1
n =1
for s,h ∈ C, (cf. [6, 8]).
(4.1)
176
q-Riemann zeta function
1
Im(x)
0
−1
60
40
n
20
0
0
1
Re(x)
2
3
Figure 3.6. Stacks of zeros of q-Bernoulli polynomials βn (x,1 | 1/2), 1 ≤ n ≤ 60, from a 3D structure.
0.7
0.6
β(s)
0.5
0.4
0.3
0.2
0.1
0
−20
−15
−10
−5
0
s
Figure 3.7. The curve β(s) runs through the points β−n (n | 1/2).
For k ∈ N, h ∈ Z, it was known that
βk
ζq(h) (1 − k) = −
h|q
,
k
(cf. [6, 8]).
(4.2)
In the special case h = s − 1, ζq(s−1) (s) will be written as ζq (s). For s ∈ C, we note that
ζq (s) =
∞
qn(s−1)
n =1
[n]sq
,
(cf. [6, 8]).
(4.3)
T. Kim et al. 177
0.7
0.08
0.6
0.5
β−n
β−n
0.06
0.04
0.4
0.3
0.2
0.02
0.1
0
−20
0
−15
−10
−5
−20
0
−15
n
−10
−5
0
n
Figure 3.8. The curve of β−n (n | q) and limn→∞ β−n = nζq (n + 1) = 0, q = 3/10, 5/10
.
3.5
1
3
0.6
β−n
β−n
0.8
2.5
0.4
2
0.2
1.5
0
−20
−15
−10
−5
−20
0
n
−15
−10
−5
0
n
Figure 3.9. The curve of β−n (n | q) and limn→∞ β−n = nζq (n + 1) = 0, q = 7/10, 9/10
.
By (4.1), (4.2), and (4.3), we easily see that
βk − k | q
ζq (1 − k) = −
,
k
for k ∈ N, (cf. [3, 4, 6]).
(4.4)
From the above analytic continuation of q-Bernoulli numbers, we consider
βn = βn − n | q −→ β(s),
βn+1 − n + 1 | q
β(s + 1)
ζ(s)
ζq (−n) = −
−→ ζq (−s) = −
=⇒ ζq (1 − s) = −
.
n+1
s+1
s
(4.5)
q-Riemann zeta function
178
20
15
15
12.5
β−n
β−n
10
7.5
5
10
5
2.5
0
0
−20
−15
−10
−5
−20
0
−15
−10
−5
0
n
n
Figure 3.10. The curve of β−n (n | q), q = 99/100,999/1000.
2
1.8
1.6
βq (s)
q
5
4
3
2
1
2
1.4
1.8
1.6
0.2
1.4
0.4
s
1.2
q
0.2
0.6
1.2
0.8
0.4
0.6
0.8
s
Figure 4.1. The plot of βq (s), 0.1 ≤ s ≤ 0.9, 1.1 ≤ q ≤ 2.
From relation (4.5), we can define the other analytic continued half of q-Bernoulli
numbers,
β(s) = −sζq (1 − s),
β(−s) = sζq (1 + s)
=⇒ β−n = β−n n | q = β(−n) = nζq (n + 1),
n ∈ N.
(4.6)
The curve β(s) runs through the points β−n and limn→∞ β−n = nζq (n + 1) = 0.
However, the curve β−n (n | q) grows ∼ n asymptotically as q → 1, (−n) → −∞.
ζq (m) =
∞
qn(m−1)
n =1
[n]m
q
=⇒ lim ζq (m) = 0.
m→∞
(4.7)
T. Kim et al. 179
2
1.75
1.5
1.25
q
4
βq (s)
0.75
2
2
0.5
1.5
0
1
1.2
1.4
1
0.25
q
0.5
1.6
1.2 1.4 1.6 1.8 2
1.8
s
s
2
Figure 4.2. The plot of βq (s), 1.03 ≤ s ≤ 2, 0.1 ≤ q ≤ 2.
0.04
0.3
ζq (n)
0.02
0.2
0.1
0.01
0
0
20
40
60
80
100
20
40
n
60
n
Figure 4.3. The curve of ζq (n), q = 7/10,9/10.
0
−2
β(s, w)
ζq (n)
0.03
−4
β(2, w)
β(3, w)
−6
−8
−10
−12
−0.4
−0.2
0
w
0.2
0.4
Figure 4.4. The curve of β(s,w), 2 ≤ s ≤ 3, −0.5 ≤ w ≤ 0.5, q = 11/10.
80
100
q-Riemann zeta function
180
5. Analytic continuation of q-Bernoulli polynomials
For consistency with the redefinition of βn = β(n) in (4.5) and (4.6),
βn (x) = βn x, −n | q =
n n
k =0
k
βk qkx [x]nq −k .
(5.1)
The analytic continuation can be then obtained as
n −→ s ∈ R,
x −→ w ∈ C,
βk −→ β k + s − [s] | q = − k + s − [s] ζq 1 − k + s − [s]
,
n
Γ(1 + s)
−→ k
Γ 1 + k + s − [s] Γ 1 + [s] − k
[s]
−k
Γ(1 + s)β k + s − [s] q(k+s−[s])w [w][s]
q
=⇒ βn (s) −→ β s,w | q =
Γ 1 + k + s − [s] Γ 1 + [s] − k
k=−1
=
[s]+1
k =0
−k
Γ(1 + s)β (k − 1) + s − [s] q((k−1)+s−[s])w [w][s]+1
q
,
Γ k + s − [s] Γ 2 + [s] − k
(5.2)
where [s] gives the integer part of s, and so s − [s] gives the fractional part.
Deformation of the curve β(2,w) into the curve β(3,w) via the real analytic continuation β(s,w), 2 ≤ s ≤ 3, −0.5 ≤ w ≤ 0.5.
Acknowledgments
The fourth author was supported by Kyungpook National University Research team
Fund, 2003. The first author would like to dedicate this paper to the memory of Katsumi
Shiratani.
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T. Kim: Science Education Research Institute, Education Science Research Center, Kongju National
University, Kongju 314-701, Korea
E-mail address: tkim@kongju.ac.kr
C. S. Ryoo: Department of Mathematics, Hannam University, Daejeon 306-791, Korea
E-mail address: ryoocs@math.hannam.ac.kr
L. C. Jang: Department of Mathematics and Computer Science, Konkuk University, Chungju-Si,
Chungcheongbuk-do 380-701, Korea
E-mail address: leechae.jang@kku.ac.kr
S. H. Rim: Department of Mathematics Education, Teachers’ College Kyungpook National University, Daegu 702-701, Korea
E-mail address: shrim@kongju.ac.kr