AND q -BERNOULLI POLYNOMIALS

AND q -BERNOULLI POLYNOMIALS

T. KIM, C. S. RYOO, L. C. JANG, AND S. H. RIM Received 15 July 2004

We study that theq-Bernoulli polynomials, which were constructed by Kim, are analytic continued toβs(z). A new formula for theq-Riemann zeta functionζq(s) due to Kim in terms of nested series ofζq(n) is derived. The new concept of dynamics of the zeros of analytic continued polynomials is introduced, and an interesting phenomenon of “scat- tering” of the zeros ofβs(z) is observed. Following the idea ofq-zeta function due to Kim, we are going to use “Mathematica” to explore a formula forζq(n).

1. Introduction

Throughout this paper,Z,R, andCwill denote the ring of integers, the field of real num- bers, and the complex numbers, respectively.

When one talks ofq-extension,qis variously considered as an indeterminate, a com- plex number, or a p-adic number. In the complex number field, we will assume that

|q|<1 or|q|>1. Theq-symbol [x]qdenotes [x]q=(1−q^x)/(1−q).

In this paper, we study that theq-Bernoulli polynomials due to Kim (see [2,8]) are analytic continued toβs(z). By those results, we give a new formula for theq-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 ofβs(z). Finally, we are going to use a software package called

“Mathematica” to explore dynamics of the zeros from analytic continuation forq-zeta function due to Kim.

2. Generatingq-Bernoulli polynomials and numbers

Forh∈Z, theq-Bernoulli polynomials due to Kim were defined as ∞

n=0

βn x,h|q

n! tⁿ= −t^∞

l=0

q^l(h+1)+xe^[l+x]qt+ (1−q)h^∞

l=0

q^lhe^[l+x]qt, (2.1)

forx,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

In the special casex=0,βn(0,h|q)=βn(h|q) are calledq-Bernoulli numbers (cf.

[1,5,7,8]).

By (2.1), we easily see that

βn

x,h|q= 1 (1−q)ⁿ

n j=0

n j

(−1)^j j+h

[j+h]_qq^jx, (cf. [2,6]), (2.2)

whereⁿ_jis a binomial coefficient.

In (2.1), it is easy to see that

q^hqβh|q+ 1ⁿ−βn h|q=







1 ifn=1,

0 ifn >1, (2.3)

with the usual convention of replacingβⁿ(h|q) byβn(h|q).

By differentiating both sides with respect totin (2.1), we have

βm

h|q= −m^∞

n=0

q^hn[n]^m_q⁻¹−(q−1)(m+h)^∞

n=0

q^hn[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, β2

2|q= 2q² [3]_q[4]_q, β3

2|q= −q²(q−1)2[3]_q+q

[3]_q[4]_q[5]_q ,..., β0

h|q= h [h]_q, β1

h|q= −

1 +q+···+q^h−1+q1 +q+···+q^h−2+···+q^h−1

[h]q[h+ 1]_q ,....

(2.5)

By (2.1), theq-Bernoulli polynomials can be written as

βm

x,h|q=^m

j=0

m j

[x]ⁿq^−jq^jxβj

h|q. (2.6)

1 0.5

−0.5 0

−1

x

−0.6

−0.4

−0.2 0 0.2 0.4 0.6

βm

Figure 3.1. The curve ofβm(x, 1|1/2), 1≤m≤10,−1≤x≤1.

In the caseh=0,βm(x, 0|q) will be symbolically written asβm,q(x).LetGq(x,t) be the generating function ofq-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

logq e^t/(1−q)−t^∞

n=0

q^h+xe^[n+x]qt, |t|<1, (cf. [2,3,4,6]). (2.8)

Forx=0,βn,q=βn,q(0) will be calledq-Bernoulli numbers.

By (2.8), we easily see that

βm,q(n)−βm,q=mⁿ⁻ 1 l=0

q^l[l]^m_q⁻¹. (2.9)

Thus, we have

n−1 l=0

q^l[l]^m_q⁻¹= 1 m

m−1 l=0

m l

q^nlβl,q[n]^m_q^−l+ 1 m

1−q^mnβm,q. (2.10)

3. Beautiful shape ofq-Bernoulli polynomials

In this section, we display the shapes of the q-Bernoulli polynomialsβm(x, 1|1/2). For m=1, 2,..., 10, we can draw a plot ofβm(x, 1|1/2), respectively. This shows the ten plots combined into one. Form=1,..., 10,q,Figure 3.1displays the shapes of theq-Bernoulli

4 3 2 1

−1 0

−2

−3

Re(x)

−2

−1 0 1 2 3

Im(x)

4 3 2 1

−1 0

−2

−3

Re(x)

−2

−1 0 1 2 3

Im(x)

Figure 3.2. Zeros ofq-Bernoulli polynomialsβm(x, 1|1/2),m=40, 60, andx∈C.

1 0.75 0.5 0.25

−0.25 0

−0.5

−0.75

Re(x)

−1.5

−1

−0.5 0 0.5 1 1.5 2

Im(x)

1 0.75 0.5 0.25

−0.25 0

−0.5

−0.75

Re(x)

−1.5

−1

−0.5 0 0.5 1 1.5 2

Im(x)

Figure 3.3. Zeros ofq-Bernoulli polynomialsβm(x, 1| −1/2),m=40, 60, andx∈C.

polynomialsβm(x, 1|1/2). We plot the zeros ofβm(x, 1|1/2),m=40,m=60, andx∈C (Figure 3.2). We plot the zeros ofβm(x, 1| −1/2),m=40,m=60, andx∈C(Figure 3.3).

We plot the zeros of βm(x, 1|11/10),m=40,m=60, andx∈C(Figure 3.4). We plot the zeros ofβm(x, 1| −11/10),m=40,m=60, and x∈C(Figure 3.5). Stacks of zeros ofβn(x, 1|1/2), 1≤n≤60, from a 3D structure are presented inFigure 3.6. The curve β(s) runs through the pointsβ−n(n|1/2) (Figure 3.7). We draw the curve ofβ−n(n|q) and limn→∞=nζq(n+ 1),q=3/10, 5/10, 7/10, 9/10, 99/100, 999/1000 (Figures3.8,3.9, and 3.10).

20 15 10 5

−5 0

−10

−15

Re(x)

−15

−10

−5 0 5 10 15 20

Im(x)

20 15 10 5

−5 0

−10

−15

Re(x)

−15

−10

−5 0 5 10 15 20

Im(x)

Figure 3.4. Zeros ofq-Bernoulli polynomialsβm(x, 1|11/10),m=40, 60, andx∈C.

1 0.75 0.5 0.25

−0.25 0

−0.5

−0.75

Re(x)

−1.5

−1

−0.5 0 0.5 1 1.5 2

Im(x)

1 0.75 0.5 0.25

−0.25 0

−0.5

−0.75

Re(x)

−1.5

−1

−0.5 0 0.5 1 1.5 2

Im(x)

Figure 3.5. Zeros ofq-Bernoulli polynomialsβm(x, 1| −11/10),m=40, 60, andx∈C.

4.q-Riemann zeta function

We display the plot ofβq(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 ofβ−q(s,w), 2≤s≤3,−0.5≤w≤0.5,q=11/10 (Figure 4.4).

Theq-Riemann zeta function due to Kim was defined as ζq^(h)(s)=1−s+h

1−s (q−1) ∞ n=1

q^nh [n]^s_q⁻¹+

∞ n=1

q^nh

[n]^s_q, fors,h∈C, (cf. [6,8]). (4.1)

3 2 1 0

Re(x) 0

20 40 60

n

−1 0 Im(x) 1

Figure 3.6. Stacks of zeros ofq-Bernoulli polynomialsβn(x, 1|1/2), 1≤n≤60, from a 3D structure.

−5 0

−10

−15

−20

s 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

β(s)

Figure 3.7. The curveβ(s) runs through the pointsβ−n(n|1/2).

Fork∈N,h∈Z, it was known that ζ_q^(h)(1−k)= −βk

h|q

k , (cf. [6,8]). (4.2)

In the special caseh=s−1,ζq^(s−1)(s) will be written asζq(s). Fors∈C, we note that

ζq(s)= ∞ n=1

q^n(s−1)

[n]^s_q , (cf. [6,8]). (4.3)

−5 0

−10

−15

−20

n 0

0.02 0.04 0.06 0.08

β−n

−5 0

−10

−15

−20

n 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

β−n

Figure 3.8. The curve ofβ−n(n|q) and limn→∞β−n=nζq(n+ 1)₌0,q=3/10, 5/10 .

−5 0

−10

−15

−20

n 0

0.2 0.4 0.6 0.8 1

β−n

−5 0

−10

−15

−20

n 1.5

2 2.5 3 3.5

β−n

Figure 3.9. The curve ofβ−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 ζq(1−k)= −βk

−k|q

k , fork∈N, (cf. [3,4,6]). (4.4) From the above analytic continuation ofq-Bernoulli numbers, we consider

βn=βn

−n|q−→β(s), ζq(−n)= −βn+1

−n+ 1|q

n+ 1 ^−→ζq(−s)= −β(s+ 1)

s+ 1 ^=⇒ζq(1−s)= −ζ(s) s .

(4.5)

−5 0

−10

−15

−20

n 0

2.5 5 7.5 10 12.5 15

β−n

−5 0

−10

−15

−20

n 0

5 10 15 20

β−n

Figure 3.10. The curve ofβ−n(n|q),q=99/100, 999/1000.

2 1.8 1.6 1.4 1.2

q 0.8

0.6 0.4 0.2

s 1

2 3 4 5 βq(s)

0.8 0.6 0.4 0.2

s 1.2

1.4 1.6 1.8 2

q

Figure 4.1. The plot ofβq(s), 0.1≤s≤0.9, 1.1≤q≤2.

From relation (4.5), we can define the other analytic continued half ofq-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 curveβ(s) runs through the pointsβ−nand lim_n→∞β−n=nζq(n+ 1)=0.

However, the curveβ−n(n|q) grows∼nasymptotically asq→1, (−n)→ −∞. ζq(m)=

∞ n=1

q^n(m−1)

[n]^m_q ^=⇒lim

m→∞ζq(m)=0. (4.7)

2 1.5 1 0.5

q 2

1.8 1.6 1.4 1.2

s 0

2 4 βq(s)

2 1.8 1.6 1.4 1.2

s 0.25

0.5 0.75 1 1.25 1.5 1.75 2

q

Figure 4.2. The plot ofβq(s), 1.03≤s≤2, 0.1≤q≤2.

100 80 60 40 20

n 0

0.01 0.02 0.03 0.04

ζq(n)

100 80 60 40 20

n 0

0.1 0.2 0.3

ζq(n)

Figure 4.3. The curve ofζq(n),q=7/10, 9/10.

0.4 0.2

−0.2 0

−0.4

w

−12

−10

−8

−6

−4

−2 0

β(s,w)

β(2, w) β(3, w)

Figure 4.4. The curve ofβ(s,w), 2≤s≤3,−0.5≤w≤0.5,q=11/10.

5. Analytic continuation ofq-Bernoulli polynomials

For consistency with the redefinition ofβn=β(n) in (4.5) and (4.6), βn(x)=βn

x,−n|q= n k=0

n k

βkq^kx[x]ⁿ_q^−k. (5.1)

The analytic continuation can be then obtained as

n−→s∈R, x−→w∈C, βk−→βk+s−[s]|q= −

k+s−[s]ζq 1−

k+s−[s], n

k

−→ Γ(1 +s)

Γ1 +k+s−[s]Γ1 + [s]−k

=⇒βn(s)−→βs,w|q= [s]

k=−1

Γ(1 +s)βk+s−[s]q^(k+s−[s])w[w]^[q^s]−k

Γ1 +k+s−[s]Γ1 + [s]−k

=

[s]+1 k=0

Γ(1 +s)β(k−1) +s−[s]q^{((k−1)+s−[s])w}[w]^[s]+1q ^−k

Γk+s−[s]Γ2 + [s]−k , (5.2) where [s] gives the integer part ofs, and sos−[s] gives the fractional part.

Deformation of the curveβ(2,w) into the curveβ(3,w) via the real analytic continua- tionβ(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:[email protected]

C. S. Ryoo: Department of Mathematics, Hannam University, Daejeon 306-791, Korea E-mail address:[email protected]

L. C. Jang: Department of Mathematics and Computer Science, Konkuk University, Chungju-Si, Chungcheongbuk-do 380-701, Korea

E-mail address:[email protected]

S. H. Rim: Department of Mathematics Education, Teachers’ College Kyungpook National Uni- versity, Daegu 702-701, Korea

E-mail address:[email protected]