On the Values of the Weighted q-Zeta and L-Functions

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Volume 2011, Article ID 476381,7pages doi:10.1155/2011/476381

Research Article

On the Values of the Weighted q-Zeta and L-Functions

T. Kim,

¹

S. H. Lee,

¹

Hyeon-Ho Han,

²

and C. S. Ryoo

³

1Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea

2Department of Information display, Kwangwoon University, Seoul 139-701, Republic of Korea

3Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Correspondence should be addressed to T. Kim,[email protected] Received 17 August 2011; Accepted 3 October 2011

Academic Editor: Binggen Zhang

Copyrightq2011 T. Kim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Recently, the modifiedq-Bernoulli numbers and polynomials are introduced inD. V. Dolgy et al., in press. These numbers are valuable to study the weightedq-zeta andL-functions. In this paper, we study the weightedq-zeta functions and weightedL-functions from the modifiedq-Bernoulli numbers and polynomials with weightα.

1. Introduction

Letq∈Cwith|q|<1. The modifiedq-Bernoulli numbers and polynomials with weightαare defined by

B^α_0,q αq−1 logq,

q^αBq^α1n

−Bn,q^α

⎧⎨

⎩ α

α_q ifn1, 0 ifn >1,

1.1

with the usual convention about replacingB^αq ⁿbyB^αn,qsee1,2.

Throughout this paper, we use the notation ofq-number as x_q 1−q^x

1−q, 1.2

see1–14.

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From1.1, we note that

B^αn,q 1 1−q^α ⁿ

n l0

n l

−1^l αl αl_q 1

1−q ⁿαⁿ_q n

l0

n l

−1^l αl αl_q.

1.3

LetFq^αt _∞

n0B^αn,qtⁿ/n!, Then, by1.3, we get Fq^αt αq−1

logqe^1/1−q^α^t− αt α_q

∞ m0

q^αme^m^qαt. 1.4

Let us define the modifiedq-Bernoulli polynomials with weightαas follows:

Bn,q^αx ⁿ

l0

n l

x^n−l_qα q^αlxB^α_l,q

x_qαq^xαB^αq

n

, 1.5

with the usual convention about replacingB^αq ⁿbyB^αn,qsee1–13.

From1.5, we can derive the following equation:

Bn,q^αx 1 1−q^α ⁿ

n l0

n l

−1^lq^αlx αl αl_q 1

1−q ⁿαⁿ_q n

l0

n l

−1^lq^αlx αl αl_q,

1.6

see2.

LetFq^αt, x _∞

n0B^αn,qxtⁿ/n!, then, by1.6, we get Fq^αt, x αq−1

logqe^1/1−q^α^t−t α α_q

∞ m0

q^αmxe^mx^qαt. 1.7

In this paper, we consider the generalizedq-Bernoulli numbers with weightα, and we study the weighted q-zeta function and q-analogue of L-function with weight αfrom the modifiedq-Bernoulli numbers and polynomials with weightα.

2. Weighted q -Zeta Function and Weighted q - L -Function

From1.7, we note that

B^αn,qx α 1−q ⁿαⁿ_q

q−1 logq

− nα α_q

∞ m0

q^αmxmxⁿ⁻¹_qα . 2.1

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Forn∈N, we have

−B^αn,qx

n

α α_q

1 1−q^α

_n−1 1 logq

α

α_q ∞ m0

q^αmxmxⁿ⁻¹_q^α . 2.2

LetΓsbe the gamma function, then we consider the following complex integral. For s∈C,

1 Γs

_∞

0

Fq^α−t, xt^s−2dt α s−1

q−1 logq

1−q^α ^s−1 α α_q

∞ m0

q^αmx

mx^s_q^α, 2.3 wherex /0,−1,−2,−3, . . ..

Now, we define the twisted Hurwitz’s typeq-zeta function as follows.

Fors∈C, define

ζ^αq s, x α α_q

1 1−s

1−q^α ^s logq α

α_q ∞ m0

q^αmx

mx^s_q^α, 2.4 wherex /0,−1,−2,−3, . . ..

Note thatζq^αs, xis meromorphic function whole in complexs-plane except fors1.

From2.3and2.4, we can derive the following equation:

ζ^αq s, x 1 Γs

_∞

0

Fq^α−t, xt^s−2dt. 2.5

By1.7,2.3,2.4,2.5, and Laurent series, we get

ζ^αq 1−k, x −B_k,q^αx

k , 2.6

wherek∈N.

Therefore, by2.6, we obtain the following theorem.

Theorem 2.1. Fork∈N, one has

ζ^αq 1−k, x −B_k,q^αx

k . 2.7

From2.4, one notes that

ζq^αs,1 α α_q

1 1−s

1−q^α ^s logq α

α_q ∞ m0

q^αm1 m1^s_q^α α

α_q 1 1−s

1−q^α ^s logq α

α_q ∞ m1

q^αm m^s_q^α.

2.8

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Now, by2.8, one defines the weightedq-zeta function as follows:

ζ^αq s α α_q

1 1−s

1−q^α ^s logq α

α_q ∞ m1

q^αm m^s_qα

.

ζ^αq s,1.

2.9

Fork∈N, by1.1and1.5, one gets

ζ^αq 1−k ζq^α1−k,1 −B_k,q^α1 k

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

− α

α_q B^α_1,q

ifk1,

−B^α_k,q

k ifk >1.

2.10

Therefore, by2.10, one obtains the following corollary.

Corollary 2.2. Fork∈N, one has

ζ^αq 1−k

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

− α

α_q B^α_1,q

ifk1,

−B^α_k,q

k ifk >1.

2.11

Letχbe the Dirichlet’s character with conductord ∈ N. Let us consider the generalizedq- Bernoulli polynomials with weightαas follows:

F^αq,χt, x α α_qt

∞ m0

χmq^αmxe^mx^qα^t

^∞

n0

Bn,χ,q^α xtⁿ n!.

2.12

The sequence B^αn,χ,qx will be called the nth generalized q-Bernoulli polynomials with weight α attached toχ.

In the special case,x0,Bn,χ,q^α 0 B^αn,χ,qare called thenth generalizedq-Bernoulli numbers with weightαattached toχ.

From1.7and2.12, one notes that Fq,χ^αt, x 1

d_q d−1 a0

χaF_q^αd

d_q^αt,xa d

. 2.13

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Thus, by2.13, one gets

Bn,χ,q^α x dⁿ_q^α d_q

d−1

a0

χaB^α_n,qd

xa d

. 2.14

Therefore, by2.14, one obtains the following theorem.

Theorem 2.3. Forn∈Z, one has

Bn,χ,q^α x dⁿ_qα

d_q

d−1

a0

χaB^α_n,qd

xa d

. 2.15

In the special case,x0, one obtains the following corollary.

Corollary 2.4. Forn∈Z, one has

B^αn,χ,q dⁿ_q^α d_q

d−1 a0

χaB_n,q^αd

a d

. 2.16

Let

Fq,χ^αt α α_qt

∞ m0

χmq^αme^m^qαt

^∞

n0

Bn,χ,q^α tⁿ n!,

2.17

then, by2.12and2.17, one easily gets

Bn,χ,q^α d−B^αn,χ,q

n α

α_q d−1

l0

χlq^αllⁿ⁻¹_q^α . 2.18

Fors∈C, consider 1 Γs

_∞

0

F^αq,χ−t, xt^s−2dt α α_q

1 Γs

_∞

0

∞ m0

χmq^αmxe^−mx^qαtt^s−1dt

α α_q

∞ m0

χmq^αmx mx^s_qα

1 Γs

_∞

0

e^−yy^s−1dy

α α_q

∞ m0

χmq^αmx mx^s_qα

,

2.19

wherex /0,−1,−2,−3, . . ..

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Now, one defines Hurwitz’s typeq-L-function with weightαas follows. Fors∈C, L^αq

s, χ|x −t, x α α_q

∞ n0

χnq^nxα

nx^s_q^α , 2.20 wherex /0,−1,−2,−3, . . ..

From2.19and2.20, one notes that L^αq

s, χ|x 1 Γs

_∞

0

Fq,χ^α−t, xt^s−2dt. 2.21

By1.7and2.21and Laurent series, one obtains the following theorem.

Theorem 2.5. Fork∈N, one has L^αq

1−k, χ|x −B_k,χ,q^α x

k . 2.22

In the special case,x0,L^αq 1−k, χ|0 L^αq 1−k, χare called theq-L-function with weightα.

Let

Fq^αs, a|F α F_qα_q

⎛

⎜⎝ ^∞

m≡amodF m>0

q^αm m^s_qα

1−q^α ^s F1−slogq

⎞

⎟⎠

α F_qα_q

_∞

n0

q^αanF anF^s_q^α

1−q^α ^s F1−slogq

F_q^α F_qF^s_qα

ζ^α_qF

s,a

F

,

2.23

whereaandFare positive integers with 0< a < F.

Then, by2.23, one gets

Hq^α1−n, a|F −Fⁿ_q^αB^αn,χ,qa/F

F_qn , n≥1, 2.24

andHq^αs, a|Fhas as simple pole ass1 with residueα/F_qq−1/logq^F. Letχbe the Dirichlet character with conductorF, then one easily sees that

L^αq

s, χ ^F

a1

χaHq^αs, a|F. 2.25

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References

1 T. Kim, “On the weightedq-Bernoulli numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 21, pp. 207–215, 2011.

2 D. V. Dolgy, T. Kim, S. H. Lee, B. Lee, and S.-H. Rim, “A note on the modifiedq-Bernoulli numbers and polynomials with weightα,” communicated.

3 S. Araci, D. Erdal, and D.-J. Kang, “Some new properties on theq-Genocchi numbers and polynomials associated withq-Bernstein polynomials,” Honam Mathematical Journal, vol. 33, pp. 261–270, 2011.

4 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.

5 T. Kim, “p-Bernoulli numbers and polynomials associated with Gaussian binomial coeﬃcients,”

Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.

6 T. Kim, “Multiplep-adicL-function,” Russian Journal of Mathematical Physics, vol. 13, no. 2, pp. 151–

157, 2006.

7 T. Kim, “Power series and asymptotic series associated with theq-analog of the two-variablep-adic L-function,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 186–196, 2005.

8 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks onq-Bernoulli numbers associated with Daehee numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.

9 L.-C. Jang, “On multiple generalizedw-Genocchi polynomials and their applications,” Mathematical Problems in Engineering, vol. 2010, Article ID 316870, 8 pages, 2010.

10 S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, “On theq-Genocchi numbers and polynomials associated withq-zeta function,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261–267, 2009.

11 Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–

278, 2008.

12 Y. Simsek, “Theorems on twistedL-function and twisted Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 11, no. 2, pp. 205–218, 2005.

13 M. Cenkci, Y. Simsek, and V. Kurt, “Multiple two-variablep-adic q-L-function and its behavior at s0,” Russian Journal of Mathematical Physics, vol. 15, no. 4, pp. 447–459, 2008.

14 S. Araci, D. Erdal, and J. J. Seo, “A study on the fermionicp-adicq-integral representation onZp

associated with weightedq-Bernstein andq-Genocchi polynomials,” Abstract and Applied Analysis. In press.

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