Identities on the Weighted q-Bernoulli Numbers of Higher Order

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Volume 2011, Article ID 918364,6pages doi:10.1155/2011/918364

Research Article

Identities on the Weighted q-Bernoulli Numbers of Higher Order

T. Kim,

¹

D. V. Dolgy,

²

B. Lee,

³

and S.-H. Rim

⁴

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

2Hanrimwon, Kwangwoon University, Seoul 139-701, Republic of Korea

3Department of Wireless of Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea

4Department of Mathematics Education, Kyungpook National University, Taegu 702-701, Republic of Korea

Correspondence should be addressed to T. Kim,[email protected] Received 6 August 2011; Accepted 17 August 2011

Academic Editor: Lee-Chae Jang

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.

We give a new construction of the weighted q-Bernoulli numbers and polynomials of higher order by using multivariatep-adicq-integral onZp.

1. Introduction

Letpbe a fixed prime number. Throughout this paper,Z,Zp,Qp,C, andCpwill, respectively, denote the ring of rational integers, the ring ofp-adic rational integers, the field of p-adic rational numbers, the complex number field, and the completion of the algebraic closure of Qp. Thep-adic norm ofCp is defined by|x|p 1/p^r wherex p^rs/twith p, s p, t s, t 1, r ∈Q. In this paper, we assumeα∈QandZ N∪ {0}. Letq∈Cpwith|q−1|p <

p^−1/p−1and letx_q 1−q^x/1−q. Note that limq→1x_q xsee1–13. Recently, the q-Bernoulli numbers with weightαare defined by

β^α_0,q 1, q

q^αβ^α_q1n

−β^αn,q

⎧⎨

⎩ α

α_q ifn1, 0 ifn >1,

1.1

with the usual convention about replacingβ^αⁿbyβ^αn,q,see4.

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Theq-Bernoulli polynomials with weightαare also defined by as

βn,q^αx ⁿ

l0

n l

x^n−l_q^α q^αlxβ^α_l,q

x_q^αq^αxβ^αq

_n

, forn≥0.

1.2

Forf∈UDZp the space of uniformly diﬀerentiable functions onZp, thep-adicq-integral onZpis defined by

Iq

f

Zp

fxdμqx lim

N→ ∞

1 p^N

q p^N−1

x0

fxq^x, 1.3

see4–12. From1.3, we note that

qⁿIq

fn

Iq

f

q−1ⁿ⁻¹

l0

q^lfl q−1 logq

n−1

l0

q^lfl, 1.4

wherefnx fxn,see1–12.

We have the Witt formula for theq-Bernoulli numbers and polynomials with weightα as followssee4,5,12:

β^αn,q

Zp

xⁿ_q^αdμqx, β^αn,qx

Zp

xy_n

q^αdμqx. 1.5

From1.4and1.5, we have

β^αn,qx 1 1−q_n

αⁿ_q n

l0

−1^lq^αlx αl1

αl1_q, 1.6

see4. By1.6, we easily get limq→1β^αn,qx Bnx, whereBnxare the Bernoulli polynomials of degreen.

To give the new construction of the weightedq-Bernoulli numbers and polynomials of higher order, we first use the multivariatep-adicq-integral onZp. The purpose of this paper is to give the higher-orderq-Bernoulli numbers and polynomials with weightαand to derive a new explicit formulas by these numbers and polynomials.

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2. On the Higher Order q -Bernoulli Numbers with Weight α

Forhi i1,2, . . . , k∈Z, we consider a sequence ofp-adic rational numbers as expansion of the weightedq-Bernoulli numbers and polynomials of orderkas follows:

β^k,αn,q h1, h2, . . . , hk

Zp

· · ·

Zp

x1· · ·xkⁿ_q^αq^k^j1^x^j^h^j⁻¹dμqx1· · ·dμqxk, 2.1 β^k,αn,q h1, h2, . . . , hk|x

Zp

· · ·

Zp

xx1· · ·xkⁿ_qαq^k^j1^x^j^h^j⁻¹dμqx1· · ·dμqxk. 2.2

From2.1and2.2, we can derive the following equations:

β^k,αn,q h1, h2, . . . , hk 1 1−qnαⁿ_q

n l0

n l

−1^l αlh1αlh2· · ·αlhk

αlh1_qαlh2_q· · ·αlhk_q, 2.3 β^k,αn,q h1, h2, . . . , hk|x

1 1−q_n

αⁿ_q n

l0

n l

−1^lq^αlx αlh1αlh2· · ·αlhk αlh1_qαlh2_q· · ·αlhk_q, ⁿ

l0

n l

x^n−l_qα q^αlxβ^k,α_l,q h1, h2, . . . , hk.

2.4

By2.3and2.4, we get n

l0

n l

−1^lq^αlx αlh1αlh2· · ·αlhk αlh1_qαlh2_q· · ·αlhk_q,

ⁿ

l0

n l

x^n−l_q^α q^αlx

1−q_n−l

α^n−l_q ^l

s0

l s

−1^s αsh1· · ·αshk αsh1_q· · ·αshk_q.

2.5

Therefore, by2.5, we obtain the following theorem.

Theorem 2.1. Forhi i1,2, . . . , k∈Z, andk∈N, we have n

l0

n l

−1^lq^αlx αlh1αlh2· · ·αlhk αlh1_qαlh2_q· · ·αlhk_q ⁿ

l0

n l

x^n−l_q^α q^αlx

1−q_n−l

α^n−l_q ^l

s0

l s

−1^s αsh1· · ·αshk αsh1_q· · ·αshk_q.

2.6

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From1.3and1.4, we note that

q^h¹

Zp

x1x1ⁿ_qαq^x¹^h¹⁻¹dμqx1

Zp

x1xⁿ_q^αq^x¹^h¹⁻¹dμqx1 xⁿ_q^αh1

q−1

nxⁿ⁻¹_q^α α α_q.

2.7

Theorem 2.2. Forn∈Z, we have

q^h¹β^1,αn,q h1|x1−β^1,αn,q h1|x h1

q−1

xⁿ_q^αnxⁿ⁻¹_q^α α

α_q. 2.8

From2.2and2.3, we have

q^αxβ^k,αn,q αh1α, αh2α, . . . , αhkα|x q^αx

Zp

· · ·

Zp

x1· · ·xkxⁿ_q^αq^k^j1^x^j^αh^j^α−1dμqx1· · ·dμqxk

q^α−1

Zp

· · ·

Zp

x1· · ·xkxⁿ¹_q^α q^k^j1^x^j^αh^j⁻¹dμqx1· · ·dμqxk

Zp

· · ·

Zp

x1· · ·xkxⁿ_qαq^k^j1^x^j^αh^j⁻¹dμqx1· · ·dμqxk.

2.9

Therefore, by2.9, we obtain the following proposition.

Proposition 2.3. Forn∈Z, we have

q^αxβn,q^k,ααh1α, αh2α, . . . , αhkα|x

q^α−1β^k,α_n1,qαh1, . . . , αhk|x β^k,αn,q αh1, . . . , αhk|x. 2.10

By2.2, we get

β^k,αn,q h1, h2, . . . , hk|x

Zp

· · ·

Zp

xx1· · ·xkⁿ_q^αq^k^j1^x^j^h^j⁻¹dμqx1· · ·dμqxk

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dⁿ_qα

d^k_q

d−1

s1,...,sk0

q^k^j1^h^j^s^j

Zp

· · ·

Zp

s1· · ·skx

d x1· · ·xk

n q^dα

×q^d^k^j1^x^j^h^j⁻¹dμ_q^dx1· · ·dμ_q^dxk dⁿ_q^α

d^k_q

d−1

s1,...,sk0q^k^j1^h^j^s^jβ^k,α_n,qd

h1, . . . , hk|s1· · ·skx d

,

2.11

whered∈N.

Theorem 2.4. Forn∈Zandd∈N, we have β^k,αn,q h1, h2, . . . , hk|x dⁿ_q^α

d^k_q

d−1

s1,...,sk0

q^k^j1^h^j^s^jβ_n,q^k,αd

h1, . . . , hk|s1· · ·skx d

. 2.12

Letd>0be a fixed integer. ForN∈N, we set

X Xdlim

←−N

Z

dp^NZ, X1Zp,

X^∗

0<a<dp a,p1

adpZp

,

adp^NZp

x∈X|x≡a

moddp^N ,

2.13

wherea∈Zsatisfies the condition 0≤a < dp^N.

Letχbe a primitive Dirichlet character with conductord ∈N. Then we consider the generalizedq-Bernoulli numbers with weightαof orderkas follows:

βn,χ,q^k,αh1, h2, . . . , hk

X

· · ·

X

k i1

χxix1· · ·xkⁿ_q^αq^kⁱ¹^xⁱ^hⁱ⁻¹dμqx1· · ·dμqxk. 2.14

From2.14, we have

β^k,αn,χ,qh1, h2, . . . , hk

dⁿ_q^α d^k_q

d−1 s1,...,sk0

q^k^j1^h^j^s^j

⎛

⎝^k

j1

χ sj

⎞

⎠β^k,α_n,qd

h1, . . . , hk| s1· · ·sk

d

.

2.15

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