the Twisted q-Bernoulli Polynomials and

doi:10.1155/2010/801580

Research Article

A Note on Symmetric Properties of

the Twisted q-Bernoulli Polynomials and

the Twisted Generalized q-Bernoulli Polynomials

L.-C. Jang,

H. Yi,

K. Shivashankara,

T. Kim,

Y. H. Kim,

and B. Lee

1Department of Mathematics and Computer Science, KonKuk University, Chungju 138-70, Republic of Korea

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

3Department of Mathematics, Yuvaraja’s College, University of Mysore, Mysore 570# 005, India

4Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

5Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to H. Yi,[email protected]

Received 11 September 2009; Revised 14 April 2010; Accepted 31 May 2010 Academic Editor: Abdelkader Boucherif

Copyrightq2010 L.-C. Jang 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 define the twistedq-Bernoulli polynomials and the twisted generalizedq-Bernoulli polynomi- als attached toχof higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between twistedq- Bernoulli numbers and polynomials and between twisted generalizedq-Bernoulli numbers and polynomials.

1. Introduction

Letpbe a fixed prime number. Throughout this paperZp,Qp, andCpwill, respectively, denote the ring ofp-adic rational integers, the field ofp-adic rational numbers, and the completion of algebraic closure ofQp. Letvpbe the normalized exponential valuation ofCpwith|p|p p^−vpp p⁻¹.When one talks ofq-extension,qis variously considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp. Ifq ∈ C,one normally assumes

|q|< 1.Ifq∈ Cp,then we assume|q−1|p < p^−1/p−1,so thatq^x expxlogqfor|x|p ≤ 1 cf.1–32.

ForN, d∈N, we set

X Xd lim_←

NZ

dp^NZ, X1Zp 1.1

see1–13. The Bernoulli numbersBnand polynomialsBnxare defined by the generating function as

t

e^t−1 ^∞

n0

Bntⁿ

n!, 1.2

t

e^t−1e^xt∞

n0

Bnxtⁿ

n! 1.3

cf.17,18,21,24,26. Let UDXbe the set of uniformly differentiable functions onX. For f∈UDX, thep-adic invariant integral onZpis defined as

I f

X

fxdx lim

N→ ∞

1 dp^N

dp^N−1 x0

fx. 1.4

Note that

Xfxdx

Zpfxdxsee27. Letfnxbe a translation withfnx fxn.

We note that

I fn

I f

ⁿ⁻¹

i0

fi 1.5

cf. 1–32. Kim 18 studied the symmetric properties of the q-Bernoulli numbers and polynomials as follows:

tlogq

qe^t−1 e^xt∞

n0

Bn^qxtⁿ

n!. 1.6

In this paper, we define the twisted q-Bernoulli polynomials and the twisted generalized q-Bernoulli polynomials attached to χ of higher order and investigate some symmetric properties of them. Furthermore, using these symmetric properties of them, we can obtain some relationships between the twisted q-Bernoulli numbers and polynomials and between the twisted generalizedq-Bernoulli numbers and polynomials attached toχof higher order.

2. The Twisted q -Bernoulli Polynomials

LetCp^∞

n≥1Cpⁿ lim_{n→ ∞}Cpⁿ be the locally constant space, whereCpⁿ {ξ|ξ^pn 1}is the cyclic group of orderpⁿ. Forw∈Cp^∞, we denote the locally constant function by

φw:Zp−→Cp, x −→w^x 2.1

cf.2,3,21,24. If we takefx φwxq^xe^tx, then

Zp

e^xtw^xq^xdx logqt

wqe^t−1. 2.2

Now we define the q-extension of twisted Bernoulli numbers and polynomials as follows:

logqt wqe^t−1 ^∞

n0

Bn,w^q tⁿ

n!, 2.3

logqt

wqe^t−1e^tx∞

n0

B^qn,wxtⁿ

n! 2.4

see31. From1.5,2.2,2.3, and2.4, we can derive

Zp

w^yq^y xy_n

dyB^qn,wx,

Zp

w^yq^yyⁿdyB^qn,w. 2.5

By1.5, we can see that

1 logqt

Zp

w^nxq^nxe^nxtdx−

Zp

w^xq^xe^xtdx wⁿqⁿe^nt−1

tlogq

Zp

w^xq^xe^xtdx wⁿqⁿe^nt−1

wqe^t−1 ⁿ⁻¹

i0

wⁱqⁱe^it

^∞

k0

_n−1

i0

i^kwⁱqⁱ t^k k!.

2.6

In1.4, it is easy to show that

1 logqt

Zp

w^nxq^nxe^nxtdx−

Zp

w^xq^xe^xtdx n

Zpw^xq^xe^xtdx

Zpw^nxq^nxe^nxtdx. 2.7

For each integerk≥0, let

S^q_k,wn 0^k1^kwq2^kw²q²· · ·n^kwⁿqⁿ. 2.8

From2.6,2.7, and2.8, we derive

1 logqt

Zp

w^nxq^nxe^nxtdx−

Zp

w^xq^xe^xtdx n

Zpw^xq^xe^xtdx

Zpw^nxq^nxe^nxtdx ^∞

k0

S^q_k,wn−1t^k k!. 2.9

From2.9, we note that

B^q_k,wn−B^q_k,w kS^q_k−1,wn−1 logqS^q_k,wn−1, 2.10

for allk, n∈N. Letu1, u2 ∈Nand w∈Cp^∞; then we have

Zpw^u1x1u2x2q^u1x1u2x2e^u1x1u2x2dx1dx2

Zpw^u1u2xq^u1u2xe^u1u2xtdx

tlogqw^u1u2q^u1u2e^u1t−1

w^u2q^u2e^u2t−1 . 2.11

By2.9, we see that u1

Zpw^xq^xe^xtdx

Zpw^u1xq^u1xe^u1xtdx ^∞

l0

_u

1−1

k0

k^lw^kq^k t^l l! ^∞

l0

S^q_l,wu1−1t^l

l!. 2.12

LetTwu1, u2;x, tbe as follows:

Twu1, u2;x, t

Zpw^u1x1u2x2q^u1x1u2x2e^{u1x1u2x2u1u2xt}dx1dx2

Zpw^u1u2xq^u1u2xe^u1u2xtdx . 2.13

Then we have

Twu1, u2;x, t

tlogq e^u1u2t

w^u1u2q^u1u2e^u1u2t−1 w^u1q^u1e^u1t−1

w^u2q^u2e^u2t−1 . 2.14

From2.13, we derive

Twu1, u2;x, t 1

u1

Zp

w^u1x1q^u1x1e^u1x1u2xtdx1

⎛

⎝ u1

Zpw^u2x2q^u2x2e^u2x2t

Zpw^u1u2xq^u1u2xe^u1u2xtdx

⎞

⎠. 2.15

By2.4,2.12, and2.15, we can see that

Twu1, u2;x, t 1 u1

_∞

i0

B_i,w^qu1u1u2xuⁱ₁tⁱ i!

∞ l0

S^q_l,w^u2u2u1−1u^l₂t^l l!

^∞

n0

_n

i0

n i

B^q_i,w^u1u1u2xS^q_n−i,w^u2 u2u1−1uⁱ⁻¹₁ uⁿ⁻ⁱ₂ tⁿ n!.

2.16

By the symmetry ofp-adic invariant integral onZp, we also see that

Twu1, u2;x, t 1

u2

Zp

w^u2x2q^u2x2e^u2x2u1xtdx2

⎛

⎝ u2

Zpw^u1x1q^u1x1e^u1x1t

Zpw^u1u2xq^u1u2xe^u1u2xtdx

⎞

⎠

^∞

n0

_n

i0

n i

B^q_i,w^u2u2u1xS^q_n−i,w^u1 u1u2−1uⁱ⁻¹₂ uⁿ⁻ⁱ₁ tⁿ n!.

2.17

By comparing the coefficients oftⁿ/n! on both sides of2.16and2.17, we obtain the following theorem.

Theorem 2.1. Letu1, u2, n∈N. Then for allx∈Zp,

n i0

n i

B_i,w^qu1u1u2xS^q_n−i,w^u2 u2u1−1uⁱ⁻¹₁ uⁿ⁻ⁱ₂ ⁿ

i0

n i

B_i,w^qu2u2u1xS^q_n−i,w^u1 u1u2−1uⁱ⁻¹₂ uⁿ⁻ⁱ₁ , 2.18

whereⁿ_iis the binomial coefficient.

FromTheorem 2.1, if we takeu21, then we have the following corollary.

Corollary 2.2. Form≥0, one we has

B^q_i,wu1x ⁿ

i0

n i

B_i,w^qu1u1xS^q_n−i,wu1−1uⁱ⁻¹₁ , 2.19

whereⁿ_iis the binomial coefficient.

By2.17,2.18, and2.19, we can see that

Twu1, u2;x, t

e^u1u2xt u1

Zp

w^u1xq^u1x1e^u1x1tdx1

⎛

⎝u1

Zpw^u2x2q^u2x2e^u2x2tdx2

Zpw^u1u2xq^u1u2xe^u1u2xtdx

⎞

⎠

e^u1u2xt u1

Zp

w^u1xq^u1x1e^u1x1tdx1 u1−1

i0

w^u2iq^u2ie^u2it

1 u1

u1−1 i0

w^u2iq^u2i

Zp

w^u1xq^u1xe^{x1u2xu2/u1itu1}dx1

^∞

n0 u1−1

i0

B^q_n,w^u1u1

u2xu2

u1i

uⁿ⁻¹₁ w^u2iq^u2itⁿ n!.

2.20

From the symmetry ofTwu1, u2;x, t, we can also derive

Twu1, u2;x, t ^∞

n0 u2−1

i0

B^q_n,w^u2u2

u1xu1

u2i

uⁿ⁻¹₂ w^u1iq^u1itⁿ

n!. 2.21

By comparing the coefficients of tⁿ/n! on both sides of 2.20 and 2.21, we obtain the following theorem.

Theorem 2.3. Form∈Z,u1, u2∈N, we have

u1−1 i0

B_n,w^qu1u1

u2xu2

u1i

uⁿ⁻¹₁ w^u2iq^u2iu2−1

i0

B_n,w^qu2u2

u1xu1

u2i

uⁿ⁻¹₂ w^u1iq^u1i. 2.22

We note that by settingu2 1 in Theorem 2.3, we get the following multiplication theorem for the twistedq-Bernoulli polynomials.

Theorem 2.4. Form∈Z,u1∈N, one has Bn,w^q u1x uⁿ⁻¹₁

u1−1 i0

B_n,w^qu1u1

x i

u1

wⁱqⁱ. 2.23

Remark 2.5. 18, Kim suggested open questions related to finding symmetric properties for Carlitzq-Bernoulli numbers. In this paper, we give the symmetric property forq-Bernoulli numbers in the viewpoint to give the answer of Kim’s open questions.

3. The Twisted Generalized Bernoulli Polynomials Attached to χ of Higher Order

In this section, we consider the generalized Bernoulli numbers and polynomials and then define the twisted generalized Bernoulli polynomials attached toχof higher order by using

multivariatep-adic invariant integrals onZp. Letχ be Dirichlet’s character with conductor d ∈ N. Then the generalized Bernoulli numbersBn,χ and polynomialsBn,χxattached toχ are defined as

t_d−1

a0χae^at e^dt−1 ^∞

n0

Bn,χtⁿ

n!, 3.1

t_d−1

a0χae^at

e^dt−1 e^xt∞

n0

Bn,χxtⁿ

n! 3.2

cf.2,18,23,27.

LetCp^∞

n≥1Cpⁿ lim_{n→ ∞}Cpⁿbe the locally constant space, whereCpⁿ {w|w^pn 1}is the cyclic group of orderpⁿ. Forw∈Cp^∞, we denote the locally constant function by

φw:Zp−→Cp, x−→w^x 3.3

cf.2,3,21,23,24. If we takefx χxe^txφwxq^x, forq∈Cpwith|q−1|p<1, then it is obvious from3.1that

X

χxe^txw^xq^xdx

tlogq ^d−1_a0χaw^aq^ae^at

w^dq^de^dt−1 . 3.4

Now we define the twisted generalized Bernoulli numbersB^qn,χ,w and polynomialsBn,χ,w^q x attached toχas follows:

tlogq ^d−1_a0χaw^aq^ae^at w^dq^de^dt−1 ^∞

n0

B^qn,χ,wtⁿ

n!, 3.5

tlogq ^d−1_a0χaw^aq^ae^ate^xt w^dq^de^dt−1 ^∞

n0

Bn,χ,w^q xtⁿ

n! 3.6

for eachw∈Cp^∞see31,32. By3.5and3.6,

X

χxxⁿw^xq^xdxBn,χ,w^q ,

X

χ y

xy_n

w^yq^ydyB^qn,χ,wx.

3.7

Thus we have

1 logqt

X

χxe^ndxtw^nxq^nxdx−

X

χxe^xtw^xq^xdx

nd

Xχxe^xtw^xq^xdx

Xe^ndxtw^ndxq^ndxdx w^ndq^nde^ndt−1

w^dq^de^dt−1 d−1

i0

χie^itwⁱqⁱ.

3.8

Then

1 logqt

X

χxe^ndxtw^nxq^nxdx−

X

χxe^xtw^xq^xdx

^nd−1

l0

χle^ltw^lq^l∞

k0 nd−1

l0

χll^kw^lq^lt^k k!.

3.9

Let us define thep-adic twistedq-functionT_k,w^q χ, nas follows:

T_k,w^q χ, n

ⁿ

l0

χll^kw^lq^l. 3.10

By3.9and3.10, we see that

1 logqt

X

χxe^ndxtw^ndxq^ndxdx−

X

χxe^xtw^xq^xdx

^∞

k0

T_k,w^q

χ, nd−1t^k

k!. 3.11

Thus,

X

χxndx^kw^nxq^nxdx−

X

χxx^kw^xq^xdx

tlogq T_k,w^q

χ, nd−1

, 3.12

for allk, n, d∈N. This means that

B_k,χ,w^q nd−B^qn,χ,w

tlogq T_k,w^q

χ, nd−1

, 3.13

for allk, n, d∈N. For allu1, u2, d∈N, we have

d

X

Xχx1χx2e^w1x1w2x2tw^u1x1u2x2q^u1x1u2x2dx1dx2

Xe^du1u2xtw^du1u2xq^du1u2xdx

tlogq

e^du1u2tw^du1u2q^du1u2−1 e^du1tw^du1q^du1−1

e^du2tw^du2q^du2−1

× _d−1

a0

χae^u1atw^u1aq^u1a

d−1

b0

χbe^u2btw^u2bq^u2b .

3.14

The twisted generalized Bernoulli numbersB^k,qn,χ,wand polynomialsBn,χ,w^k,qxattached toχof orderkare defined as

tlogq ^d−1_a0χaw^aq^ae^at w^dq^de^dt−1

k

^∞

n0

B^k,qn,χ,wtⁿ

n!, 3.15

tlogq ^d−1_a0χaw^aq^ae^at w^dq^de^dt−1

k

e^xt∞

n0

Bn,χ,w^k,qxtⁿ

n! 3.16

for eachw∈Cp^∞. Foru1, u2∈N, we set

Kw^q

m, χ;u1, u2

d

X^m

_m

i1χxie^mi1xiu2xu1tw^mi1xiu2xu1q^mi1xiu2xu1dx1· · ·dxm

Xe^du1u2xtw^du1u2xq^du1u2xdx

×

X^m

m i1

χxie^mi1xiu1yu2tw^mi1xiu1yu2q^mi1xiu1yu1dx1· · ·dxm,

3.17

where

X^mfx1· · ·xmdx1· · ·dxm

X· · ·

Xfx1, . . . , xmdx1· · ·dxm. In3.17, we note that K^q_wm, χ;u1, u2is symmetric inu1, u2. From3.17, we have

K^qw

m, χ;u1, u2

X^m

m i1

χxie^mi1xiu2tw^mi1xiu2q^mi1xiu2dx1· · ·dxm

×e^u1u2xtw^u1u2xq^u1u2x d

Xχx me^u2xmtw^u2xmq^u2xmdxm Xe^du1u2xq^du1u2xdx

×

X^m−1 m−1

i1

χxie^{m−1i1 xiu2t}w^{m−1i1 xiu2}q^{m−1i1 xiu2}dx1· · ·dxm−1

×e^u1u2ytw^u1u2yq^u1u2y.

3.18

Thus we can obtain

u1d

Xχxe^xtw^xq^xdx

Xe^du2xtw^du2xq^du2xdx ^∞

k0

_u

1d−1

i0

χii^kwⁱqⁱ t^k k! ^∞

k0

T_k,w^q

χ, u1d−1t^k k!, e^u1u2xtw^u1u2xq^u1u2x

X^m

m i1

χxie^mi1xiu1tw^mi1xiu1q^mi1xiu1dx1· · ·dxm

e^u1u2xtw^u1u2xq^u1u2x

u1

e^du1tw^du1q^du1−1 d−1 a0

χae^u1atw^u1aq^u1a

^∞

n0

B^m,qn,χ,wu2xuⁿ₁tⁿ n!.

3.19

From3.19, we derive

K^qw

m, χ;u1, u2

^∞

l0

B^m,q_l,χ,wu1xu^l₁t^l l!

∞ k0

T_k,w^q

χ, u1d−1t^k k!

_∞

i0

B^m−1,q_i,χ,w

u1yuⁱ₂tⁱ i!

1 u1

^∞

n0

n j0

n j

u^j₂u^n−j−1₁ B_n−j,χ,w^m,q u2x× j k0

T_k,w^q

χ, u1d−1j k

B_j−k,χ,w^m−1,q u1ytⁿ

n!.

3.20

By the symmetry ofK^qwm, χ;u1, u2inu1andu2, we can see that

K^qw

m, χ;u1, u2

^∞

n0

n j0

n j

u^j₁u^n−j−1₂ B_n−j,χ,w^m,q u1x× j k0

T_k,w^q

χ, u2d−1j k

B_j−k,χ,w^m−1,q u2ytⁿ

n!. 3.21

By comparing the coefficients on both sides of3.20and3.21, we see the following theorem.

Theorem 3.1. Ford, u1, u2, m∈N,n∈Z, one has

n j0

n j

u^j₂u^n−j−1₁ B^m,q_n−j,χ,wu2x j k0

T_k,w^q

χ, u1d−1j k

B_j−k,χ,w^m−1,q u1y

ⁿ

j0

n j

u^j₁u^n−j−1₂ B_n−j,χ,w^m,q u1x^j

k0

T_k,w^q

χ, u2d−1j k

B_j−k,χ,w^m−1,q u2y

.

3.22

Remark 3.2. If we takey0 andm1 in3.22, then we have

n j0

n j

u^j₂u^n−j−1₁ B_n−j,χ,w^q u2x j k0

T_k,w^q

χ, u1d−1j k

ⁿ

j0

n j

u^j₁u^n−j−1₂ B^q_n−j,χ,wu1x j k0

T_k,w^q

χ, u2d−1j k

.

3.23

Now we can also calculate

K^qw

m, χ;u1, u2

^∞

n0

_n

k0

n k

u^k−1₁ u^n−k₂ B_n−k,χ,w^m−1,q

u1y^du1−1

i0

B_i,χ,w^m,q

u2xu2

u1i tⁿ

n!. 3.24

From the symmetric property ofKw^qm, χ;u1, u2inu1andu2, we derive

K^qw

m, χ;u1, u2

^∞

n0

_n

k0

n k

u^k−1₂ u^n−k₁ B_n−k,χ,w^m−1,q

u2y^du2−1

i0

B_i,χ,w^m,q

u1xu1

u2i tⁿ

n!. 3.25

By comparing the coefficients on both sides of 3.24 and 3.26, we obtain the following theorem.

Theorem 3.3. Ford, u1, u2, m∈N,n∈Z, we have

n k0

n k

u^k−1₁ u^n−k₂ B_n−k,χ,w^m−1,q

u1y^du1−1

i0

B^m,q_k,χ,w

u2xu2

u1i

ⁿ

k0

n k

u^k−1₂ u^n−k₁ B^m−1,q_n−k,χ,w

u2y^du2−1

i0

B_k,χ,w^m,q

u1xu1

u2i

.

3.26

Remark 3.4. If we takey0 andm1 in3.26, then one has

uⁿ⁻¹₁

du1−1 i0

B^qn,χ,w

u2xu2

u1i

uⁿ⁻¹₂

du2−1 i0

Bn,χ,w^q

u1xu1

u2i

. 3.27

Remark 3.5. In our results forq 1, we can also derive similar results, which were treated in 27. In this paper, we used the p-adic integrals to derive the symmetric properties of theq-Bernoulli polynomials. By using the symmetric properties ofp-adic integral onX, we can easily derive many interesting symmetric properties related to Bernoulli numbers and polynomials.