2. Twisted h, q-Euler Numbers and Polynomials

Volume 2009, Article ID 946569,17pages doi:10.1155/2009/946569

Research Article

On Interpolation Functions of the Generalized Twisted h, q -Euler Polynomials

Kyoung Ho Park

Department of Mathematics, Sogang University, Seoul 121-742, South Korea

Correspondence should be addressed to Kyoung Ho Park,[email protected] Received 5 November 2008; Accepted 14 January 2009

Recommended by Vijay Gupta

The aim of this paper is to constructp-adic twisted two-variable Euler-h,q-L-functions, which interpolate generalized twistedh,q-Euler polynomials at negative integers. In this paper, we treat twistedh,q-Euler numbers and polynomials associated withp-adic invariant integral onZp. We will construct two-variable twistedh,q-Euler-zeta function and two-variableh,q-L-function in Complexs-plane.

Copyrightq2009 Kyoung Ho Park. 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.

1. Introduction

Tsumura and Young treated the interpolation functions of the Bernoulli and Euler polynomials in 1,2. Kim and Simsek studied onp-adic interpolation functions of these numbers and polynomials3–48. In49, Carlitz originally constructedq-Bernoulli numbers and polynomials. Many authors studied these numbers and polynomials 4, 28, 38, 41, 50. After that, twistedh, q-Bernoulli and Euler numberspolynomials were studied by several authors 1–32, 32–65. In 62, Whashington constructed one-variable p-adic-L- function which interpolates generalized classical Bernoulli numbers at negative integers.

Fox introduced the two-variable p-adi L-functions 53. Young defined p-adic integral representation for the two-variable p-adicL-functions64. Furthermore, Kim constructed the two-variable p-adic q-L-function, which is interpolation function of the generalized q-Bernoulli polynomials 8. This function is the q-extension of the two-variable p-adic L-function. Kim constructed q-extension of the generalized formula for two-variable of Diamond and Ferrero and Greenberg formula for two-variablep-adicL-function in the terms of thep-adic gamma and log-gamma functions8. Kim and Rim introduced twistedq-Euler numbers and polynomials associated with basic twistedq--functions28. Also, Jang et al.

investigated thep-adic analogue twistedq--function, which interpolates generalized twisted

q-Euler numbers En,q,ξ,χ attached to Dirichlet’s character χ 55. Kim et al. have studied two-variablep-adicL-functions, which interpolate the generalized Bernoulli polynomials at negative integers. In this paper, we will construct two-variale p-adic twisted Euler h, q- L-functions. This functions interpolation functions of the generalized twisted h, q-Euler polynomials.

Let p be a fixed odd prime number. Throughout this paper Z,Zp,Qp and Cp will respectively denote the ring of rational integers, the ring ofp-adic rational integers, the field ofp-adic rational numbers and the completion of the algebraic closure ofQp. Letvp be the normalized exponential valuation ofCp such that|p|p p^−vpp p⁻¹. Ifs ∈ C, then|q| <1.

If q ∈ Cp, we normally assume |1 −q|p < p^−1/p−1, so that q^x explogq for|x|p ≤ 1.

Throughout this paper we use the following notationscf.1–32,32–48,50,51,54–65:

x_q x:q 1−q^x

1−q, x_−q 1−−q^x

1q . 1.1

Hence, lim_q→1x_qx, for anyxwith|x|p≤1 in the presentp-adic case.

Forda fixed positive integer withp, d 1, set

X Xd lim_←

N

Z

dp^NZ, X1Zp,

X^∗

0<a<dp, a,p1

adpZp

,

adp^NZp

x∈X |x≡a

mod dp^N ,

1.2

wherea∈Zsatisfies the condition 0≤a < dp^N. The distribution is defined by

μq

adp^NZp

q^a

dp^N_q. 1.3 We say thatf is uniformly differential function at a pointa∈ Zp, and we writef ∈ UDZp, if the difference quotients, Ffx, y fx−fy/x−yhave a limitfaas x, y → a, a.

Forf∈UDZp, thep-adic invariantq-integral onZpis defined as4,18

Iqf

Zp

fxdμqx lim

N→ ∞

1 p^N_q

p^N−1 x0

fxq^x. 1.4

The fermionicp-adicq-measures onZpis defined ascf.14–16,18,22,28

μ−q

adp^NZp

−q^a

dp^N_−q, 1.5

forf ∈UDZp. Forf ∈UDZp, the ferminoicp-adic invariantq-integral onZpis defined as

I−qf

Zp

fxdμ−qx lim

N→ ∞

1 p^N_−q

p^N−1 x0

fx−q^x, 1.6

which has a sense as we see readily that the limit is convergent. Forf ∈UDZp,Cp, we note thatcf.14,16,18,22,28

Zp

fxdμ−1x

X

fxdμ−1x. 1.7

From the fermionic invariant integral onZp, we derive the following integral equation cf.14,35:

I−1 f1

I−1f 2f0, 1.8

wheref1x fx1.

2. Twisted h, q-Euler Numbers and Polynomials

In this section, we will treat some properties of twistedh, q-Euler numbers and polynomials associated withp-adic invariant integral onZp. From now on, we takeh∈Zandq∈Cpwith

|q−1|p< p^−1/p−1. LetCpⁿ be the space of primitivepⁿth root of unity, Cpⁿ

w∈Cpⁿ |w^pn 1

. 2.1

Then, we denote

Tp lim

n→ ∞Cpⁿ

n≥0

Cpⁿ. 2.2

HenceTpis ap-adic locally constant space. Forξ∈Tp, we denote byφξ :Zp → Cpdefined by φξx ξ^x, the locally constant function. If we takefx ξ^xe^xt, then we havecf.35

En,ξ

Zp

xⁿξⁿdμ₋₁x. 2.3

By induction in1.8, Kim constructed the following useful identitycf.14,28:

I₋₁ fn

−1ⁿ⁻¹I₋₁f 2

n−1

0

−1ⁿ⁻¹⁻f, 2.4

wheren∈N, fnfxn. From2.4, ifnis odd, then we have

I−1 fn

I−1f 2 n−1 0

−1f. 2.5

If we replacenbydoddinto2.5, we obtain

I−1 fd

I−1f 2

d−1

0

−1f. 2.6

Letξ∈Tp. Letχbe a Dirichlet’s character of conductord, whichdis any multiple ofp withp≡1mod 2. By substitutingfx χxξ^xe^xtinto2.6, we have

I−1

χxξ^xe^{xt ∞}

n0

En,ξ,χtⁿ

n!. 2.7

Remark 2.1. In complex case, the generating function of the Euler numbersEn,ξ,χ is given by cf.28

2 ^d−1₀−1χξe^t ξ^de^dt1 ^∞

n0

En,ξ,χtⁿ

n!, |t|< π

d. 2.8

By using Taylor series ofe^xt, then we can define the generalized twisted Euler numbersEn,ξ,χ

attached toχas followscf.55:

En,ξ,χ

X

ξⁿxⁿχxdμ−1x. 2.9

In8,h, q-Euler numbers were defined by

E^h,1n,q x

Zp

q^h−1yxyⁿ_qdμ_−qy, 2.10

whereh∈Zandx∈Zp. In particular, if we takex0, thenEn,q^h,10 E^h,1n,q . These numbers are calledh, q-Euler numbers.

By using iterative method ofp-adic invariant integral onZpin the sense of fermionic, we define twistedh, q-Euler numbers as followscf.55:

E^h,1_n,q,ξx

Zp

q^h−1yφξyxyⁿ_qdμ−qy. 2.11

Forh∈Zandn∈N, we have thatcf.55 E^h,1_n,q,ξx 1q

1−qⁿ n

i0

n i

−1ⁱq^xi 1

1ξq^hi, 2.12

E_n,q,ξ^h,1x 1q 1q^d

d−1

a0

−1^aq^haξ^aE^h,1_n,ξd,q^d

xa d

dⁿ_q, 2.13

whered∈Nwithd≡1mod 2.

LetF_q,ξ^h,1t, xbe the generating function ofE_n,q,ξ^h,1xin complex plane as followscf.

55:

F_q,ξ^h,1t, x 1q^∞

n0

−1ⁿq^hnξⁿe^tnxq

^∞

n0

E^h,1_n,q,ξxtⁿ n!.

2.14

Letχbe the Dirichlet’s character with conductord∈Nwithd≡1 mod 2. Then the generalized twistedh, q-Euler polynomials attached toχis given by as follows:

Forn∈ZN∪ {0},

E_n,q,ξ,χ^h,1 x

X

χyq^h−1yξ^yxyⁿ_qdμ_−qy, 2.15 whereh∈Z, dis any multiple ofpwithp≡1mod 2andx∈Cp.

Then the distribution relation of the generalized twistedh, q-Euler polynomials is given by as followscf.14:

E^h,1_n,q,ξ,χx 1q 1q^d

d a1

χa−1^aq^haξ^aE^h,1_n,qd,ξ^d

xa d

dⁿ_q. 2.16

3. Two-Variable Twisted h, q-Euler-Zeta Function and h, q - L -Function

In this section, we will construct two-variable twisted h, q-Euler-zeta function and two- variableh, q-L-function in Complexs-plane. We assumeq∈Cwith|q|<1.

Firstly, we consider twisted q-Euler numbers and polynomials in C as followscf.

55:

F_q,ξ^h,1t, x 1q^∞

n0

−1ⁿq^hnξⁿe^tnxq

^∞

n0

E^h,1_n,q,ξxtⁿ n!,

3.1

whereq, x ∈ C, r ∈ Z N∪ {0}andξ isrth root of unity. In particular, if we takex 0, then we haveE^h,1_n,q,ξ0 E^h,1_n,q,ξ. These numbers are called twisted Euler numbers. By using derivative operator, we haved^k/dt^kFq,ξt, x|_t0E^h,1_n,q,ξx.

From3.1, we can define Hurwitz-type twistedh, q-Euler-zeta function as follows cf.55:

ζ^h,1_E,q,ξs, x 1q^∞

k0

−1^kq^hkξ^k

xk^s_q , 3.2 whereq∈C,|q|<1, s∈C, h∈Zandx∈R,0< x≤1. Note that ifx1 in3.2, then we see that the twistedh, q-Euler-zeta function is defined bycf.28,55

ζ^h,1_E,q,ξs 1q^∞

k1

−1^kq^hkξ^k

k^s_q , s∈C,Res>1. 3.3

Forn∈N, we knowcf.28

ζ^h,1_E,q,ξ−n, x E^h,1_n,q,ξx. 3.4

From now on, we will define the two-variableh, q-L-functionsL^h,1_E,q,ξs, x :χwhich interpolates the generalizedh, q-Euler polynomials.

Definition 3.1. Let χbe the Dirichlet’s character with conductor dwith d ≡ 1 mod 2. For s∈C, h∈Zandx∈R,0< x≤1, we define

L^h,1_E,q,ξs, x:χ 1q^∞

n0

χn−1ⁿq^hnξⁿ

nx^s_q . 3.5 By substitutingnajd, d≡1mod 2,1≤a≤dandn0,1,2, . . .into3.5, then using3.2, we have

L^h,1_E,q,ξs, x:χ1q^d

a1

∞ j0

χajd−1^ajdq^hajdξ^ajd ajdx^s_q

1q^d

a1

χa−1^aq^haξ^a d^s_q

∞ j0

−1^jdq^hjd j ax/d^s_qd

1q 1q^d

d a1

χa−1^aq^haξ^aζ^h,1_E,qd,ξ^d

s,ax

d

d^−s_q .

3.6

Thus, we see the functionL^h,1_E,q,ξs, x:χwhich interpolates the generalizedh, q-Euler polynomials as follows.

Theorem 3.2. For s ∈ C, h ∈ Z, letχ be the Dirichlet’s character with conductor dwith d ≡ 1mod 2. Then one has

L^h,1_E,q,ξs, x:χ 1q 1q^d

d a1

χa−1^aq^haξ^aζ_E,q^h,1d,ξ^d

s,ax

d

d^−s_q . 3.7

By substitutings−nwithn >0, into3.7, we obtain

L^h,1_E,q,ξ−n, x:χ 1q 1q^d

d a1

χa−1^aq^haξ^aζ_E,q^h,1d,ξ^d

−n,ax d

dⁿ_q 1q

1q^d d a1

χa−1^aq^haξ^aE^h,1_n,qd,ξ^d

ax d

dⁿ_q E_n,q,ξ,χ^h,1 x,

3.8

whered≡1 mod 2, d∈N.

Thus, we have the following theorem.

Theorem 3.3. Forn ∈N, letχbe the Dirichlet’s character with conductordwithd ≡ 1mod 2.

Then one has

L^h,1_E,q,ξ−n, x:χ E^h,1_n,q,ξ,χx. 3.9

Remark 3.4. If we takex1 in3.5, then we havecf.28,55

L^h,1_E,q,ξs, χ 1q^∞

n1

χn−1ⁿq^hnξⁿ

n^s_q , fors∈C. 3.10 From3.9and3.10, we have the following corollary.

Corollary 3.5. Letχbe the Dirichlet’s character with conductordwithd ≡ 1mod 2. Then one has

E^h,1_n,q,ξ,χx 1q 1q^d

d a1

χa−1^aq^haξ^aE^h,1_n,qd,ξ^d

ax d

dⁿ_q. 3.11

Secondly, we will define two-variable twisted Eulerh, q-L-function as follows.

Definition 3.6. Letχbe the Dirichlet’s character with conductordwithd≡1 mod 2, d∈N.

Fors∈C, h∈Z, x∈R,0< x≤1 andξ^r 1 withξ /1, we define

L^h,1_E,q,ξs, x:χ 1q^∞

k0

χk−1^kq^hkξ^k

kx^s_q . 3.12

We consider the well-known identitycf.44,65 1

1−x^{s ∞}

j0

sj−1 j

x^j. 3.13

By using3.12, we define two-variable twisted Eulerh, q-L-function as follows:

L^h,1_E,q,ξs, x:χ 1q1−q^s∞

j0

∞ k0

sj−1 j

χk−1^kξ^kq^hkjkx. 3.14

We will investigate the relations betweenL^h,1_E,q,ξs, x:χandL^h,1_E,q,ξs, χas follows.

Substitutingk ajd, a 1,2, . . . , dwithd ≡1 mod 2, j 0,1,2, . . . ,into3.12, we have

L^h,1_E,q,ξs, x:χ 1q^d

a1

∞ j0

χajd−1^ajdq^hajdξ^ajd

ajdx^s_q , 3.15

Thus we obtain the following theorem.

Theorem 3.7. Fors ∈ Cwithh ∈ Z, letχbe the Dirichlet character with conductordwithd ≡ 1mod 2andx∈R,0< x≤1, ξ^r 1 withξ /1. Then one has

L^h,1_E,q,ξs, x:χ 1q 1q^d

d a1

χa−1^aq^haξ^aζ^h,1_E,qd,ξ^d

s,ax

d

d^−s_q . 3.16

By substitutings−nwithn∈Ninto3.16and using3.4, we can obtain

L^h,1_E,q,ξ−n, x:χ 1q 1q^d

d a1

χa−1^aq^haξ^aζ_E,q^h,1d,ξ^d

−n,ax d

dⁿ_q 1q

1q^d d a1

χa−1^aq^haξ^aE^h,1_n,qd,ξ^d

ax d

dⁿ_q E_n,q,ξ,χ^h,1 x.

3.17

Thus, we see that the functionL^h,1_E,q,ws, x:χinterpolates generalizedh, q-Euler polynomi- als attached toχat negative integer values ofsas followings.

Theorem 3.8. Forn∈N, letχbe the Dirichlet’s character with odd conductord. Then one has

L^h,1_E,q,ξ−n, x:χ E^h,1_n,q,ξ,χx. 3.18

Note that if we takex1, thenTheorem 3.8reduces toTheorem 3.3.

LetaandFbe integers withF≡1mod 2and 0< a < F. Fors∈C, we define partial h, q-Hurwitz type zeta functionH_E,q,ξ^h,1s, a, x|Fas follows:

H_E,q,ξ^h,1s, a, x|F

m≡amodF, m>0

−1^mq^hmξ^m

mx^s_q . 3.19

By substitutingmajF, we have

H_E,q,ξ^h,1s, a, x|F ^∞

j0

−1^ajFq^hajFξ^ajF ajFx^s_q

−1^aq^haξ^aF^−s_q ^∞

j0

−1^jFq^Fhjξ^Fj ax/F j^s_qF

F^−s_q −1^aq^haξ^a 1 1q^F

∞ j0

−1^jFq^Fhjξ^Fj ax/F j^s_qF

F^−s_q −1^aq^haξ^a 1q^F ζ^h,1_E,qF,ξ^F

s,ax

F

.

3.20

By substituting3.2, fors−n, we get

H_E,q,ξ^h,1s, a, x|F Fⁿ_q−1^aq^haξ^a 1q^F E^h,1_n,qF,ξ^F

ax F

. 3.21

Equation 3.20 means that the function H_E,q,ξ^h,1s, a, x | F interpolates E_n,q,ξ^h,1s, a, x | F polynomials at negative integers.

From3.16and3.20, we have the following theorem.

Theorem 3.9. Fors∈C, ξ^r 1 withξ /1, letχbe the Dirichlet’s character with conductord∈N withd≡1mod 2andx∈R,0< x≤1, Fis any multiple ofd. Then one has

L^h,1_E,q,ξs, x:χ 1q^F

a1

χa−1^aH_E,q,ξ^h,1s, a, x|F. 3.22

Remark 3.10. If we takes0 in3.22, then we have

L^h,1_E,q,ξ0, x:χ 1q^F

a1

χaH_E,q,ξ^h,10, a, x|F 1q

1q^F F a1

χa−1^aq^haξ^aE^h,1_0,qF,ξ^F

ax F

.

3.23

From2.12, if we takes0, then we have the following corollary.

Corollary 3.11. Fors∈C, ξ^r 1 withξ /1, letχbe the Dirichlet’s character with conductord∈N withd≡1 mod 2andx∈R,0< x≤1,Fis any multiple ofd. Then one has

L^h,1_E,q,ξ0, x:χ 1q² 1q^F1ξq^h

F a1

χa−1^aq^haξ^a. 3.24

4. p -Adic Twisted Two-Variable Euler h, q-L-Functions

In 62, Washington constructed one-variable p-adic-L-function which interpolates gen- eralized classical Bernoulli numbers negative integers. Kim 22 investigated the p-adic analogues of two-variables Euler q-L-function. In this section, we will construct p-adic twisted two-variable Euler-h, q-L-functions, which interpolate generalized twisted h, q- Euler polynomials at negative integers. Our notations and methods are essentially due to Kim and Washingtoncf.22,62.

We assume thatq ∈ Cpwith |1−q|p < p^−1/p−1, so thatq^x expxlogq. Letpbe an odd prime number. Let ωdenote the Teichm ¨uller character having conductorp. For an arbitrary character χ, we define χn χω⁻ⁿ, wheren ∈ Z, in the sense of the product of characters. Let a a : q ω⁻¹aa_q a_q/ωa. Thena ≡ 1modp^11/p−1. Hence we see that

aptω⁻¹aptapt_q ω⁻¹aa_qω⁻¹aq^apt_q

≡1

mod p^11/p−1

,

4.1

wheret∈Cpwith|t|p≤1,a, p 1.

We denote the subsetDofC^∗_pbycf.62

D{s∈Cp:|s|p≤p^1−1/p−1}. 4.2

Let

Ajx ^∞

j0

an,jxⁿ, an,j ∈Cp, j0,1,2, . . . , 4.3

be a sequence of power series, each of which converges in a fixed subsetDsuch that 1an,j → an,0asj → ∞for alln, jand

2for eachs∈Dandε >0, there existsn0n0s, εsuch that

n≥n0

an,jsⁿ p

< ε, for∀j. 4.4

Then lim_{j→ ∞}Ajs A0sfor alls∈Dcf.2,22,50,51,60,62.

Letχbe the Dirichlet’s character with conductordwithd≡1 mod 2and letF be a positive multiple ofpandd.

Now we set

L^h,1_E,p,q,ξs, x:χ 1q 1q^F

F a1,pa

χa−1^aξ^aapt^−s

·^∞

j0

−s j

E^h,1_j,qF,ξ^Fq^japt F

apt j

q^apt

.

4.5

ThenL^h,1_E,p,q,ξs, x:χis analytic fort∈Cpwith|t|p ≤1, whens∈D. Fort∈Cpwith|t|p ≤1, we have

∞ j0

−s j

E_j,q^h,1F,ξ^Fq^japt F

apt j

q^apt

4.6

is analytic fors∈D. It readily follows that

apt^sω^−saapt^s_qa^s∞

m0

s

m q^aa⁻¹_q pt_qm

4.7

is analytic fors∈Cpwith|t|p≤1 whens∈D. Thus we see that

L^h,1_E,p,q,ξ0, x:χ 1q 2

F a1

−1^aχnaξ^a. 4.8

Letn∈Zand fixedt∈Cpwith|t|p≤1. Then we have that

E^h,1_n,q,ξ,χ

npt Fⁿ_q 1q 1q^F

F a0

χna−1^aξ^aE^h,1_n,qF,ξ^F

apt F

. 4.9

Ifχnp/0, thenp, dχn 1, soF/pis a multiple ofdχn. Therefore, we have χnppⁿ_qE^h,1_n,qF,ξ^F,χnt

χnppⁿ_q F p

_n

q^p

1q^p 1q^pF/p

F/p−1

a0

χna−1^aξ^aE^h,1_n,qpF/p,ξ^pF/p

at F/p

Fⁿ_q1q^p 1q^F

F a0pa

χna−1^aξ^aE^h,1_n,qF,ξ^F

apt F

.

4.10

Then we note that 1q

1q^pχnppⁿ_qE_n,q^h,1F,ξ^F,χnt 1q

1q^FFⁿ_q^F

a0p|a

χna−1^aξ^aE_n,q^h,1F,ξ^F

apt F

. 4.11

The difference of these equations yields

E^h,1_n,q,ξ,χ

npt− 1q

1q^pχnppⁿ_qE^h,1_n,qF,ξ^F,χnt 1q

1q^FFⁿ_q^F

a0pa

χna−1^aξ^aE^h,1_n,qF,ξ^F

apt F

. 4.12

Using distribution forh, q-Euler polynomials, we easily see that

E_n,q^h,1F,ξ^F

apt F

F⁻ⁿ_q aptⁿ_qⁿ

k0

n k

q^aptkξ^a F

apt k

q^apt

E^h,1_k,qF,ξ^F. 4.13

Sinceχna χaω⁻ⁿa, fora, p 1, andt∈Cp, with|t|p≤1, we have

E^h,1_n,q,ξ,χ

npt− 1q

1q^pχnppⁿ_qE_n,q^h,1F,ξ^F,χnt 1q

1q^F

F−1

a0

χna−1^aξ^aE_n,q^h,1F,ξ^F

apt F

1q 1q^p

F−1

a0,pa

χna−1^aξ^aaptⁿⁿ

k0

n k

q^aptk F

apt _k

q^apt

E_k,q^h,1F,ξ^F.

4.14

From4.5–4.14, we can derive that

E^h,1_n,q,ξ,χnpt− 1q

1q^pχnppⁿ_qE^h,1_n,qp,ξ^p,χnt L^h,1_E,p,q,ξ−n, t:χ. 4.15 Therefore we obtain the following theorem.

Theorem 4.1. LetFbe a positive integral multiple ofpandddχwithF≡1mod 2, and let

L^h,1_E,p,q,ξs, t:χ 1q 1q^d

F a1,pa

χa−1^aξ^aapt^−s∞

m0

−s m

q^aptm F

apt _m

q^apt

E^h,1_m,qF,ξ^F. 4.16

ThenL^h,1_E,p,q,ξs, t : χis analytic for t ∈ Cp,|t|p ≤ 1, provides s ∈ D when χ 1.

Furthermore, for eachn∈Z, we have

L^h,1_E,p,q,ξ−n, t:χ E^h,1_n,q,ξ,χnpt− 1q

1q^pχnppⁿ_qE_n,q^h,1p,ξ^p,χnt. 4.17

Thus we note thatL^h,1_E,p,q,ξs,0 : χ L^h,1_E,p,q,ξs, χfor alls ∈ D, whereL^h,1_E,p,q,ξs, χis twisted p-adic Eulerh, q-L-function,cf.15,22.

We now generalized to two-variable p-adic Euler h, q-L-function, L^h,1_E,p,q,ξs, t : χ which is first defined by the interpolation function

H_E,p,q,ξ^h,1 s, a, x|F −1^a

1q^Fq^haξ^aapt^−s

·^∞

j0

−s j

q^japt

F_q apt_q

^j E^h,1_j,qF,ξ^F,

4.18

fors∈Zp.

From4.18, we have that

H_E,p,q,ξ^h,1 −n, a, x|F −1^a

1q^Fξ^aq^haapt^na

j0

n j

q^aptj

F_q a_q

_j E^h,1_j,qF,ξ^F

−1^a

1q^Fq^haξ^aω⁻ⁿaFⁿ_qEn,q^F,ξ^F

a F

ω⁻ⁿaH_E,q,ξ^h,1−n, a, x|F.

4.19

By using the definition of H_E,p,q,ξ^h,1 s, a, x | F, we can express L^h,1_E,p,q,ξs, t : χ for all a ∈ Z,a, p 1 andt∈Cpwith|t| ≤1 as follows:

L^h,1_E,p,q,ξs, t:χ ^F

a1,pa

χaH_E,p,q,ξ^h,1 s, apt|F. 4.20

We know thatH_E,p,q,ξ^h,1 s, apt |Fis analytic fort ∈ Cp,|t| ≤1, whens ∈ D. The value of

∂/∂sL^h,1_E,p,q,ξs, t:χis the coefficients ofsin the expansion ofL^h,1_E,p,q,ξs, t:χats0. Using the Taylor expansion ats0, we see that

apt^−s1−slogapt· · ·,

−s m

−1^m

m s· · ·. 4.21

Thep-adic logarithmic function, log_p, is the unique functionC^∗_p → Cpthat satisfies

log_p1x ^∞

n1

−1ⁿ

n xⁿ, |x|p<1, log_pxy log_px log_py, ∀x, y∈C^∗_p,

log_pp 0.

4.22

By employing these expansion and some algebraic manipulations, we evaluate the derivative

∂/∂sL^h,1_E,p,q,ξ0, t:χ. It follows from the definition ofLE,p,q,ξs, t:χthat

L^h,1_E,p,q,ξs, t:χ 1q 1q^F

F a1,pa

χa−1^aξ^aapt^−s

·^∞

m0

−s m

q^aptm F

apt m

q^apt

E^h,1_m,qF,ξ^F.

4.23

Thus, we have

∂

∂sL^h,1_E,p,q,ξs, t:χ|_s0 1q 1q^F

F a1,pa

χa−1^aξ^a

·

−logaptE_0,q^h,1F,ξ^F∞

m1

−1^m m q^aptm

F apt

m q^apt

E^h,1_m,qF,ξ^F

. 4.24

Sinceωais a root of unity fora, p 1, we have

log_paptlog_papt log_pω⁻¹a log_papt. 4.25 Thus we have the following theorem.

Theorem 4.2. Letχbe a primitive Dirichlet’s character with odd conductord, d∈Nand letFbe a odd positive integral multiple ofpandd. Then for anyt∈Cpwith|t| ≤1, one has

∂

∂sL^h,1_E,p,q,ξs, t:χ 1q 1q^F

F a1,pa

χa−1^aξ^a ∞ m1

−1^m m q^aptm

F_q apt_q

m

E^h,1_m,qF,ξ^F

−1q 2

F a1pa

χa−1^aξ^alogapt.

4.26

References

1 H. Tsumura, “On ap-adic interpolation of the generalized Euler numbers and its applications,” Tokyo Journal of Mathematics, vol. 10, no. 2, pp. 281–293, 1987.

2 P. T. Young, “Congruences for Bernoulli, Euler, and Stirling numbers,” Journal of Number Theory, vol.

78, no. 2, pp. 204–227, 1999.

3 T. Kim, “On aq-analogue of thep-adic log gamma functions and related integrals,” Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999.

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

5 T. Kim, “On Euler-Barnes multiple zeta functions,” Russian Journal of Mathematical Physics, vol. 10, no.

3, pp. 261–267, 2003.

6 T. Kim, “q-Riemann zeta function,” International Journal of Mathematics and Mathematical Sciences, vol.

2004, no. 12, pp. 599–605, 2004.

7 T. Kim, “Analytic continuation of multipleq-zeta functions and their values at negative integers,”

Russian Journal of Mathematical Physics, vol. 11, no. 1, pp. 71–76, 2004.

8 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.

9 T. Kim, “q-generalized Euler numbers and polynomials,” Russian Journal of Mathematical Physics, vol.

13, no. 3, pp. 293–298, 2006.

10 T. Kim, “A new approach top-adicq-L-functions,” Advanced Studies in Contemporary Mathematics, vol.

12, no. 1, pp. 61–72, 2006.

11 T. Kim, “A note onp-adic invariant integral in the rings of p-adic integers,” Advanced Studies in Contemporary Mathematics, vol. 13, no. 1, pp. 95–99, 2006.

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

157, 2006.

13 T. Kim, “q-extension of the Euler formula and trigonometric functions,” Russian Journal of Mathematical Physics, vol. 14, no. 3, pp. 275–278, 2007.

14 T. Kim, “On the analogs of Euler numbers and polynomials associated withp-adicq-integral onZpat q−1,” Journal of Mathematical Analysis and Applications, vol. 331, no. 2, pp. 779–792, 2007.

15 T. Kim, “On p-adic q-l-functions and sums of powers,” Journal of Mathematical Analysis and Applications, vol. 329, no. 2, pp. 1472–1481, 2007.

16 T. Kim, “On theq-extension of Euler and Genocchi numbers,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 1458–1465, 2007.

17 T. Kim, “A note onp-adicq-integral onZp associated withq-Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 133–137, 2007.

18 T. Kim, “q-Euler numbers and polynomials associated withp-adicq-integrals,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007.

19 T. Kim, “Euler numbers and polynomials associated with zeta functions,” Abstract and Applied Analysis, vol. 2008, Article ID 581582, 11 pages, 2008.

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

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

21 T. Kim, “Note on the Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 17, no. 2, pp. 109–115, 2008.

22 T. Kim, “On p-adic interpolating function for q-Euler numbers and its derivatives,” Journal of Mathematical Analysis and Applications, vol. 339, no. 1, pp. 598–608, 2008.

23 T. Kim, “A note onq-Euler numbers and polyomials,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161–170, 2008.

24 T. Kim, “The modified q-Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 161–170, 2008.

25 T. Kim, “On a p-adic interpolation function for the q-extension of the generalized Bernoulli polynomials and its derivative,” Discrete Mathematics, vol. 309, no. 6, pp. 1593–1602, 2009.

26 T. Kim, “Note on the Eulerq-zeta functions,” Journal of Number Theory. In press.

27 T. Kim, J. Y. Choi, and J. Y. Sug, “Extendedq-Euler numbers and polynomials associated with fermionicp-adicq-integral onZp,” Russian Journal of Mathematical Physics, vol. 14, no. 2, pp. 160–163, 2007.

28 T. Kim and S.-H. Rim, “On the twistedq-Euler numbers and polynomials associated with basicq-l- functions,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 738–744, 2007.

29 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on sum of products of h,q-twisted Euler polynomials and numbers,” Journal of Inequalities and Applications, vol. 2008, Article ID 816129, 8 pages, 2008.

30 H. Ozden, Y. Simsek, and I. N. Cangul, “Euler polynomials associated withp-adicq-Euler measure,”

General Mathematics, vol. 15, no. 2, pp. 24–37, 2007.

31 H. Ozden and Y. Simsek, “A new extension ofq-Euler numbers and polynomials related to their interpolation functions,” Applied Mathematics Letters, vol. 21, no. 9, pp. 934–939, 2008.

32 I. N. Cangul, H. Ozden, and Y. Simsek, “Generating functions of theh,qextension of twisted Euler polynomials and numbers,” Acta Mathematica Hungarica, vol. 120, no. 3, pp. 281–299, 2008.

33 H. Ozden, I. N. Cangul, and Y. Simsek, “Multivariate interpolation functions of higher-orderq-Euler numbers and their applications,” Abstract and Applied Analysis, vol. 2008, Article ID 390857, 16 pages, 2008.

34 Y. Simsek, O. Yurekli, and V. Kurt, “On interpolation functions of the twisted generalized Frobenius- Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 15, no. 2, pp. 187–194, 2007.

35 S.-H. Rim and T. Kim, “A note onq-Euler numbers associated with the basicq-zeta function,” Applied Mathematics Letters, vol. 20, no. 4, pp. 366–369, 2007.

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

37 Y. Simsek, “q-analogue of twistedl-series andq-twisted Euler numbers,” Journal of Number Theory, vol. 110, no. 2, pp. 267–278, 2005.

38 Y. Simsek, “On p-adic twisted q-L-functions related to generalized twisted Bernoulli numbers,”

Russian Journal of Mathematical Physics, vol. 13, no. 3, pp. 340–348, 2006.

39 Y. Simsek, “Hardy character sums related to Eisenstein series and theta functions,” Advanced Studies in Contemporary Mathematics, vol. 12, no. 1, pp. 39–53, 2006.

40 Y. Simsek, “Remarks on reciprocity laws of the Dedekind and Hardy sums,” Advanced Studies in Contemporary Mathematics, vol. 12, no. 2, pp. 237–246, 2006.

41 Y. Simsek, “Twistedh,q-Bernoulli numbers and polynomials related to twistedh,q-zeta function andL-function,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 790–804, 2006.

42 Y. Simsek, “On twistedq-Hurwitz zeta function andq-two-variableL-function,” Applied Mathematics and Computation, vol. 187, no. 1, pp. 466–473, 2007.

43 Y. Simsek, “The behavior of the twisted p-adic h,q-L-functions at s 0,” Journal of the Korean Mathematical Society, vol. 44, no. 4, pp. 915–929, 2007.

44 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.

45 Y. Simsek, D. Kim, and S.-H. Rim, “On the two-variable Dirichlet q-L-series,” Advanced Studies in Contemporary Mathematics, vol. 10, no. 2, pp. 131–142, 2005.

46 Y. Simsek and A. Mehmet, “Remarks on Dedekind eta function, theta functions and Eisenstein series under the Hecke operators,” Advanced Studies in Contemporary Mathematics, vol. 10, no. 1, pp. 15–24, 2005.

47 H. Ozden, I. N. Cangul, and Y. Simsek, “On the behavior of two variable twistedp-adic Eulerq-l- functions,” Nonlinear Analysis. In press.

48 Y. Simsek and S. Yang, “Transformation of four Titchmarsh-type infinite integrals and generalized Dedekind sums associated with Lambert series,” Advanced Studies in Contemporary Mathematics, vol.

9, no. 2, pp. 195–202, 2004.

49 L. Carlitz, “q-Bernoulli and Eulerian numbers,” Transactions of the American Mathematical Society, vol.

76, pp. 332–350, 1954.

50 M. Cenkci and M. Can, “Some results onq-analogue of the Lerch zeta function,” Advanced Studies in Contemporary Mathematics, vol. 12, no. 2, pp. 213–223, 2006.

51 M. Cenkci, Y. Simsek, and V. Kurt, “Further remarks on multiplep-adicq-l-function of two variables,”

Advanced Studies in Contemporary Mathematics, vol. 14, no. 1, pp. 49–68, 2007.

52 A. Da¸browski, “A note onp-adicq-ζ-functions,” Journal of Number Theory, vol. 64, no. 1, pp. 100–103, 1997.

53 G. J. Fox, “Ap-adicL-function of two variables,” L’Enseignement Math´ematique, IIe S´erie, vol. 46, no.

3-4, pp. 225–278, 2000.

54 L.-C. Jang, S.-D. Kim, D.-W. Park, and Y.-S. Ro, “A note on Euler number and polynomials,” Journal of Inequalities and Applications, vol. 2006, Article ID 34602, 5 pages, 2006.

55 L.-C. Jang, V. Kurt, Y. Simsek, and S. H. Rim, “q-analogue of thep-adic twistedl-function,” Journal of Concrete and Applicable Mathematics, vol. 6, no. 2, pp. 169–176, 2008.

56 N. Koblitz, “On Carlitz’sq-Bernoulli numbers,” Journal of Number Theory, vol. 14, no. 3, pp. 332–339, 1982.

57 K. H. Park and Y.-H. Kim, “On some arithmetical properties of the Genocchi numbers and polynomials,” Advances in Difference Equations, 14 pages, 2008.

58 S.-H. Rim, K. H. Park, and E. J. Moon, “On Genocchi numbers and polynomials,” Abstract and Applied Analysis, vol. 2008, Article ID 898471, 7 pages, 2008.

59 W. H. Schikhof, Ultrametric Calculus: An Introduction to p-Adic Analysis, vol. 4 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1984.

60 K. Shiratani and S. Yamamoto, “On a p-adic interpolation function for the Euler numbers and its derivatives,” Memoirs of the Faculty of Science, Kyushu University. Series A, vol. 39, no. 1, pp. 113–125, 1985.

61 H. M. Srivastava, T. Kim, and Y. Simsek, “q-Bernoulli numbers and polynomials associated with multipleq-zeta functions and basicL-series,” Russian Journal of Mathematical Physics, vol. 12, no. 2, pp. 241–268, 2005.

62 L. C. Washington, Introduction to Cyclotomic Fields, vol. 83 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1997.

63 C. F. Woodcock, “Specialp-adic analytic functions and Fourier transforms,” Journal of Number Theory, vol. 60, no. 2, pp. 393–408, 1996.

64 P. T. Young, “On the behavior of some two-variablep-adicL-functions,” Journal of Number Theory, vol.

98, no. 1, pp. 67–88, 2003.

65 J. Zhao, “Multipleq-zeta functions and multipleq-polylogarithms,” Ramanujan Journal, vol. 14, no. 2, pp. 189–221, 2007.