On Interpolation Functions of the Generalized Twisted Euler 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.

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 qx explogq for|x|p ≤ 1.

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

xq x:q

1−qx

1−q, x−q

1−−qx

1q . 1.1

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

Forda fixed positive integer withp, d 1, set

X Xd lim_← N

Z

dpN_Z, X1Zp,

X∗

0<a<dp,

a,p1

adpZp

,

adpN_Z p

x∈X |x≡amod dpN_,

1.2

wherea∈Zsatisfies the condition 0≤a < dpN. The distribution is defined by

μq

adpN_Z p

qa dpN

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

Zpfxdμqx Nlim→ ∞

1 pN

q pN₋₁

x0

fxqx. 1.4

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

μ−q

adpNZp

−qa dpN

−q

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

as

I−qf

Zpfxdμ−qx Nlim→ ∞

1 pN

−q pN₋₁

x0

fx−qx, 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

Zpfxdμ−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∈C_pn |wp n

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 ξx, the locally constant function. If we takefx ξxext, then we havecf.35

En,ξ

Zpx

n_ξn_dμ

−1x. 2.3

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

I−1

fn

−1n−1_I

−1f 2

n−1

0

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

I−1

fn

I−1f 2

n−1

0 −1

f. 2.5

If we replacenbydoddinto2.5, we obtain

I−1

fd

I−1f 2

d−1

0 −1

f. 2.6

Letξ∈Tp. Letχbe a Dirichlet’s character of conductord, whichdis any multiple ofp

withp≡1mod 2. By substitutingfx χxξx_ext_{into2.6, we have}

I−1

χxξxext

∞

n0 En,ξ,χt

n

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−₀1−1χξ_et

ξd_edt₁

∞

n0 En,ξ,χt

n

n!, |t|< π

d. 2.8

By using Taylor series ofext, 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

En,qh,1x

Zpq

h−1y_xyn

qdμ−qy, 2.10

whereh∈Zandx∈Zp. In particular, if we takex0, thenEn,qh,10 En,qh,1. 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_n,q,ξh,1x

Zpq

h−1y_φ

Forh∈Zandn∈N, we have thatcf.55

E_n,q,ξh,1x 1q 1−qn

n

i0

n i

−1i

qxi 1

1ξqhi, 2.12

E_n,q,ξh,1x 1q 1qd

d−1

a0 −1a

qha_ξa_Eh,1 n,ξd_,qd

xa d

dnq, 2.13

whered∈Nwithd≡1mod 2.

LetF_q,ξh,1t, xbe the generating function ofE_n,q,ξh,1xin complex plane as followscf. 55:

F_q,ξh,1t, x 1q

∞

n0 −1n

qhnξnetnxq

∞

n0

E_n,q,ξh,1xt

n

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,χh,1 x

X

χyqh−1y_ξy_xyn

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_n,q,ξ,χh,1 x 1q 1qd

d

a1

χa−1aqhaξaE_n,qh,1d_,ξd

xa d

dnq. 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,ξh,1t, x 1q

∞

n0

−1n_qhn_ξn_etnxq

∞

n0

E_n,q,ξh,1xt

n

n!,

whereq, x ∈ C, r ∈ Z N∪ {0}andξ isrth root of unity. In particular, if we takex 0, then we haveE_n,q,ξh,10 E_n,q,ξh,1. These numbers are called twisted Euler numbers. By using derivative operator, we havedk_/dtk_F

q,ξt, x|t0En,q,ξh,1x.

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

ζ_E,q,ξh,1s, x 1q

∞

k0

−1k_qhk_ξk

xksq

, 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_E,q,ξh,1s 1q

∞

k1

−1k_qhk_ξk

ksq

, s∈C,Res>1. 3.3

Forn∈N, we knowcf.28

ζ_E,q,ξh,1−n, x E_n,q,ξh,1x. 3.4

From now on, we will define the two-variableh, q-L-functionsL_E,q,ξh,1s, 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_E,q,ξh,1s, x:χ 1q

∞

n0

χn−1nqhn_ξn

nxsq

. 3.5

By substitutingnajd, d≡1mod 2,1≤a≤dandn0,1,2, . . .into3.5, then

using3.2, we have

L_E,q,ξh,1s, x:χ1q

d

a1

∞

j0

χajd−1ajdqhajd_ξajd

ajdxs_q

1q

d

a1

χa−1aqha_ξa

dsq

∞

j0

−1jd

qhjd

j ax/ds_qd

1q 1qd

d

a1

χa−1aqha_ξa_ζh,1

E,qd_,ξd

s,ax d

d−_qs.

3.6

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

L_E,q,ξh,1s, x:χ 1q 1qd

d

a1

χa−1aqhaξaζ_E,qh,1d_,ξd

s,ax d

d−qs. 3.7

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

L_E,q,ξh,1−n, x:χ 1q 1qd

d

a1

χa−1aqhaξaζ_E,qh,1d_,ξd

−n,ax d

dnq

1q 1qd

d

a1

χa−1aqhaξaE_n,qh,1d_,ξd

ax d

dnq

E_n,q,ξ,χh,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_E,q,ξh,1−n, x:χ E_n,q,ξ,χh,1 x. 3.9

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

L_E,q,ξh,1s, χ 1q

∞

n1

χn−1nqhn_ξn

nsq

, 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_n,q,ξ,χh,1 x 1q 1qd

d

a1

χa−1aqhaξaE_n,qh,1d_,ξd

ax d

dnq. 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 andξr _{1 withξ /1, we define}

L_E,q,ξh,1s, x:χ 1q

∞

k0

χk−1kqhkξk

We consider the well-known identitycf.44,65

1 1−xs

∞

j0

sj−1 j

xj. 3.13

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

L_E,q,ξh,1s, x:χ 1q1−qs

∞

j0

∞

k0

sj−1 j

χk−1kξkqhkjkx. 3.14

We will investigate the relations betweenL_E,q,ξh,1s, x:χandL_E,q,ξh,1s, χas follows. Substitutingk ajd, a 1,2, . . . , dwithd ≡1 mod 2, j 0,1,2, . . . ,into3.12,

we have

L_E,q,ξh,1s, x:χ 1q

d

a1

∞

j0

χajd−1ajdqhajd_ξajd

ajdxs_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, ξr _{1withξ /1. Then one has}

L_E,q,ξh,1s, x:χ 1q 1qd

d

a1

χa−1aqhaξaζ_E,qh,1d_,ξd

s,ax d

d−qs. 3.16

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

L_E,q,ξh,1−n, x:χ 1q 1qd

d

a1

χa−1aqhaξaζ_E,qh,1d_,ξd

−n,ax d

dnq

1q 1qd

d

a1

χa−1aqhaξaE_n,qh,1d_,ξd

ax d

dnq

E_n,q,ξ,χh,1 x.

3.17

Thus, we see that the functionL_E,q,wh,1s, 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_E,q,ξh,1−n, x:χ E_n,q,ξ,χh,1 x. 3.18

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

H_E,q,ξh,1s, a, x|F

m≡amodF, m>0

−1m_qhm_ξm

mxsq

. 3.19

By substitutingmajF, we have

H_E,q,ξh,1s, a, x|F

∞

j0

−1ajF_qhajF_ξajF

ajFxs_q

−1a_qha_ξa_F−s q

∞

j0

−1jF_qFhj_ξFj

ax/F js_qF

F−qs−1aqhaξa

1 1qF

∞

j0

−1jF_qFhj_ξFj

ax/F js_qF

F−qs

−1a_qha_ξa

1qF ζ

h,1

E,qF_,ξF

s,ax F

.

3.20

By substituting3.2, fors−n, we get

H_E,q,ξh,1s, a, x|F Fnq

−1a

qha_ξa

1qF E

h,1

n,qF_,ξF

ax F

. 3.21

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

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

Theorem 3.9. Fors∈C, ξr _{1withξ /1, letχbe the Dirichlet’s character with conductord∈N}

withd≡1mod 2andx∈R,0< x≤1, Fis any multiple ofd. Then one has

L_E,q,ξh,1s, x:χ 1q

F

a1

χa−1aH_E,q,ξh,1s, a, x|F. 3.22

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

L_E,q,ξh,10, x:χ 1q

F

a1

χaH_E,q,ξh,10, a, x|F

1q 1qF

F

a1

χa−1aqha_ξa_Eh,1

0,qF_,ξF

ax F

.

3.23

Corollary 3.11. Fors∈C, ξr _{1withξ /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_E,q,ξh,10, x:χ 1q 2

1qF1_ξqh F

a1

χa−1aqhaξa. 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 thatqx 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−1_aa

q aq/ωa. Thena ≡ 1modp11/p−1.

Hence we see that

aptω−1aptaptq

ω−1aaqω−1aqaptq

≡1mod p11/p−1,

4.1

wheret∈Cpwith|t|p≤1,a, p 1.

We denote the subsetDofC∗_pbycf.62

D{s∈Cp:|s|p≤p1−1/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

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

Now we set

L_E,p,q,ξh,1 s, x:χ 1q 1qF

F

a1, pa

χa−1aξaapt−s

·∞

j0

₋

s j

E_j,qh,F1_,ξFqjapt

F apt

j

qapt .

4.5

ThenL_E,p,q,ξh,1 s, x:χis analytic fort∈Cpwith|t|p ≤1, whens∈D. Fort∈Cpwith|t|p ≤1,

we have

∞

j0

₋

s j

E_j,qh,F1_,ξFqjapt

F apt

j

qapt

4.6

is analytic fors∈D. It readily follows that

aptsω−saaptsqas

∞

m0

s

m q

a_a−1

q ptq

m

4.7

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

L_E,p,q,ξh,1 0, x:χ 1q 2

F

a1 −1a_χ

naξa. 4.8

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

E_n,q,ξ,χh,1

npt F

n q

1q 1qF

F

a0

χna−1aξaE_n,qh,1F_,ξF

apt F

. 4.9

Ifχnp/0, thenp, dχn 1, soF/pis a multiple ofdχn. Therefore, we have

χnppnqE h,1

n,qF_,ξF_,χ nt

χnppnq

F p

n

qp 1qp 1qpF/p

F/p−1

a0

χna−1aξaE_n,h,_q1pF/p_,ξpF/p

at F/p

Fnq

1qp

1qF F

a0

pa

χna−1aξaE_n,qh,1F_,ξF

apt F

.

Then we note that

1q

1qpχnpp n qE

h,1

n,qF_,ξF_,χ nt

1q 1qFF

n q

F

a0

p|a

χna−1aξaE_n,qh,1F_,ξF

apt F

. 4.11

The difference of these equations yields

E_n,q,ξ,χh,1 npt−

1q

1qpχnpp n

qEn,qh,1F_,ξF_,χnt

1q 1qFF

n q

F

a0

pa

χna−1aξaE_n,qh,1F_,ξF

apt F

.

4.12

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

E_n,qh,1F_,ξF

apt F

F−qnaptnq n

k0

n k

qaptkξa

F apt

k

qapt

E_k,qh,1F_,ξF. 4.13

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

E_n,q,ξ,χh,1 npt−

1q

1qpχnpp n

qE_n,qh,1F_,ξF_,χnt

1q 1qF

F−1

a0

χna−1aξaE_n,qh,1F_,ξF

apt F

1q 1qp

F−1

a0, pa

χna−1aξaaptn n

k0

n k

qaptk

F apt

k

qapt E_k,qh,1F_,ξF.

4.14

From4.5–4.14, we can derive that

E_n,q,ξ,χh,1 _npt− 1q

1qpχnpp n

qEn,qh,1p_,ξp_,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_E,p,q,ξh,1 s, t:χ 1q 1qd

F

a1, pa

χa−1aξaapt−s

∞

m0

−s m

qaptm

F apt

m

qapt

E_m,qh,1F_,ξF.

ThenL_E,p,q,ξh,1 s, t : χis analytic for t ∈ Cp,|t|p ≤ 1, provides s ∈ D when χ 1.

Furthermore, for eachn∈Z, we have

L_E,p,q,ξh,1 −n, t:χ E_n,q,ξ,χh,1 _npt− 1q

1qpχnpp n qE

h,1

n,qp_,ξp_,χ

nt. 4.17

Thus we note thatL_E,p,q,ξh,1 s,0 : χ L_E,p,q,ξh,1 s, χfor alls ∈ D, whereL_E,p,q,ξh,1 s, 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_E,p,q,ξh,1 s, t : χ which is first defined by the interpolation function

H_E,p,q,ξh,1 s, a, x|F −1

a

1qFq

ha_ξa_apt−s

·∞

j0

−s j

qjapt

_F

q

apt_q

j

E_j,qh,F1_,ξF,

4.18

fors∈Zp.

From4.18, we have that

H_E,p,q,ξh,1 −n, a, x|F −1

a

1qFξ

a_qha_aptna

j0

n j

qaptj

Fq

aq

j

E_j,qh,F1_,ξF

−1a 1qFq

ha_ξa_ω−n_aFn qEn,qF_,ξF

a F

ω−naH_E,q,ξh,1−n, a, x|F.

4.19

By using the definition of H_E,p,q,ξh,1 s, a, x | F, we can express L_E,p,q,ξh,1 s, t : χ for all a ∈

Z,a, p 1 andt∈Cpwith|t| ≤1 as follows:

L_E,p,q,ξh,1 s, t:χ

F

a1, pa

χaH_E,p,q,ξh,1 s, apt|F. 4.20

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

∂/∂sL_E,p,q,ξh,1 s, t:χis the coefficients ofsin the expansion ofL_E,p,q,ξh,1 s, t:χats0. Using the Taylor expansion ats0, we see that

apt−s1−slogapt· · ·,

−s m

−1m

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

log_p1x

∞

n1 −1n

n x

n_{, |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_E,p,q,ξh,1 0, t:χ. It follows from the definition ofLE,p,q,ξs, t:χthat

L_E,p,q,ξh,1 s, t:χ 1q 1qF

F

a1, pa

χa−1aξaapt−s

·∞

m0

₋

s m

qaptm

F apt

m

qapt

E_m,qh,1F_,ξF.

4.23

Thus, we have

∂ ∂sL

h,1

E,p,q,ξs, t:χ|s0 1q 1qF

F

a1, pa

χa−1aξa

·

−logaptE₀h,_,q1F_,ξF

∞

m1 −1m

m q

aptm

F apt

m

qapt E_m,qh,1F_,ξF

.

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 1qF

F

a1, pa

χa−1aξa

∞

m1 −1m

m q

aptm

_F

q

apt_q

m

E_m,qh,1F_,ξF

−1q

2

F

a1

pa

χa−1aξalogapt.

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