A Note on the q-Euler Measures

Advances in Difference Equations Volume 2009, Article ID 956910,8pages doi:10.1155/2009/956910

Research Article

A Note on the q-Euler Measures

Taekyun Kim,

Kyung-Won Hwang,

and Byungje Lee

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139701, South Korea

2General Education Department, Kookmin University, Seoul 136702, South Korea

3Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139701, South Korea

Correspondence should be addressed to Kyung-Won Hwang,[email protected] Received 6 March 2009; Accepted 20 May 2009

Recommended by Patricia J. Y. Wong

Properties ofq-extensions of Euler numbers and polynomials which generalize those satisfied byEk and Ekx are used to construct q-extensions of p-adic Euler measures and define p- adicq--series which interpolateq-Euler numbers at negative integers. Finally, we give Kummer Congruence for theq-extension of ordinary Euler numbers.

Copyrightq2009 Taekyun 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.

1. Introduction

Letp be a fixed prime number. Throughout this paperZp,Qp,C,andCp will, respectively, denote the ring ofp-adic rational integers, the field ofp-adic rational numbers, the complex number field, and the completion of algebraic closure of Qp. Let vp be the normalized exponential valuation of Cp with |p|_p p^−vpp 1/p. When one talks of q-extension,qis variously considered as an indeterminate, a complex numberq∈Corp-adic numbersq∈Cp. Ifq ∈C, one normally assumes|q|< 1. Ifq∈Cp, one normally assumes|1−q|_p < 1. In this paper, we use the notations ofq-number as followssee1–37:

x_q 1−q^x

1−q, x_−q 1−

−q_x

1q . 1.1

The ordinary Euler numbers are defined assee1–37 ∞

k0

Ekt^k k! 2

e^t1, |t|< π, 1.2

where 2/e^t1is written as e^Et whenE^k is replaced by Ek. From the definition of Euler number, we can derive

E01, E1ⁿEn0, if n >0, 1.3

with the usual convention of replacingEⁱbyEi.

Remark 1.1. The second kind Euler numbers are also defined as followssee25:

secht 2

e^te^−t 2e^t e^2t1 ^∞

k0

E^∗_kt^k k!

|t|< π 2

. 1.4

The Euler polynomials are also defined by

2

e^t1e^xte^Ext∞

n0Enxtⁿ

n!, |t|< π. 1.5

Thus, we have

Enx ⁿ

k0

n k

Ekx^n−k. 1.6

In7,q-Euler numbers,Ek,q, can be determined inductively by

E0,q1, q

qEq1_k

Ek,q 0 if k >0, 1.7

whereE^k_q must be replaced byEk,q, symbolically. Theq-Euler polynomialsEk,qxare given byq^xEq x_q^k,that is,

Ek,qx

q^xEq xq

_k ^k

i0

k i

Ei,qq^ixx^k−i_q . 1.8

Letdbe a fixedodd positive integer. Then we havesee7 2_q

2_q^ddⁿ_q^d−1

a0

q^a−1^aEn,q

xa d

En,qx, forn∈Z. 1.9

We use1.9to get boundedp-adicq-Euler measures and finally take the Mellin transform to definep-adicq--series which interpolateq-Euler numbers at negative integers.

2. p -adic q -Euler Measures

Letdbe a fixed odd positive integer, and letpbe a fixed odd prime number. Define

XXdlim_←−

N

Z dp^NZ

, X1Zp,

X^∗

0<a<dp, a,p1

adpZp

,

adp^NZp

x∈X |x≡a

mod dp^N ,

2.1

wherea∈Zlies in 0≤a < dp^N,see1–37.

Theorem 2.1. Letμ^E_k,qbe given by

μ^E_k,q

adp^NZp

dp^Nk

q

dp^N

−q

q^a−1^aE_k,qdpN

a dp^N

, fork∈Z, N∈N. 2.2

Thenμ^E_k,q extends to aQq-valued measure on the compact open setsU⊂X. Note thatμ^E_0,q μ−q, whereμ−qadp^NZp −q^a/dp^N_−qis fermionic measure onX(see [7]).

Proof. It is sufficient to show that

p−1 i0

μ^E_k,q

aidp^Ndp^N1Zp

μ^E_k,q

adp^NZp

. 2.3

By1.9and2.2, we see that

p−1 i0

μ^E_k,q

aidp^Ndp^N1Zp

dp^N1_k q

dp^N1

−q

p−1 i0

q^aidpN−1^aidpNE_k,qdpN1

aidp^N dp^N1

dp^N1k

q

dp^N

−q

q^a−1^ap−1

i0

q^dpN_i

−1ⁱE_k,qdpN^p

a/dp^Ni p

dp^Nk

q

dp^N

−q

q^a−1^a 2_qdpN

2_q^dpN1 p_k

q^dpN

p−1 i0

q^dpN_i

−1ⁱE_k,qdpN^p

a/dp^Ni p

dp^Nk

q

dp^N

−q

q^a−1^a 2_qdpN

2_qdpN^p

pk q^dpN

p−1 i0

q^dpN_i

−1ⁱE_k,qdpN^p

a/dp^Ni p

dp^Nk

q

dp^N

−q

q^a−1^aE_k,qdpN

a dp^N

μ^E_k,q

adp^NZp

,

2.4 and we easily see that|μ^E_k,q|

p≤Mfor some constantM.

Let χ be a Dirichlet character with conductord ∈ Nwith d ≡ 1mod 2. Then we define the generalizedq-Euler numbers attached toχas follows:

Ek,χ,q 2_q

2_q^dd^k_q ^d−1

x0

q^x−1^xχxEk,q^d

x d

. 2.5

The locally constant functionχonXcan be integrated by thep-adic boundedq-Euler measure μ^E_k,q as follows:

X

χxdμ^E_k,qx lim

N→ ∞

0≤x<dp^N

χxμ^E_k,q

xdp^NZp

lim

N→ ∞

dp^Nk

q

dp^N

−q

0≤a<d

0≤x<p^N

χadxq^adx−1^adxE_k,qdpN

axd dp^N

2_q 2_qd

d^k_q

0≤a<d

χa−1^aq^a lim

N→ ∞

p^N_k

q^d

p^N

−q^d

×

0≤x<p^N

q^dx

−1^xE

k,^qdpN a/dx p^N

2_q

2_q^dd^k_q

0≤a<d

χa−1^aq^aEk,q^d

a d

Ek,χ,q,

pX

χxdμ^E_k,qx p_n

q

2_q 2_q^p

2_q^p 2_qpd

dⁿ_q^p

0≤a<d

χ pa

q^pa−1^aEn,q^dp

a d

χ p

p_n

q

2_q 2_q^p

2_q^p 2_qpd

dⁿ_q^p

0≤a<d

χaq^pa−1^aEn,q^dp

a d

χ p

p_n

q

2_q 2_q^pEn,χ,q^p.

2.6 Therefore, we obtain the following theorem.

Theorem 2.2. Letχbe the Dirichlet character with conductord∈Nwithd≡1mod 2. Then one has

X

χxdμ^E_k,qx Ek,χ,q,

pX

χxdμ^E_k,qx χ p

pk q

2_q 2_qp

Ek,χ,q^p,

X^∗

χxdμ^E_k,qx Ek,χ,q−χ p

pk q

2_q 2_q^pEk,χ,q^p.

2.7

Letk∈Z. From2.2, we note that

μ^E_k,q

adp^NZp

dp^Nk

q

dp^N

−q

q^a−1^aE_k,qdpN

a dp^N

dp^Nk

q

dp^N

−q

q^a−1^ak

i0

k i

E_i,qdpNq^ai a

dp^N _k−i

q^dpN

dp^Nk

q

dp^N

−q

q^a−1^ak

i0

k i

E_i,qdpNq^ai a^k−i_q dp^N_k−i

q

−qa

dp^N

−q

a^k_q

dp^Nk

q

dp^N

−q

q^a−1^ak

i1

k i

E_i,qdpNq^ai a^k−i_q dp^N_k−i

q

.

2.8

Thus, we have

dμ^E_k,qx x^k_qdμ_−qx. 2.9

Therefore, we obtain the following theorem and corollary.

Theorem 2.3. Fork≥0, one has

dμ^E_k,qx x^k_qdμ_−qx. 2.10

Corollary 2.4. Fork≥0, one has

X

dμ^E_k,qx

X

x^k_qdμ−qx Ek,q. 2.11

3. p -adic q - -Series

In this section, we assume thatq∈Cpwith|1−q|_p < p^−1/p−1. Letωdenote the Teichm ¨uller character modp. Forx∈X^∗, we setx_q x_q/ωx. Note that|x_q−1|_p< p^−1/p−1, and x^s_qis defined by expslog_px_q, for|s|_p≤1. Fors∈Zp,we define

p,q

s, χ

X^∗

x^−s_q χxdμ_−qx. 3.1

Thus, we have

p,q

−k, χω^k

X^∗

x^k_qχxdμ_−qx

X^∗

χxdμ^E_k,qx Ek,χ,q−χ

p p_k

q

2_q

2_q^pEk,χ,q^p, for k∈Z .

3.2

Since|x_q−1|_p < p^−1/p−1 forx∈X^∗, we havex^pn ≡1mod pⁿ.Letk ≡k modpⁿp− 1. Then we have

p,q

−k, χω^k

≡p,q

−k, χω^k

mod pⁿ

. 3.3

Therefore, we obtain the following theorem.

Theorem 3.1. Letk≡kmod p−1pⁿ. Then one has

Ek,χ,q− 2_q 2_q^pχ

p p_k

qEk,χ,q^p ≡E_k,χ,q− 2_q 2_q^pχ

p p_k

qE_k,χ,q^p

mod pⁿ

. 3.4

Acknowledgments

This paper was supported by Jangjeon Mathematical Society.

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