On Carlitz's Type Euler Numbers Associated with the Fermionic Adic Integral on

Volume 2010, Article ID 358986,13pages doi:10.1155/2010/358986

Research Article

On Carlitz’s Type

q

-Euler Numbers Associated with

the Fermionic

P

-Adic Integral on

Z

p

Min-Soo Kim,

_{Taekyun Kim,}

_{and Cheon-Seoung Ryoo}

1_{Department of Mathematics, Korea Advanced Institute of Science and Technology,}

373-1 Guseong-dong, Yuseong-Gu, Daejeon 305-701, Republic of Korea

2_{Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea} 3_{Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea}

Correspondence should be addressed to Taekyun Kim,[email protected]

Received 2 August 2010; Accepted 28 September 2010

Academic Editor: Jewgeni Dshalalow

Copyrightq2010 Min-Soo 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 consider the following problem in the paper of Kim et al.2010: “Find Witt’s formula for Carlitz’s typeq-Euler numbers.” We give Witt’s formula for Carlitz’s typeq-Euler numbers, which is an answer to the above problem. Moreover, we obtain a newp-adic q-l-functionlp,qs, χfor Dirichlet’s characterχ, with the property thatlp,q−n, χ En,χn,q−χnpp

n

qEn,χn,qpforn0,1, . . . using the fermionicp-adic integral onZp.

1. Introduction

Throughout this paper, letpbe an odd prime number. The symbol,Zp,Qp,andCpdenote the

rings ofp-adic integers, the field ofp-adic numbers, and the field ofp-adic completion of the algebraic closure ofQp,respectively. The p-adic absolute value inCpis normalized in such

way that|p|_pp−1_{.LetNbe the set of natural numbers andZN∪ {0}.}

As the definition ofq-number, we use the following notations:

x_q 1−q

x

1−q, x−q

1−−qx

1q . 1.1

Note that limq→1xqxforx∈Zp,whereqtends to 1 in the region 0<|q−1|p<1.

|t|p<1.We will further suppose that ordpt>1/p−1,so thatqxexpxlogqfor|x|p≤1.

Ifq∈C,then we assume that|q|<1.

After Carlitz 1, 2 gave q-extensions of the classical Bernoulli numbers and polynomials, theq-extensions of Bernoulli and Euler numbers and polynomials have been studied by several authors cf. 1–21. The Euler numbers and polynomials have been studied by researchers in the field of number theory, mathematical physics, and so oncf.

1,2,9,11,13–16,22,23. Recently, variousq-extensions of these numbers and polynomials have been studied by many mathematicianscf.6–8,10,12,17,18,20. Also, some authors have studied in the several area ofq-theorycf.3,4,16,19,24.

It is known that the generating function of Euler numbersFtis given by

Ft 2

et₁

∞

n0

Ent n

n!. 1.2

From1.2, we know the recurrence formula of Euler numbers is given by

E01, E1nEn0 ifn >0, 1.3

with the usual convention of replacingEn_byE

nsee7,18.

In17, theq-extension of Euler numbersE∗n,qare defined as

E∗0,q1,

qE∗1nE∗n,q

⎧ ⎨ ⎩

2 ifn0,

0 ifn >0,

1.4

with the usual convention of replacingE∗nbyE∗n,q.

As the same motivation of the construction in18, Carlitz’s typeq-Euler numbersEn,q

are defined as

E0,q 2

2_q, q

qE1nEn,q

⎧ ⎨ ⎩

2 ifn0,

0 ifn >0,

1.5

with the usual convention of replacingEn_byE

n,q.It was shown that limq→1En,q En,where

En is thenth Euler number. In the complex case, the generating function of Carlitz’s type

q-Euler numbersFqtis given by

Fqt

∞

n0

En,qt n

n! 2

∞

n0

−qn

enqt_{, 1.6}

whereqis a complex number with|q|< 1see18. The remark point is that the series on the right-hand side of1.6is uniformly convergent in the wider sense. Inp-adic case, Kim et al.18could not determine the generating function of Carlitz’s typeq-Euler numbers and Witt’s formula for Carlitz’s typeq-Euler numbers.

is a partial answer to the problem in 18. Moreover, we obtain a new p-adicq-l-function

lp,qs, χfor Dirichlet’s characterχ,with the property that

lp,q

−n, χEn,χn,q−χn

pp n_qEn,χn,qp, 1.7

forn∈Zusing the fermionicp-adic integral onZp.

2. Carlitz’s Type

q

-Euler Numbers in the

p

-Adic Case

Let UDZp be the space of uniformly differentiable functions on Zp. Then, the p-adic q

-integral of a functionf∈UDZponZpis defined by

Iq

f

Zp

fadμqa lim

N→ ∞

1

pN

q pN₋₁

a0

faqa, 2.1

cf.5–17,19,20,22. The bosonicp-adic integral onZpis considered as the limitq → 1,that

is,

I1

f

Zp

fadμ1a. 2.2

From2.1, we have the fermionicp-adic integral onZpas follows:

I−1

f lim

q→ −1Iq

f

Zp

fadμ−1a. 2.3

Using2.3, we can readily derive the classical Euler polynomials,Enx,namely

2

Zp

exytdμ−1

y 2e

xt

et₁

∞

n0

Enxt n

n!. 2.4

In particular, whenx0, En0 Enis the well-known the Euler numberscf.7,16,19.

By definition ofI−1f,we show that

I−1

f1

I−1

f2f0, 2.5

wheref1x fx1 see7. By2.5and induction, we obtain

I−1

fn

−1n−1I−1

f2

n−1

i0

wheren1,2, . . .andfnx fxn.From2.6, we note that

I−1

fn

I−1

f2

n−1

i0

−1ifi ifnis odd

I−1

fn

−I−1

f2

n−1

i0

−1i1fi ifnis even.

2.7

Forx∈Zpand any integeri≥0,we define

x

i

⎧ ⎪ ⎨ ⎪ ⎩

xx−1· · ·x−i1

i! ifi≥1, 1, ifi0.

2.8

It is easy to see thatxi∈Zpsee23, page 172. We putx ∈Cpwith ordpx > 1/p−1

and|1−q|p<1.We defineqxforx∈Zpby

qx ∞

i0

x i

q−1i, xq

∞

i1

x i

q−1i−1. 2.9

If we setfx qx_{in2.7, we have}

I−1

qx 2 qn₁

n−1

i0

−1iqi 2

q1 if nis odd

I−1

qx 2 qn₋₁

n−1

i0

−1i1qi 2

q1 ifnis even.

2.10

From2.10, we note that iffx qx_{,then I−}

1qx 2/q1,hence there is no need to

consider bothodd and evencases. Thus, for eachl ∈ N,we obtainI−1qlx 2/ql1.

Therefore, we have

I−1

qxxnq

1

1−qn

n

l0

n l

−1lI−1

ql1x

1

1−qn

n

l0

n l

−1l 2

ql1₁.

2.11

Also, iffx qlx_{in2.5, then}

I−1

qlx1_I

−1

On the other hand, by2.12, we obtain that

I−1

qx1x1n_qI−1

qxxnq

1

1−qn

n

l0

n l

−1l

I−1

ql1x1

I−1

ql1x

2

1−qn

n

l0

n

l

−1l0

2.13

is equivalent to

0I−1

qx1_x1n q

I−1

qx_xn q

qI−1

qx1qxnI−1

qxxnq

qI−1

qx

n

l0

n l

qlxl

I−1

qxxnq

q

n

l0

n

l

qlI−1

qxxlI−1

qxxnq

.

2.14

From the definition of fermionicp-adic integral onZpand2.11, we can derive

I−1

qxxnq

Zp

xnqqxdμ−1x

lim

N→ ∞

pN₋₁

a0

1

1−qn

n

i0

n

i

−1iqia_−qa

1

1−qn

n

i0

n i

−1i lim

N→ ∞

pN₋₁

a0

−1aqi1a

1

1−qn

n

i0

n i

−1i 2

1qi1

2.15

is equivalent to

∞

n0

I−1

qxxnq

_tn

n!

∞

n0

1

1−qn

n

i0

n i

−1i 2

1qi1

tn

n!

2

∞

n0

−qn

enqt_.

From2.12,2.13,2.14,2.15, and2.16, it is easy to show that

q

n

l0

n l

qlEl,qEn,q

⎧ ⎨ ⎩

2 ifn0,

0 ifn >0,

2.17

whereEn,qare Carlitz’s typeq-Euler numbers defined bysee18

Fqt 2

∞

n0

−qn

enqt

∞

n0

En,qt n

n!. 2.18

Therefore, we obtain the recurrence formula for the Carlitz’s typeq-Euler numbers as follows:

qqE1nEn,q

⎧ ⎨ ⎩

2 ifn0,

0 ifn >0,

2.19

with the usual convention of replacingEn_byE

n,q.Therefore, by2.16,2.18, and2.19, we

obtain the following theorem, which is a partial answer to the problem in18.

Theorem 2.1Witt’s formula forEn,q. Forn∈Z,

En,q 1

1−qn

n

i0

n i

−1i 2

1qi1

Zp

xnqqxdμ−1x. 2.20

Carlitz’s typeq-Euler numbersEnEn,qcan be determined inductively by

qqE1nEn,q

⎧ ⎨ ⎩

2 ifn0,

0 ifn >0, 2.21

with the usual convention of replacingEn_byE n,q.

Carlitz type q-Euler polynomials En,qx are defined by means of the generating

functionFqx, tas follows:

Fqx, t 2

∞

k0

−1kqkekxqt

∞

n0

En,qxt n

n!. 2.22

In the casesx0, En,q0 En,qwill be called Carlitz typeq-Euler numberscf.8,19. One

also can see that the generating functionsFqx, tare determined as solutions of

Fqx, t 2exqt−qetFq

x, qt. 2.23

Lemma 2.2. 1Fqx, t 2et/1−q∞j01/q−1jqxj1/1qj1tj/j!.

2En,qx 2∞k0−1kqkkxnq.

It is clear from1and2of Lemma2.2that

En,qx

2

1−qn

n

k0

n

k

−1k 1qk1q

xk_,

m−1

k0

−1kqkkxnq

∞

k0

−1kqkkxnq −

∞

k0

−1kmqkmkmxnq

1

2

En,qx −1m1qmEn,qxm

.

2.24

From2.24, we may state the following.

Proposition 2.3. Ifm∈Nandn∈Z,then (1)En,qx 2/1−qn

_n

k0nk−1k/1qk1qxk,

(2)m_k−₀1−1kqk_kxn

q 1/2En,qx −1m1qmEn,qxm.

Proposition 2.4. Forn∈Z,the value of_Z pxy

n

qqydμ−1yisn!times the coefficient oftnin the

formal expansion of2∞_k0−1kqkekxqt_{in powers oft.That is,E}

n,qx

Zpxy

n

qqydμ−1y.

Proof. From2.3, we have

Zp

qkxyqydμ−1

yqxk lim

N→ ∞

pN₋₁

a0

−qk1a 2qxk

1qk1, 2.25

which leads to

Zp

xy n_qqydμ−1

y2

n

k0

n k

1

1−qn−1

k

Zp

qkxyqydμ−1

y

2

1−qn

n

k0

n k

−1k 1qk1q

xk_.

2.26

The result now follows by using1of Proposition2.3.

Corollary 2.5. Ifn∈Z,then

En,qx n

k0

n k

Letd∈Nwithd≡1mod 2andpbe a fixed odd prime number. One sets

Xlim

←

N

_Z

dpN_Z

, X∗

0<a<dp a,p1

adpZp,

adpNZp

x∈X|x≡amoddpN,

2.28

wherea∈ Zwith 0 ≤ a < dpN_{cf. 7,9. Note that the natural mapZ/dp}N_{Z → Z/p}N_Z

induces

π:X−→Zp. 2.29

Hereafter, iffis a function onZp,one denotes by the samefthe functionf◦πonX.Namely

one considersfas a function onX.

Let χ be the Dirichlet character with an odd conductor d dχ ∈ N. Then, the

generalized Carlitz typeq-Euler polynomials attached toχare defined by

En,χ,qx

X

χaxy n_qqydμ−1

y, 2.30

wheren∈Zandx∈Zp.Then, one has the generating function of generalized Carlitz type

q-Euler polynomials attached toχ

Fq,χx, t 2

∞

m0

χm−1mqmemxqt

∞

n0

En,χ,qxt n

n!. 2.31

Now, fixed anyt∈Cpwith ordpt>1/p−1and|1−q|p<1.From2.31, one has

Fq,χx, t 2

∞

m0

χm−qm ∞

n0

1

1−qn

n

i0

n i

−1iqimxt

n

n!

2

∞

n0

1

1−qn

n

i0

n i

−1iqix

×d−1

j0

∞

l0

χjdl−qjdlqijdlt

n

n!

2

∞

n0

1

1−qn

d−1

j0

χj−qj

n

i0

n i

−1i q

ixj

1qdi1 tn

n!,

wherex∈Zpandd∈Nwithd≡1mod 2.By2.31and2.32, one can derive

En,χ,qx 1

1−qn

d−1

j0

χj−qj

n

i0

n i

−1iqixj 2

1qdi1

1

1−qn

d−1

j0

χj−qj

n

i0

n

i

−1iqixj_{× lim}

N→ ∞

pN₋₁

l0

−1lqdi1l

lim

N→ ∞

d−1

j0 pN₋₁

l0

χjdl 1

1−qn

n

i0

n i

−1iqijdlx×−1jdlqjdl

lim

N→ ∞

dpN₋₁

a0

χa 1

1−qn

n

i0

n i

−1iqiax−qa

X

χyxy n_qqydμ−1

y,

2.33

wherex∈Zpandd∈Nwithd≡1mod 2.Therefore, one obtains the following.

Theorem 2.6.

En,χ,qx

1

1−qn

d−1

j0

χj−qj

n

i0

n i

−1iqixj 2

1qdi1, 2.34

wheren∈Zandx∈Zp.

Letωdenote the Teichm ¨uller character modp.Forx∈X∗,one sets

x xqω−1x

xq

ωx. 2.35

Note that since|x −1|p< p−1/p−1,xsis defined by expslog_pxfor|s|p ≤1cf.10,12,

21. One notes thatxsis analytic fors∈Zp.

One defines an interpolation function for Carlitz typeq-Euler numbers. Fors∈Zp,

lp,q

s, χ

X∗x

−s

χxqxdμ−1x. 2.36

Then,lp,qs, χis analytic fors∈Zp.

Theorem 2.7. For integersn≥0,

lp,q

−n, χEn,χn,q−χn

pp n_qEn,χn,qp, 2.37

whereχnχω−n.In particular, ifχωn,thenlp,q−n, ωn En,q−pnqEn,qp.

Proof.

lp,q

−n, χ

X∗x

n_χxqx_dμ− 1x

X

xnqχnxqxdμ−1x−

X

px n_qχn

pxqpxdμ−1

px

X

xnqχnxqxdμ−1x−

p n_qχn

p

X

xnqpχ_nxqpxdμ₋₁x.

2.38

Therefore by2.30, the theorem is proved.

Letχbe the Dirichlet character with an odd conductorddχ ∈N.LetFbe a positive

integer multiple ofpandd.Then, by2.22and2.31, we have

Fq,χx, t 2

∞

m0

χm−1mqmemxqt

2

F−1

a0

χa−qa ∞

k0

−qFk

eFqkxa/FqFt

∞

n0

Fnq F−1

a0

χa−qaEn,qF

_xa

F _tn

n!.

2.39

Therefore, we obtain the following

En,χ,qx Fnq F−1

a0

χa−qaEn,qF

_xa

F

. 2.40

Ifχnp/0,thenp, dχn 1,so thatF/pis a multiple ofdχn.From2.40, we derive

χn

pp n_qEn,χn,qp χn

pp n_q

F p

n

qp

F/p−1

a0

χna

−qpa

E_n,qpF/p

a F/p

Fnq F

a0 p|a

χna

−qa

En,qF

_a F

.

Thus, we have

En,χn,q−χn

pp n_qEn,χn,qp F

n q

F−1

a0 pa

χna

−qa

En,qF

_a F

. _2.42

By Corollary2.5, we easily see that

En,qF

_a F

n

k0

n k

_a

F n−k

qF q

ka_E k,qF

F−qnanq n

k0 n k F a k

qaq

ka_E k,qF.

2.43

From2.42and2.43, we have

En,χn,q−χn

pp n_qEn,χn,qp F

n q

F−1

a0 pa

χna

−qa

En,qF

_a F

F−1 a0 pa

χaan−qa ∞

k0 n k F a k

qaq

ka_E k,qF,

2.44

sinceχna χaω−na.From Theorem2.7and2.44, we have

lp,q

−n, χF−1 a0 pa

χaan−qa ∞

k0 n k F a k

qaq

ka_E

k,qF, 2.45

forn∈Z.Therefore, we have the following theorem.

Theorem 2.8. LetFbe a positive integer multiple ofpandddχ,and let

lp,q

s, χ

X∗

x−s_χxqx_dμ

−1x, s∈Zp. 2.46

Then,lp,qs, χis analytic fors∈Zpand

lp,q

s, χ

F−1

a0 pa

χaa−s−qa ∞

k0 −s k F a k qa

qkaEk,qF. 2.47

Furthermore, forn∈Z

lp,q

−n, χEn,χn,q−χn

Acknowledgments

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science, and Technology 2010-0001654. The second author was supported by the research grant of Kwangwoon University in 2010.

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