On Carlitz’s Type q-Euler Numbers Associated with the Fermionic P -Adic Integral on Z

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

Min-Soo Kim,

Taekyun Kim,

and Cheon-Seoung Ryoo

1Department of Mathematics, Korea Advanced Institute of Science and Technology, 373-1 Guseong-dong, Yuseong-Gu, Daejeon 305-701, Republic of Korea

2Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

3Department 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 new p-adic q-l-functionlp,qs, χfor Dirichlet’s characterχ, with the property thatlp,q−n, χ En,χn,q−χnppⁿ_qEn,χn,q^pforn0,1, . . . using the fermionic p-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−

−q_x

1q . 1.1

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

When one talks of q-analogue, q is variously considered as an indeterminate, a complex numberq∈C,or ap-adic numberq∈Cp.Ifq1t∈Cp,one normally assumes

|t|_p<1.We will further suppose that ordpt>1/p−1,so thatq^xexpxlogqfor|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

e^t1 ^∞

n0

Entⁿ

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 replacingEⁿbyEnsee7,18.

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

E^∗_0,q1,

qE^∗1_n

E^∗_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

qE1_n

En,q

⎧⎨

⎩

2 ifn0,

0 ifn >0, 1.5

with the usual convention of replacingEⁿbyEn,q.It was shown that lim_q→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,qtⁿ n! 2

∞ n0

−q_n

e^nqt, 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.

In this paper, we obtain the generating function of Carlitz’s type q-Euler numbers in thep-adic case. Also, we give Witt’s formula for Carlitz’s typeq-Euler numbers, which

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, χ

En,χn,q−χn

p

p ⁿ_qEn,χn,q^p, 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 p^N _q

p^N−1 a0

faq^a, 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

e^xytdμ−1 y

2e^xt e^t1 ^∞

n0

Enxtⁿ

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 f

2f0, 2.5

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

I−1 fn

−1ⁿ⁻¹I−1 f

2 n−1

i0

−1ⁿ⁻ⁱ⁻¹fi, 2.6

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

I−1 fn

I−1 f

2 n−1

i0

−1ⁱfi ifnis odd

I₋₁ fn

−I₋₁ f

2 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 that^xi∈Zpsee23, page 172. We putx ∈Cpwith ordpx > 1/p−1 and|1−q|p<1.We defineq^xforx∈Zpby

q^{x ∞}

i0

x i

q−1_i

, x_q^∞

i1

x i

q−1_i−1

. 2.9

If we setfx q^xin2.7, we have

I−1 q^x

2 qⁿ1

n−1

i0

−1ⁱqⁱ 2

q1 if nis odd

I−1 q^x

2 qⁿ−1

n−1

i0

−1ⁱ¹qⁱ 2

q1 ifnis even.

2.10

From2.10, we note that iffx q^x,then I−1q^x 2/q1,hence there is no need to consider bothodd and evencases. Thus, for eachl ∈ N,we obtainI₋₁q^lx 2/q^l1.

Therefore, we have

I−1

q^xxⁿ_q 1

1−qn

n l0

n l

−1^lI−1

q^l1x

1 1−qn

n l0

n l

−1^l 2 q^l11.

2.11

Also, iffx q^lxin2.5, then

I₋₁ q^lx1

I₋₁ q^lx

2f0 2. 2.12

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

I₋₁

q^x1x1ⁿ_q I₋₁

q^xxⁿ_q 1

1−q_nⁿ

l0

n l

−1^l

I₋₁

q^l1_x1

I₋₁

q^l1_x

2 1−q_nⁿ

l0

n l

−1^l0

2.13

is equivalent to

0I₋₁

q^x1x1ⁿ_q I₋₁

q^xxⁿ_q qI−1

q^x

1qxⁿ I−1

q^xxⁿ_q qI₋₁

q^x

n l0

n l

q^lx^l

I₋₁

q^xxⁿ_q

q n

l0

n l

q^lI−1

q^xx^l

I−1

q^xxⁿ_q .

2.14

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

I−1

q^xxⁿ_q

Zp

xⁿ_qq^xdμ−1x

lim

N→ ∞ p^N−1

a0

1

1−q_nⁿ

i0

n i

−1ⁱq^ia

−q_a 1

1−q_nⁿ

i0

n i

−1ⁱ lim

N→ ∞ p^N−1

a0

−1^a qⁱ¹a

1 1−q_nⁿ

i0

n i

−1ⁱ 2 1qⁱ¹

2.15

is equivalent to ∞ n0

I−1

q^xxⁿ_qtⁿ n! ^∞

n0

1

1−q_nⁿ

i0

n i

−1ⁱ 2 1qⁱ¹

tⁿ n!

2 ∞ n0

−q_n e^nqt.

2.16

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

q n

l0

n l

q^lEl,qEn,q

⎧⎨

⎩

2 ifn0,

0 ifn >0, 2.17

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

Fqt 2 ∞ n0

−q_n

e^nqt∞

n0

En,qtⁿ

n!. 2.18

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

q

qE1_n

En,q

⎧⎨

⎩

2 ifn0,

0 ifn >0, 2.19

with the usual convention of replacingEⁿbyEn,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−q_nⁿ

i0

n i

−1ⁱ 2 1qⁱ¹

Zp

xⁿ_qq^xdμ−1x. 2.20 Carlitz’s typeq-Euler numbersEnEn,qcan be determined inductively by

q

qE1_n

En,q

⎧⎨

⎩

2 ifn0,

0 ifn >0, 2.21

with the usual convention of replacingEⁿbyEn,q.

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

Fqx, t 2 ∞ k0

−1^kq^ke^kxqt∞

n0

En,qxtⁿ

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 2e^xqt−qe^tFq

x, qt

. 2.23

From2.22, one gets the following.

Lemma 2.2. 1Fqx, t 2e^t/1−q_∞

j01/q−1^jq^xj1/1q^j1t^j/j!.

2En,qx 2_∞

k0−1^kq^kkxⁿ_q.

It is clear from1and2of Lemma2.2that

En,qx 2 1−q_nⁿ

k0

n k

−1^k 1q^k1q^xk,

m−1

k0

−1^kq^kkxⁿ_q ^∞

k0

−1^kq^kkxⁿ_q −^∞

k0

−1^kmq^kmkmxⁿ_q 1

2

En,qx −1^m1q^mEn,qxm .

2.24

From2.24, we may state the following.

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

k0ⁿ_k−1^k/1q^k1q^xk, (2)_m−1

k0−1^kq^kkxⁿ_q 1/2En,qx −1^m1q^mEn,qxm.

Proposition 2.4. Forn∈Z,the value of

Zpxyⁿ_qq^ydμ−1yisn! times the coefficient oftⁿin the formal expansion of 2_∞

k0−1^kq^ke^kxqtin powers oft.That is,En,qx

Zpxyⁿ_qq^ydμ−1y.

Proof. From2.3, we have

Zp

q^kxyq^ydμ−1 y

q^xk lim

N→ ∞ p^N−1

a0

−q^k1_a

2q^xk

1q^k1, 2.25

which leads to

Zp

xy ⁿ_qq^ydμ−1 y

2 n k0

n k

1

1−qn−1^k

Zp

q^kxyq^ydμ−1 y

2 1−q_nⁿ

k0

n k

−1^k 1q^k1q^xk.

2.26

The result now follows by using1of Proposition2.3.

Corollary 2.5. Ifn∈Z,then

En,qx ⁿ

k0

n k

x^n−k_q q^kxEk,q. 2.27

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

Xlim

←N

Z dp^NZ

, X^∗

0<a<dp

^a,p1

adpZp,

adp^NZp

x∈X|x≡a

moddp^N ,

2.28

wherea∈ Zwith 0 ≤ a < dp^Ncf. 7,9. Note that the natural mapZ/dp^NZ → Z/p^NZ 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

χa

xy ⁿ_qq^ydμ−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−1^mq^me^mxqt∞

n0

En,χ,qxtⁿ

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

−q_m^∞

n0

1 1−qn

n i0

n i

−1ⁱq^imxtⁿ n!

2 ∞ n0

1

1−q_nⁿ

i0

n i

−1ⁱq^ix

×^d−1

j0

∞ l0

χ

jdl

−q_jdl

q^ijdltⁿ n!

2 ∞ n0

1

1−q_n^d−1

j0

χ j

−qjn i0

n i

−1ⁱ q^ixj 1q^di1

tⁿ n!,

2.32

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

En,χ,qx 1

1−qn

d−1 j0

χ j

−q_jⁿ

i0

n i

−1ⁱq^ixj 2 1q^di1

1 1−q_n^d−1

j0

χ j

−q_jⁿ

i0

n i

−1ⁱq^ixj× lim

N→ ∞ p^N−1

l0

−1^l q^di1l

lim

N→ ∞ d−1

j0 p^N−1

l0

χ

jdl 1 1−qn

n i0

n i

−1ⁱq^ijdlx×−1^jdlq^jdl

lim

N→ ∞ dp^N−1

a0

χa 1

1−q_nⁿ

i0

n i

−1ⁱq^iax

−q_a

X

χ y

xy ⁿ_qq^ydμ−1 y

,

2.33

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

Theorem 2.6.

En,χ,qx 1

1−q_n^d−1

j0

χ j

−qjn i0

n i

−1ⁱq^ixj 2

1q^di1, 2.34

wheren∈Zandx∈Zp.

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

x x_qω⁻¹x x_q

ωx. 2.35

Note that since|x −1|p< p^−1/p−1,x^sis defined by expslog_pxfor|s|p ≤1cf.10,12, 21. One notes thatx^sis analytic fors∈Zp.

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

lp,q

s, χ

X^∗

x^−sχxq^xdμ−1x. 2.36

Then,lp,qs, χis analytic fors∈Zp.

The values of this function at nonpositive integers are given by the following.

Theorem 2.7. For integersn≥0,

lp,q

−n, χ

En,χn,q−χn

p

p ⁿ_qEn,χn,q^p, 2.37

whereχnχω⁻ⁿ.In particular, ifχωⁿ,thenlp,q−n, ωⁿ En,q−pⁿ_qEn,q^p. Proof.

lp,q

−n, χ

X^∗

xⁿχxq^xdμ−1x

X

xⁿ_qχnxq^xdμ−1x−

X

px ⁿ_qχn

px

q^pxdμ−1 px

X

xⁿ_qχnxq^xdμ₋₁x− p ⁿ_qχn

p

X

xⁿ_q^pχnxq^pxdμ₋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−1^mq^me^mxqt 2

F−1

a0

χa

−q_a^∞

k0

−q_Fk

e^Fqkxa/FqFt

^∞

n0

Fⁿ_q^F−1

a0

χa

−q_a En,q^F

xa F

tⁿ n!.

2.39

Therefore, we obtain the following

En,χ,qx Fⁿ_q^F−1

a0

χa

−q_a En,q^F

xa F

. 2.40

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

χn

p

p ⁿ_qEn,χn,q^p χn

p p ⁿ_q

F p

n q^p

F/p−1

a0

χna

−q^p_a

E_n,q^pF/p a

F/p

Fⁿ_q^F

a0p|a

χna

−q_a En,q^F

a F

.

2.41

Thus, we have

En,χn,q−χn

p

p ⁿ_qEn,χn,q^p Fⁿq F−1

a0pa

χna

−q_a En,q^F

a F

. 2.42

By Corollary2.5, we easily see that

En,q^F

a F

ⁿ

k0

n k

a F

_n−k

q^F q^kaEk,q^F

F⁻ⁿq aⁿ_qⁿ

k0

n k

F a

_k

q^a

q^kaEk,q^F.

2.43

From2.42and2.43, we have

En,χn,q−χn

p

p ⁿ_qEn,χn,q^p Fⁿq F−1

a0pa

χna

−q_a En,q^F

a F

^F−1

a0pa

χaaⁿ

−q_a^∞

k0

n k

F a

k q^a

q^kaEk,q^F,

2.44

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

lp,q

−n, χ ^F−1

a0pa

χaaⁿ

−qa∞ k0

n k

F a

_k

q^a

q^kaEk,q^F, 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^−sχxq^xdμ₋₁x, s∈Zp. 2.46 Then,lp,qs, χis analytic fors∈Zpand

lp,q

s, χ ^F−1

a0pa

χaa^−s

−qa∞ k0

−s k

F a

_k

q^a

q^kaEk,q^F. 2.47

Furthermore, forn∈Z

lp,q

−n, χ

En,χn,q−χn

p

p ⁿ_qEn,χn,q^p. 2.48

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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