New Approach to q-Euler Numbers and Polynomials

doi:10.1155/2010/431436

Research Article

New Approach to q-Euler Numbers and Polynomials

Taekyun Kim,

Lee-Chae Jang,

Young-Hee Kim,

and Seog-Hoon Rim

1Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea

2Department of Mathematics and Computer Science, Konkuk University, Chungju 380-701, South Korea

3Department of Mathematics Education, Kyungpook National University, Taegu 702-701, South Korea

Correspondence should be addressed to Young-Hee Kim,[email protected] Received 11 January 2010; Accepted 14 March 2010

Academic Editor: Binggen Zhang

Copyrightq2010 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.

We give a new construction of theq-extensions of Euler numbers and polynomials. We present new generating functions which are related to theq-Euler numbers and polynomials. We also consider the generalizedq-Euler polynomials attached to Dirichlet’s characterχand have the generating functions of them. We obtain distribution relations for theq-Euler polynomials and have some identities involvingq-Euler numbers and polynomials. Finally, we derive theq-extensions of zeta functions from the Mellin transformation of these generating functions, which interpolate theq- Euler polynomials at negative integers.

1. Introduction

LetCbe the complex number field. We assume thatq∈Cwith|q|<1 and that theq-number is defined byx_q 1−q^x/1−qin this paper.

Recently, many mathematicians have studied forq-Euler andq-Bernoulli polynomials and numbers see 1–18. Specially, there are papers for the q-extensions of Euler polynomials and numbers approaching with two kinds of viewpoint among remarkable paperssee7,10. It is known that the Euler polynomials are defined by2/e^t1e^xt _∞

n0Enxtⁿ/n!, for |t| < π, and En En0 are called the nth Euler numbers. The recurrence formula for the original Euler numbersEnis as follows:

E01, E1ⁿEn0, ifn >0 1.1

see7,10. As for theq-extension of the recurrence formula for the Euler numbers, Kim10 had the following recurrence formula:

E_0,q^∗ 2_q

2 , and

qE^∗1_n

E^∗_n,q

⎧⎨

⎩

2_q ifn0,

0 ifn≥1, 1.2

with the usual convention of replacingE^∗n byE^∗_n,q. Many researchers have made a wider and deeper study of theq-number up to recentlysee1–18. In the field of number theory and mathematical physics, zeta functions and l-functions interpolating these numbers in negative integers have been studied by Cenkci and Can3, Kim 4–12, and Ozden et al.

16–18.

This research for q-Euler numbers seems to be motivated by Carlitz who had constructed the q-Bernoulli numbers and polynomials for the first time. In 1,2, Carlitz considered the recurrence formulae for theq-extension of the Bernoulli numbers as follows:

B0,q1,

qB1k−Bk,q

⎧⎨

⎩

1 ifk1,

0 ifk >1, 1.3

with the usual convention of replacingB^kbyBk,q. These numbers diverge whenq1, and so Carlitz modified and constructed them as following:

β0,q1, q

qβ1_k

−βk,q

⎧⎨

⎩

1 ifk1,

0 ifk >1, 1.4

with the usual convention of replacingβ^kbyβk,q. From this, it was shown that lim_q→1βk,q Bk. HereBkare the Bernoulli numbers.

Lately, Carlitz’sq-Bernoulli numbers have been studied actively by many mathemati- cians in the field of number theory, discrete mathematics, analysis, mathematical physics, and so onsee3–18.

The purpose of this paper is to give a new construction of the q-extensions of Euler numbers and polynomials. It is expected that new constructedq-Euler numbers and polynomials in this paper are more useful to be applied to various areas related to number theory. In this paper, we present new generating functions which are related to q-Euler numbers and polynomials. We also consider the generalizedq-Euler polynomials attached to Dirichlet’s characterχwith an odd conductor and have the generating functions of them. We obtain distribution relations for theq-Euler polynomials, and have some identities involving theq-Euler numbers and polynomials. Finally, we derive theq-extensions of zeta functions from the Mellin transformation of these generating functions. Using the Cauchy residue theorem and Laurent series, we show that theseq-extensions of zeta functions interpolate theq-Euler polynomials at negative integers.

2. New Approach to q -Euler Numbers and Polynomials

LetNbe the set of natural numbers andZN∪ {0}. Forq∈Cwith|q|<1, let us define the q-Euler polynomialsEn,qxas follows:

Fqt, x 2 ∞ m0

−1^me^mxqt∞

n0En,qxtⁿ

n!· 2.1

Note that

qlim→1Fqt, x 2

e^t1e^xt∞

n0

Enxtⁿ

n!, for|t|< π, 2.2

whereEnxare called thenth Euler polynomials. In the special casex0,En,qEn,q0are called thenthq-Euler numbers. That is,

Fqt Fqt,0 2 ∞ m0

−1^me^mqt∞

n0

En,qtⁿ

n!· 2.3

From2.1and2.3, we note that Fqt,1 Fqt e^tFq

qt Fqt ^∞

l0

t^l l!

∞ m0

q^mEm,qt^m m!

^∞

n0

En,qtⁿ n!

^∞

nlmn0

n l0

n!q^lEl,q

l!n−l!

tⁿ n!^∞

n0

En,qtⁿ n!

^∞

n0

n l0

n l

q^lEl,q

tⁿ n! ^∞

n0

En,qtⁿ n!.

2.4

From2.1and2.3, we can easily derive the following equation:

Fqt,1 Fqt 2. 2.5

By2.4and2.5, we see thatE0,q1 and

n l0

n l

q^lEl,qEn,q

⎧⎨

⎩

2 ifn0,

0 ifn >0. 2.6

Therefore, we obtain the following theorem.

Theorem 2.1. Forn∈Z, one has

E0,q1,

qE1_n

En,q

⎧⎨

⎩

2 ifn0,

0 ifn >0, 2.7

with the usual convention of replacingEⁱbyEi,q.

Theorem 2.1of this paper seems to be more interesting and valuable than theq-Euler numbers which are introduced in7,10.

From2.1, we note that

Fqt, x e^xqtFq

q^xt ^∞

n0

n l0

n l

q^lxx^n−l_q El,q

tⁿ

n!. 2.8

Therefore, we obtain the following theorem.

Theorem 2.2. Forn∈Z, one has

En,qx ⁿ

l0

n l

x^n−l_q q^lxEl,q. 2.9

By2.1, we see that

Fqt, x ^∞

n0

2 ∞ m0

−1^mmxⁿ_q tⁿ

n!

^∞

n0

2 1−qn

n l0

n l

−1^lq^lx 1 1q^l

tⁿ n!.

2.10

By2.1and2.10, we obtain the following theorem.

Theorem 2.3. Forn∈Z, one has

En,qx 2 1−q_nⁿ

l0

n l

−1^lq^lx 1

1q^l. 2.11

From2.1, we can derive that, forf∈Nwithf≡1 mod2,

Fqt, x

f−1

a0

−1^aFq^f

t

f

q,xa f

· 2.12

By2.12, we see that, forf∈Nwithf≡1 mod2,

∞ n0

En,qxtⁿ n! ^∞

n0

⎛

⎝f_n

q f−1

a0

−1^aEn,q^f

xa f

⎞⎠tⁿ

n!. 2.13

Therefore, we obtain the following theorem.

Theorem 2.4Distribution relation forEn,qx. Forn∈Z,f ∈Nwithf ≡1mod2, one has

En,qx fn

q

f−1 a0

−1^aE_n,q^f xa

f

. 2.14

By2.1, we observe the following equations:

Fqt, n Fqt 2

n−1

l0

−1^le^lqt ifnodd,

Fqt, n−Fqt 2

n−1

l0

−1^l−1e^lqt ifneven.

2.15

By2.15, we obtain the following result.

Theorem 2.5. Letn∈Nwithn≡1mod2. Then one has

Em,qn Em,q2

n−1

l0

−1^ll^m_q, 2.16

wherem∈Z.

Let χ be Dirichlet’s character with an odd conductor f ∈ N. Then we define the generalizedq-Euler polynomials attached toχas follows:

Fq,χt, x 2 ∞ m0

χm−1^me^mxqt

^∞

n0

En,χ,qxtⁿ n!.

2.17

In the special casex 0,En,χ,q En,χ,q0are called thenth generalizedq-Euler numbers attached toχ. Thus the generating functions of the generalizedq-Euler numbers attached to χare as follows:

Fq,χt 2 ∞ m0

χm−1^me^mqt

^∞

n0

En,χ,q tⁿ n!.

2.18

By2.1and2.17, we see that

Fq,χt, x ^f−1

a0

−1^aχaF_q^f

t f

q,xa f

^∞

n0

⎛

⎝f_n

q f−1

a0

−1^aχaEn,q^f

xa f

⎞⎠tⁿ n!.

2.19

Therefore, we obtain the following theorem.

Theorem 2.6. Forn∈Z,f ∈Nwithf ≡1mod2, one has

En,χ,qx f_n

q f−1

a0

−1^aχaEn,q^f

xa f

. 2.20

By2.17and2.18, we see that

Fq,χt, x e^xqtFq,χ

q^xt ^∞

n0

n l0

n l

q^lxx^n−l_q El,χ,q

tⁿ

n!. 2.21

Hence

En,χ,qx ⁿ

l0

n l

q^lxx^n−l_q El,χ,q. 2.22

From2.17, we note that

Fq,χt, x ^∞

n0

⎛

⎜⎝ 2 1−q_n

f−1 a0

−1^aχaⁿ

l0

_n

l

−1^lq^lxa 1q^lf

⎞

⎟⎠tⁿ

n!. 2.23

From2.17and2.23, we have

En,χ,qx 2

1−q_n

f−1

a0

−1^aχaⁿ

l0

_n

l

−1^lq^lxa

1q^lf 2

∞ m0

χm−1^mmxⁿ_q.

2.24

In2.19, it is easy to show that

qlim→1Fq,χt, x

⎛

⎝2f−1

a0−1^aχae^at e^ft1

⎞

⎠e^xt∞

n0

En,χxtⁿ

n!, 2.25

whereEn,χxare called thenth generalized Euler polynomials attached toχ.

Fors∈ C, we now consider the Mellin transformation for the generating function of Fqt, x. That is,

1 Γs

_∞

0

Fq−t, xt^s−1dt2 ∞ n0

−1ⁿ

nx^s_q, 2.26

fors∈C, andx /0,−1,−2, . . . .

From2.26, we define the zeta function as follows:

ζ^∗s, x ^∞

n0

−1ⁿ

nx^s_q, s∈C, x /0,−1,−2, . . . . 2.27

Note thatζ^∗s, xis analytic function in whole complexs-plane. Using the Laurent series and the Cauchy residue theorem, we have

ζ^∗−n, x En,qx, forn∈Z. 2.28

By the same method, we can also obtain the following equation:

1 Γs

_∞

0

Fq,χ−t, xt^s−1dt2 ∞ n0

χn−1ⁿ

nx^s_q . 2.29 Fors∈C, we define Dirichlet typeq-l-function as

lq

s, x|χ 2

∞ n0

χn−1ⁿ

nx^s_q , 2.30

wherex /0,−1,−2, . . . .Note thatlqs, x| χis also holomorphic function in whole complex s-plane. From the Laurent series and the Cauchy residue theorem, we can also derive the following equation:

lq

−n, x|χ

En,χ,qx, forn∈Z. 2.31

Remark 2.7. It is easy to see that

En,qx

Zp

xyn qdμ₋₁

y ,

En,X,qx

X

xy_n

qX y

dμ−1 y

,

2.32

see19, Lemma 1.

Acknowledgment

The present research has been conducted by the Research Grant of Kwangwoon University in 2010.

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