Volume 2010, Article ID 682072,6pages doi:10.1155/2010/682072
Research Article
Generalized
q
-Euler Numbers and Polynomials of
Higher Order and Some Theoretic Identities
T. Kim and Y. H. Kim
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Correspondence should be addressed to Y. H. Kim,[email protected]
Received 25 February 2010; Accepted 26 April 2010
Academic Editor: Yeol J. E. Cho
Copyrightq2010 T. Kim and Y. H. Kim. 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-Euler numbers and polynomials of higher order attached to Dirichlet’s characterχ. We derive some theoretic identities involving the generalizedq-Euler numbers and polynomials of higher order.
1. Introduction
Let C be the complex number field. We assume that q ∈ C with |q| < 1 and the q
-number is defined by xq 1 − qx/1 −q in this paper. The q-factorial is given by
nq! nqn−1q· · ·2q1qand theq-binomial formulae are known as
x:qn n
i1
1−xqi−1
n
i0
n i qq
i
2
−xi ,
1
x:qn n
i1
1
1−xqi−1
∞
i0
ni−1
i qx
i,
1.1
whereniq nq!/n−iq!iq! nqn−1q· · ·n−i1q/iq!see1–3.
After Carlitz had constructed the q-Bernoulli numbers and polynomials, many
mathematicians have studied forq-Bernoulli andq-Euler numbers and polynomialssee1–
29. Since theq-extensions of Euler numbers and polynomials contain interesting properties
to study various fields of mathematical physics and number theory, many researchers
identities from themsee2–5,8–19. The purpose of this paper is to give a new approach
to theq-Euler numbers and polynomials of higher order attached to Dirichlet’s characterχ.
From this, we will derive some theoretic identities involving generalized q-Euler numbers
and polynomials of higher order.
In Section 2, we present new generating functions which are related to q-Euler
numbers and polynomials of higher order attached to χ. We obtain distribution relations
for theq-Euler polynomials attached toχ, and have some identities involving theseq-Euler
polynomials. Using the Cauchy residue theorem, we show that theseq-extensions of theq
-l-function of orderr attached toχinterpolate theq-Euler polynomials of orderrat negative
integers.
2.
q
-Euler Polynomials of Higher Order Attached to
χ
LetNbe the set of natural numbers andZ N∪ {0}. Ford∈Nwithd≡1mod2, letχbe
Dirichlet’s character with conductord. Forr ∈ Zandh∈ Z, we will study the generalized
q-Euler andh, q-Euler polynomials and numbers of orderrattached toχ, respectively.
It is known that the Euler polynomials are defined by 2/et 1ext
∞
n0Enxtn/n!, for|t| < π. In the special casex 0, En En0are called thenth Euler
numberssee28,29.
First, we define the generalizedq-Euler polynomials attached toχas follows:
∞
n0
En,χ,qxt n
n! 2
∞
m0
χm−1memxqt, 2.1
whereEn,χ,qxare called thenth generalizedq-Euler polynomials attached toχ. In the special
casex0,En,χ,qEn,χ,q0are called thenth generalizedq-Euler numbers attached toχ.
By2.1, we see that
En,χ,qx 2 ∞
m0
χm−1mmxnq
2
1−qn
d−1
a0
χa−1a n
l0
n l
−1l qlxa
1qld .
2.2
Now we consider theq-Euler polynomials of orderrattached toχas follows:
Fq,χrt, x 2r ∞
m1,...,mr0
r
i1
χmi −1m1···mrem1···mrxqt
∞
n0
En,χ,qr x tn n!,
2.3
whereEn,χ,qr xare called thenth generalizedq-Euler polynomials of orderr attached toχ.
In the special casex0,En,χ,qr En,χ,qr 0are called thenth generalizedq-Euler numbers of
From2.3, we note that
En,χ,qr x 2r ∞
m1,...,mr0
r
i1
χmi −1m1···mrm1· · ·mrxnq
2r
1−qn
d−1
a1,...,ar0
r
i1
χai −1a1···ar n
l0
n l
−1lqlxrj1aj
1qldr .
2.4
Thus we have
En,χ,qr x 2r d−1
a1,...,ar0
r
i1
χai −1a1···ar
×∞ m0
mr−1
m
−1m
xa1· · ·ar dmnq.
2.5
That is,
Fq,χrt, x 2r d−1
a1,...,ar0
r
i1
χai −1a1···ar × ∞
m0
mr−1
m
exa1···ardmqt. 2.6
In the viewpoint ofh-extension ofEn,χ,qr x, we can define the generalizedh, q-Euler
polynomials of orderrattached toχas follows:
Fq,χh,rt, x 2r ∞
m1,...,mr0
r
i1
χmi −1m1···mrq r
j1h−jmjem1···mrxqt
∞
n0
En,χ,qh,rxt n
n!,
2.7
whereEn,χ,qh,rxare called the nth generalized h, q-Euler polynomials of order r attached
toχ. In the special casex 0,En,χ,qh,r En,χ,qh,r0are called thenth generalizedh, q-Euler
numbers of orderrattached toχ.
By2.7, we see that
En,χ,qh,rx 2rdnq ∞
m0
mr−1
m
q
−1m qdh−rm
× d−1 a1,...,ar0
r
i1
χai −1a1···arq r
j1h−jajmxa1· · ·ar d
n
qd.
That is,
Fq,χh,rt, x 2r ∞
m0
mr−1
m
q
−1m qdh−rm
× d−1 a1,...,ar0
r
i1
χai −1a1···arq r
j1h−jajexa1···ardmqt.
2.9
Lethr. Then we have
En,χ,qr,rx
2r
1−qn
d−1
a1,...,ar0
r
i1
χai −1 r
j1ajqjr1ajr−j×
n
l0
n l
−1l
qlxrj1aj
−qld :qd r
2rdnq ∞
m0
mr−1
m
q
−1m
× d−1 a1,...,ar0
r
i1
χai −1 r
j1ajqrj1r−jajm xa1· · ·ar d
n
qd.
2.10
By2.3,2.9, and2.10, we obtain the following equations:
2rqmxd−1
a1,...,ar0
r i1χai
qrj1m−jaj−1rj1aj
−qdm−r:qd r
m
l0
m l
q−1lEl,χ,q0,rx,
qdh−1En,χ,qh,rxd En,χ,qh,rx 2 d−1
l0
χl−1lEn,qh−1,r−1x.
2.11
In the special caser 1, we note that
Fq,χh,1t, x ∞
n0
En,χ,qh,1xt n
n!. 2.12
By2.12, we see that
Fq,χh,1t, x 2 ∞
n0
χnqh−1n−1nenxqt. 2.13
Hence
En,χ,qh,1x 2 ∞
m0
χmqh−1m−1mmxnq
2
1−qn
d−1
a0
χa−1a n
l0
n l
−1l qlxa
1qld .
Fors∈Randx∈CwithRx>0, we have
1
Γs
∞
0
Fq,χr−t, xts−1dt2r ∞
m1,...,mr0
−1m1···mrqrj1h−jmjr
i1χmi
m1· · ·mrxsq
. 2.15
By2.15, we can define the followingq-l-function of orderr.
Definition 2.1. Fors∈C,x∈RwithRx>0, we define theq-l-function as
lqh,r
s, x|χ2r
∞
m1,...,mr0
−1m1···mrqrj1h−jmjr
i1χmi
m1· · ·mrxsq
. 2.16
Note thatlqh,rs, x |χis analytic in whole complexs-plane. By2.7,2.15, and the
Cauchy residue theorem, we obtain the following theorem.
Theorem 2.2. Forn∈Z, one has
lqh,r
−n, x|χEn,χ,qh,rx. 2.17
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