Volume 2008, Article ID 371295,9pages doi:10.1155/2008/371295
Research Article
Note on
q
-Extensions of Euler Numbers and
Polynomials of Higher Order
Taekyun Kim,1Lee-Chae Jang,2 and Cheon-Seoung Ryoo3
1The School of Electrical Engineering and Computer Science (EECS), Kyungpook National University,
Taegu 702-701, South Korea
2Department of Mathematics and Computer Science, KonKuk University,
Chungju 143-701, South Korea
3Department of Mathematics, Hannam University, Daejeon 306-791, South Korea
Correspondence should be addressed to Cheon-Seoung Ryoo,[email protected]
Received 1 November 2007; Accepted 22 December 2007
Recommended by Paolo Emilio Ricci
In 2007, Ozden et al. constructed generating functions of higher-order twistedh, q-extension of Euler polynomials and numbers, by usingp-adic,q-deformed fermionic integral onZp. By
apply-ing their generatapply-ing functions, they derived the complete sums of products of the twistedh, q -extension of Euler polynomials and numbers. In this paper, we consider the newq-extension of Eu-ler numbers and polynomials to be different which is treated by Ozden et al. From ourq-Euler num-bers and polynomials, we derive some interesting identities and we constructq-Euler zeta functions which interpolate the newq-Euler numbers and polynomials at a negative integer. Furthermore, we study Barnes-typeq-Euler zeta functions. Finally, we will derive the new formula for “sums of prod-ucts ofq-Euler numbers and polynomials” by using fermionicp-adic,q-integral onZp.
Copyrightq2008 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 and notations
Throughout this paper we use the following notations. ByZp we denote the ring ofp-adic
ra-tional integers,Qdenotes the field of rational numbers,Qp denotes the field ofp-adic rational
numbers,Cdenotes the complex number field, andCpdenotes the completion of algebraic
clo-sure ofQp. Letνpbe the normalized exponential valuation ofCpwith|p|p p−νppp−1.When
one talks of q-extension, q is considered in many ways such as an indeterminate, a complex
numberq ∈C,orp-adic numberq ∈Cp.Ifq∈C, one normally assumes that|q|<1.Ifq ∈Cp,
we normally assume that|q−1|p < p−1/p−1so thatqxexpxlogqfor|x|
p ≤1.In this paper,
xq x:q 1−q
x
1−q 1.1
cf.1–5,22.
Hence, limq→1x xfor anyxwith|x|p ≤1 in the presentp-adic case. Letdbe a fixed
integer and letpbe a fixed prime number. For any positive integerN, we set
Xlim←
N Z
dpNZ
,
X∗
0<a<dp
a,p1
adpZp,
adpNZpx∈X|x≡amoddpN,
1.2
wherea∈Zlies in 0≤a < dpN. For any positive integerN,
μqadpNZp q
a
dpN
q
1.3
is known to be a distribution onX cf.1–20. From this distribution, we derive thep-adic,
q-integral onZpas follows:
Zp
fxdμqx lim
N→∞
1
pN
q pN−1
x0
qxfx, f ∈UDZp, 1.4
see1–23.
Higher-order twisted Bernoulli and Euler numbers and polynomials are studied by
many authors see for detail 1–21. In 14 Ozden et al. constructed generating functions
of higher-order twistedh, q-extension of Euler polynomials and numbers, by usingp-adic,
q-deformed fermionic integral onZp. By applying their generating functions, they derived the
complete sums of products of the twistedh, q-extension of Euler polynomials and numbers,
see14,15. In this paper, we consider the newq-extension of Euler numbers and polynomials
to be different which is treated by Ozden et al. From our q-Euler numbers and polynomials,
we derive some interesting identities and we constructq-Euler zeta functions which
interpo-late the newq-Euler numbers and polynomials at a negative integer. Furthermore, we study
Barnes-typeq-Euler zeta functions. Finally, we will derive the new formula for “sums of
prod-ucts ofq-Euler numbers and polynomials” by using fermionicp-adic,q-integral onZp.
2.q-extension of Euler numbers
In this section we assume thatq ∈ Cwith|q| < 1. Now we consider theq-extension of Euler
polynomials as follows:
Fqx, t
2q
qet1e
xt∞
n0
En,qx
n! t
Note that
lim
q→1Fqx, t Fx, t
2 et1e
xt∞
n0
Enx
n! t
n. 2.2
In the special casex0,theq-Euler polynomialEn,q0 En,q will be calledq-Euler numbers.
It is easy to see thatFqx, tis analytic function inC. Hence we have
∞
n0
En,qx
n! t
n 2q
qet1e
xt 2 q
∞
n0
−1nqnenxt. 2.3
If we take thekth derivative att0 on both sides in2.3, then we have
Ek,qx 2q ∞
n0
−1nqnnxk. 2.4
From2.4we can defineq-zeta function which interpolatingq-Euler numbers at negative
in-teger as follows.
Fors∈C,we define
ζqs, x 2q
∞
n0
−1nqn
nxs, s∈C. 2.5
Note that ζqs, xis analytic in complex s-plane. If we take s −kk ∈ Z, then we have
ζq−k, x Ek,qx.
By2.4and2.5, we obtain the following.
Theorem 2.1. Fork∈Z,
Ek,qx 2q ∞
n0
−1nqnnxk. 2.6
LetFq0, t Fqt.Then
2q
n−1
k0
−1kqkekt 2q
1qet −2q
−1nqnent
1qet
Fqt−−1nqnFqn, t.
2.7
From2.7, derive
∞
k0
2q
n−1
l0
−1lqllk
tk
k!
∞
k0
Ek,q−−1nqnEk,qn tk
k!. 2.8
Theorem 2.2. Letn∈N, k∈Z. Ifn≡0 mod 2, then
Ek,q−qnEk,qn 2q n−1
l0
−1lqllk. 2.9
Ifn≡1 mod 2, then
Ek,qqnEk,qn 2q n−1
l0
−1lqllk. 2.10
Forw1, w2, . . . , wr ∈C,consider the multipleq-Euler polynomials of Barnes-type as follows:
Fqrw1, w2, . . . , wr |x, t
2
r qext
qew1t1qew2t1· · ·qewrt1
∞
n0
En,q
w1, . . . , wr |x tn
n!, wheremax1≤i≤r |witlogq|< π.
2.11
For x 0, En,qw1, . . . , wr | 0 En,qw1, . . . , wr will be called the multiple q-Euler
numbers of Barnes type. It is easy to see thatFr
qw1, w2, . . . , wr |x, tis analytic function in the
given region. From2.11, we derive
2rq
∞
n1,...,nr0
−qri1nieri1niwixt
∞
n0
En,q
w1, . . . , wr |x tn
n!. 2.12
By thekth differentiation on both sides in2.12, we see that
2rq ∞
n1,...,nr0
−qri1ni
r
i1
niwix k
Ek,q
w1, . . . , wr |x
. 2.13
From2.12, we can derive the following Barnes-type multipleq-Euler zeta function as follows.
Fors∈C, define
ζr,qw1, w2, . . . , wr |s, x
2rq
∞
n1,...,nr0
−1n1···nrqn1···nr
n1w1n2w2· · · nrwrx
s. 2.14
By2.13and2.14, we obtain the following.
Theorem 2.3. Fork∈Z, w1, w2, . . . , wr ∈C,
ζr,qw1, w2, . . . , wr | −k, x
Ek,q
w1, w2, . . . , wr |x
. 2.15
Letχbe the primitive Drichlet character with conductorfodd∈N. Then we consider
generalized Euler numbers attached toχas follows:
Fχ,qt
2qfa−10−1aqaχaeat
qfeft1
∞
n0
En,χ,q
tn
where |logq t| < π/f. The numbers En,χ,q will be called the generalized q-Euler numbers
attached toχ. From2.16, note that
Fχ,qt
2qfa−10−1aqaχaeat
qfeft1
2q
f−1
a0
−1aqaχa
∞
n0
qnf−1neanft
2q
∞
n0
f−1
a0
−1anfqanfχanfeanft
2q
∞
n0
−1nqnχnent
∞
n0
En,χ,qt n
n!.
2.17
Thus,
Ek,χ,q d k
dtkFχ,qt t
0
2q
∞
n1
−1nqnχnnk, k∈N. 2.18
Therefore, we can define the Dirichlet-typel-function which interpolates at negative integer as
follows.
Fors∈C, we definelqs, χas
lqs, χ 2q ∞
n1
−1nqnχn
ns . 2.19
By2.18and2.19, we obtain the following.
Theorem 2.4. Fork∈Z,
lq−k, χ Ek,χ,q. 2.20
From2.1and the definition ofq-Euler numbers, derive
Fqt, x
2q
qet1e
xt∞
n0
En,qt n
n!
∞
l0
xl
l!t
l
∞
m0
m
n0
En,q
m n
xm−n
tm m!.
2.21
By2.21it is shown that
En,qx n
m0
Em,q
n m
Forf(=odd)∈N, note that ∞
n0
En,qx
tn
n!
2q
qet1e
xt
2q 1
qfeft1 f−1
a0
−1aqaeax/fft
2q
2qf
f−1
a0
−1aqa
2qfeax/fft qfeft1
2q 2qf
f−1
a0
−1aqa
∞
n0
En,qf
ax
f
fntn
n! .
2.23
Thus, we have the distribution relation forq-Euler polynomials as follows.
Theorem 2.5. Forf(=odd)∈N,
En,qx
fn2 q
2qf
f−1
a0
−1aqaEn,qf
ax
f
. 2.24
Fork, n∈Nwithn≡0(mod 2), it is easy to see that
2q n−1
l0
−1l−1qllk qnEk,qn−Ek,q
qn
k
m0
k m
nk−mEm,q−Ek,q
qn
k−1
m0
k m
Em,qnk−m
qn−1Ek,q.
2.25
Therefore, we obtain the following.
Theorem 2.6. Fork, n∈Nwithn≡0 (mod 2),
2q
n−1
l0
−1l−1qllk qn
k−1
m0
k m
Em,qnk−m
qn−1Ek,q. 2.26
3. Witt-type formulae onZp inp-adic number field
In this section, we assume thatq∈Cp with|1−q|p <1.gis a uniformly differentiable function
at a pointa∈Zp, and writeg ∈UDZpif the difference quotient
Fgx, y
gx−gy
x−y 3.1
has a limitfaasx, y→a, a. Forg∈UDZp, an invariantp-adic,q-integral is defined as
Iqg
Zp
gxdμqx lim
N→∞
1
pN
q pN−1
x0
The fermionicp-adic,q-integral is also defined as
I−qg
Zp
gxdμ−qx lim
N→∞
2q
1qpN
pN−1
x0
gx−1xqx 3.3
see4.
From3.3, we have the integral equation as follows:
qI−qg1 I−qg 2qg0, g1x gx1. 3.4
If we takegx etx, then we have
Iq
etx
Zp
extdμ−qx 2q
qet1. 3.5
From3.5, we note that
∞
n0
Zp
xndμ−qxt
n
n!
2q
qet1
∞
n0
En,qt n
n!. 3.6
By comparing the coefficient on both sides, we see that
Zp
xndμ−qx En,q, n∈Z. 3.7
By the same method, we see that
Zp
exytdμ−qy 2q
qet1e
xt∞
n0
En,qxt n
n!. 3.8
Hence, we have the formula of Witt’s type forq-Euler polynomial as follows:
Zp
xyndμ−qy En,qx, n∈Z. 3.9
Forn∈Z, letgnx gxn. Then we have
qnI−q
gn
−1n−1I−qg 2q n−1
l0
−1n−1−lqlgl. 3.10
Ifnis odd positive integer, then we have
qnI−q
gn
I−qg 2q n−1
l0
−1lqlgl. 3.11
Let χ be the primitive Drichlet character with conduct f odd ∈ N and let gx
χxext. From3.5we derive
I−q
χxext
X
χxetxdμ−qx
2q
f−1
a0−1
aqaχaeat
qfeft1
∞
n0
En,χ,qt n
n!.
3.12
Thus, we have the Witt formula for generalizedq-Euler numbers attached toχas follows:
X
4. Higher-orderq-Euler numbers and polynomials
In this section we also assume thatq ∈ Cp with |1−q|p < 1. Now we study on higher-order
q-Euler numbers and polynomials and sums of products ofq-Euler numbers. First, we try to
consider the multivariate fermionicp-adic,q-integral onZpas follows:
Zp
· · ·
Zp
rtimes
ea1x1a2x2···arxrtextdμ
−q
x1
· · ·dμ−qxr
2
r q
qea1t1qea2t1· · ·qeart1e
xt,
4.1
wherea1, a2, . . . , ar ∈Zp.
From4.1we consider the multipleq-Euler polynomials as follows:
2rq
qea1t1qea2t1· · ·qeart1e
xt∞
n0
En,q
a1, a2, . . . , ar |x tn
n!. 4.2
In the special casea1, a2, . . . , ar 1,1, . . . ,1,we write
En,q
a1, . . . , ar
rtimes
|xErn,qx. 4.3
Forx0, the multipleq-Euler polynomials will be called asq-Euler numbers of orderr.
From4.2we note that
En,q
a1, a2, . . . , ar |x
Zp
· · ·
Zp
rtimes
a1x1· · · arxrx nr
j1
dμ−qxj
. 4.4
It is easy to check that
En,q
a1, a2, . . . , ar |x
n
l0
n l
xn−lEl,q
a1, a2, . . . , ar
, 4.5
whereEn,qa1, a2, . . . , ar En,qa1, a2, . . . , ar | 0. Multinomial theorem is well known as
fol-lows:
r
j1
xj n
l1,...,lr≥0
l1···lrn
n l1, . . . , lr
r
a1
xla
a, 4.6
where
n l1, . . . , lr
n!
l1!l2!· · ·lr!. 4.7
By4.2and4.6we easily see that
Ern,qx n
m0
l1,...,lr≥0
l1···lrm
n m
m l1, . . . , lr
xn−m
r
j1
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