Volume 2009, Article ID 136532,8pages doi:10.1155/2009/136532
Research Article
Note on the
q
-Extension of Barnes’ Type Multiple
Euler Polynomials
Leechae Jang,
1Taekyun Kim,
2Young-Hee Kim,
2and
Kyung-Won Hwang
31Department of Mathematics and Computer Science, Konkuk University, Chungju 130-701, South Korea 2Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea 3Department of General Education, Kookmin University, Seoul 136-702, South Korea
Correspondence should be addressed to Young-Hee Kim,[email protected] Kyung-Won Hwang,[email protected]
Received 30 August 2009; Accepted 28 September 2009
Recommended by Vijay Gupta
We construct theq-Euler numbers and polynomials of higher order, which are related to Barnes’ type multiple Euler polynomials. We also derive many properties and formulae for ourq-Euler polynomials of higher order by using the multiple integral equations onZp.
Copyrightq2009 Leechae Jang 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
Letpbe a fixed odd prime number. Throughout this paper, symbolsZ,Zp,Qp, andCpwill denote the ring of rational integers, the ring ofp-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of Qp, respectively. Let N be the set of natural numbers and Z N∪ {0}. Let νpbe the normalized exponential valuation of Cp with |p|p p−νpp 1/p. When one talks of q-extension, q is variously considered as an indeterminate, a complex numberq∈C,or ap-adic numberq∈Cp. Ifq∈C,one normally assumes|q| < 1.Ifq ∈ Cp,then one normally assumes|1−q|p < 1.We use the following notations:
xq 1−q x
1−q, x−q
1−−qx
1q , 1.1
Letda fixed positive odd integer withp, d 1. ForN∈N, we set
XXd lim−→NZ
dpNZ, X1Zp, X∗
0<a<dp
a,p1
adpZp
,
adpNZp
x∈X|x≡amoddpN,
1.2
wherea∈Zlies in 0≤a < dpN. The fermionicp-adicq-measures onZ
pare defined as
μ−q
adpNZp
−qa dpN
−q
, 1.3
see5.
We say thatf is a uniformly differentiable function at a pointa∈ Zp and writef ∈ UDZp, if the difference quotientsFfx, y fx−fy/x−yhave a limitfaas
x, y → a, a. Forf∈UDZp, let us begin with expression
1 pN
−q
0≤j<pN
−qjfj
0≤j<pN
fjμ−q
jpNZp
, 1.4
which represents aq-analogue of Riemann sums forfin the fermionic sensesee4,5. The integral off onZpis defined by the limit of these sumsasn → ∞if this limit exists. The fermionic invariantp-adicq-integral of functionf ∈UDZpis defined as
I−q
f
Zp
fxdμ−qx lim N→ ∞
2q 1qpN
pN−1
x0
fx−qx. 1.5
Note that iffn → finUDZp, then
Zp
fnxdμ−qx−→
Zp
fxdμ−qx,
X
fxdμ−qx
Zp
fxdμ−qx. 1.6
The Barnes’ type Euler polynomials are considered as follows:
2k k
j1
1 ewjt1
ext
∞
n0
Enkx|w1, . . . , wkt
n
n!, 1.7
From1.5, we can derive the fermionic invariant integral onZpas follows:
lim q→1I−q
fI−1
f
Zp
fxdμ−1x. 1.8
Forn∈N, letfnx fxn, one has
I−1
fn −1nI−1
f2
n−1
l0
−1n−1−lfl. 1.9
By1.9, we see that
ext
Zp
· · ·
Zp
k-times
ew1x1···wkxktdμ
−1x1· · ·dμ−1xk 2k
k
j1
1 ewjt1
. 1.10
From1.10, we note that
Zp
· · ·
Zp
k-times
xw1x1· · ·wkxkndμ−1x1· · ·dμ−1xk Enkx|w1, . . . , wk. 1.11
In the view point of1.11, we try to study theq-extension of Barnes’ type Euler polynomials by using theq-extension of fermionicp-adic invariant integral onZp.
The purpose of this paper is to construct the q-Euler numbers and polynomials of higher order, which are related to Barnes’ type multiple Euler numbers and polynomials. Also, we give many properties and formulae for ourq-Euler polynomials of higher order. Finally, we give the generating function for theseq-Euler polynomials of higher order.
2. Barnes’ Type Multiple
q
-Euler Polynomials
Leta1, a2, . . . , ak, b1, b2, . . . , bk ∈ Z. Forw ∈ Zpandq ∈ Cpwith|1−q|p < 1, we define the Barnes’ type multipleq-Euler polynomials as follows:
En,qkw|a1, . . . , ak;b1, . . . , bk
Zk p
qkj1bj−1xj ⎡ ⎣wk
j1
ajxj ⎤ ⎦ n
q
where
Zk p
fx1, . . . , xkdμ−qx Zp · · · Zp k-times
fx1, . . . , xkdμ−qx1· · ·dμ−qxk 2.2
see1,5.
In the special casew0,Enka1, . . . , ak;b1, . . . , bkare called the Barnes’ type multiple
q-Euler numbers. From2.1, one has
Zk p
qkj1bj−1xj ⎡ ⎣wk
j1
ajxj ⎤ ⎦ n
q
dμ−qx
1
1−qn n
r0
n
r
−qwr lim N→ ∞
1q 1qpN
pN−1
x1,...,xk0
qkj1ajrbjxj−1x1···xk
1
1−qn n
r0
n r
−qwr2kq k
j1
1 1qajrbj
.
2.3
Therefore, we obtain the following theorem.
Theorem 2.1. Letw∈Zpandk∈N. Fora1, a2, . . . , ak, b1, b2, . . . , bk∈Z, one has
En,qkw|a1, . . . , ak;b1, . . . , bk
2kq
1−qn n
r0
n r
−qwr k
j1
1 1qajrbj
. 2.4
By1.7, we easily see that
lim q→1E
k
n,qw|a1, . . . , ak;b1, . . . , bk Enkw|a1, . . . , ak. 2.5
From1.7, we can derive
Zk p
qkj1bj−1xj ⎡ ⎣k
j1
ajxj ⎤ ⎦ n
q
dμ−qx
q−1
Zk p
qkj1bj−aj−1xj ⎡ ⎣k
j1
ajxj ⎤ ⎦
n1
q
dμ−qx
Zk p
qkj1bj−aj−1xj ⎡ ⎣k
j1
ajxj ⎤ ⎦ n
q
dμ−qx.
By2.6, one has
En,qka1, . . . , ak;b1, . . . , bk
q−1Enk1,qa1, . . . , ak;b1−a1, . . . , bk−ak En,qka1, . . . , ak;b1−a1, . . . , bk−ak.
2.7
Hence we obtain the following theorem.
Theorem 2.2. Fork∈Nandn∈Z, one has
En,qka1, . . . , ak;b1, . . . , bk
q−1Enk1,qa1, . . . , ak;b1−a1, . . . , bk−ak En,qka1, . . . , ak;b1−a1, . . . , bk−ak.
2.8
It is not difficult to show that the following integral equation is satisfied:
i
j0
i j
q−1j
Zk p
a1x1· · ·akxknq−ijq k
l1bl−1xldμ −qx
i
j0
i−m
j
q−1j
Zk p
a1x1· · ·akxknq−ijq k
l1blmal−1xldμ −qx,
2.9
wherem∈Nwithi≥m. By2.9, we obtain the following theorem.
Theorem 2.3. Letn∈Nandm∈N. Fori∈Nwithi≥m, one has
i
j0
i
j
q−1jEnk−ij,qa1, . . . , ak;b1, . . . , bk
i
j0
i−m
j
q−1jEnk−ij,qa1, . . . , ak;b1ma1, . . . , bkmak.
2.10
For the special casek1 inTheorem 2.3, one has
n
j0
n
j
q−1jEj,q1a1;b1
n
j0
n−m
j
q−1jEj,q1a1;b1ma1
Zp
qna1b1−1xdμ −qx
2q 1qna1b1.
2.11
Corollary 2.4. Forn, k∈Nandw∈Zp, one has
En,qkw|a1, . . . , ak;b1, . . . , bk
2kq
1−qn n
r0
n r
−qwr k
j1
1 1qajrbj
n
i0
n
i
wnq−iqwiE k
i,qw|a1, . . . , ak;b1, . . . , bk.
2.12
From2.3, we note that
qw
Zk p
⎡ ⎣wk
j1
ajxj ⎤ ⎦ m
q
qkj1bj−1xjdμ −qx
q−1
Zk p ⎡ ⎣wk
j1
ajxj ⎤ ⎦
m1
q
qkj1bj−aj−1xjdμ −qx
Zk p
⎡ ⎣wk
j1
ajxj ⎤ ⎦ m
q
qkj1bj−aj−1xjdμ −qx,
Xk ⎡ ⎣wk
j1
ajxj ⎤ ⎦ m
q
qkj1bj−1xjdμ −qx
dmq2kq d−1
i1,...,ik0
qkj1bjij
×−1i1···ik Zk p ⎡ ⎣w k j1ajij
d
k
j1
ajxj ⎤ ⎦ m
qd
qdkj1bj−1xjdμ −qdx,
2.13
wheredis an odd positive integer. By2.13, we obtain the following theorem.
Theorem 2.5. Ford∈Nwithd≡1(mod2), one has
Em,qkw|a1, . . . , ak;b1, . . . , bk
dmq2kq d−1
i1,...,ik0
qkj1bjij−1i1···ikEk
m,qd ⎛ ⎝w
k j1ajij
d |a1, . . . , ak;b1, . . . , bk ⎞ ⎠,
qwEm,qkw|a1, . . . , ak;b1, . . . , bk
q−1Emk1,qw|a1, . . . , ak;b1−a1, . . . , bk−ak Em,qkw|a1, . . . , ak;b1−a1, . . . , bk−ak.
Remark 2.6. Let
Fqw, t ∞
n0
En,qkw|a1, . . . , ak;b1, . . . , bkt
n
n!. 2.15
From2.4, we can easily derive the following equation:
Fqw,q−1t ∞
m0
q−1mEm,qkw|a1, . . . , ak;b1, . . . , bkt
m m!
2kq e−t ∞
i0
⎛ ⎝k
j1
1 1qajibj
⎞ ⎠qwiti
i!.
2.16
By differentiating both sides of2.16with respect totand comparing coefficients on both sides, one has
qwEk
m,qw|a1, . . . , ak;b1, . . . , bk−Em,qkw|a1, . . . , ak;b1−a1, . . . , bk−ak q−1Emk1,qw|a1, . . . , ak;b1−a1, . . . , bk−ak.
2.17
The inversion formula of Equation2.4atw0 is given by
m
i0
m
i
q−1i
Zk p
a1x1· · ·akxkiqq k
j1bj−1xjdμ −qx
Zk p
qkj1majbj−1xjdμ −qx.
2.18
Thus, one has
m
i0
m
i
q−1iEi,qka1, . . . , ak;b1, . . . , bk 2kq k
j1
1 1qmajbj
. 2.19
Acknowledgment
This paper was supported by Konkuk University2009.
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