Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 875098, 11 pages
doi:10.1155/2010/875098
Research Article
On the Twisted q-Analogs of the Generalized Euler
Numbers and Polynomials of Higher Order
Lee Chae Jang,1 Byungje Lee,2 and Taekyun Kim3
1
Department of Mathematics and Computer Science, KonKuk University,
Chungju 138-701, Republic of Korea
2
Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
3
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Correspondence should be addressed to Lee Chae Jang, leechae.jang@kku.ac.kr
Received 12 April 2010; Accepted 28 June 2010
Academic Editor: Istvan Gyori
Copyright q 2010 Lee Chae 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.
We consider the twisted q-extensions of the generalized Euler numbers and polynomials attached
to χ.
1. Introduction and Preliminaries
n
Let p be an odd prime number. For n ∈ Z N ∪ {0}, let Cpn {ζ | ζp 1} be the
cyclic group of order pn , and let Tp limn → ∞ Cpn n≥0 Cpn Cp∞ be the space of locally
constant functions in the p-adic number field Cp . When one talks of q-extension, q is variously
considered as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ Cp . If q ∈ C,
one normally assumes that |q| < 1. If q ∈ Cp , one normally assumes that |1 − q|p < 1. In this
paper, we use the notation
1 − qx
,
xq 1−q
x−q
x
1 − −q
.
1q
Let d be a fixed positive odd integer. For N ∈ N, we set
X Xd −Z
lim←
N
dpN Z
,
X1 Zp ,
1.1
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Advances in Difference Equations
a dpZp ,
X∗ 0<a<dp
a,p1
a dpn Zp x ∈ X | x ≡ a mod dpn ,
1.2
where a ∈ Z lies in 0 ≤ a < dpn ; compared to 1–16.
Let χ be the Dirichlet’s character with an odd conductor d ∈ N. Then the generalized
ζ-Euler polynomials attached to χ, En,χ,ζ x, are defined as
Fχ,ζ x, t 2
d−1
−1l χlζl elt xt
e
ζd edt 1
l0
∞
tn
En,χ,ζ x ,
n!
n0
1.3
for ζ ∈ Tp .
In the special case x 0, En,χ,ζ En,χ,ζ 0 are called the nth ζ-Euler numbers attached to χ.
For f ∈ UDZp , the p-adic fermionic integral on Zp is defined by
I−q f fxdμ−q x lim
N →∞
Zp
−1
p
fxμ−q x pN Zp
N
p
−1
x0
1.4
N
qx
lim
fx−1x N ,
N →∞
p −q
x0
see 7–17.
Let I−1 limq → 1 I−q f. Then, we see that
Zp
fxdμ−1 x fxdμ−1 x.
X
1.5
For n ∈ N, let fn x fx n. Then, we have
Zp
fx ndμ−1 x −1n
Zp
n−1
fxdμ−1 x 2 −1n−1−l fl.
1.6
l0
Thus, we have
n−1
I−1 fn −1n−1 I−1 f 2 −1n−1−l fl,
l0
see 7–17.
1.7
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3
By 1.7, we see that
2
χ y ζy exyt dμ−1 y X
d−1
∞
−1l χlζl elt xt tn
.
e
E
x
n,χ,ζ
n!
ζd edt 1
n0
l0
1.8
From 1.8, we can derive the Witt’s formula for En,χ,ζ x as follows:
χxxn ζx dμ−1 x En,χ,ζ ,
X
1.9
n
χ y y x ζy dμ−1 y En,χ,ζ x,
for ζ ∈ Tp , see 5–17.
X
k
The nth generalized ζ-Euler polynomials of order k, En,χ,ζ , are defined as
k
d−1
∞
2 l0 ζl −1l χlelt xt
tn
k
.
e
E
x
n,χ,ζ
n!
ζd edt 1
n0
k
1.10
k
In the special case x 0, En,χ,ζ En,χ,ζ 0 are called the nth ζ-Euler numbers of order k
attached to χ.
Now, we consider the multivariate p-adic invariant integral on X as follows:
···
X
k
X
χxi ex1 ···xk xt ζx1 ···xk dμ−1 x1 · · · dμ−1 xk i1
k
d−1
∞
2 l0 −1l χlelt
tn
k
xt
.
e
E
x
n,χ,ζ
n!
ζd edt 1
n0
1.11
k
By 1.10 and 1.11, we see the Witt’s formula for En,χ,ζ x as follows:
X
k
k
···
χxi x1 · · · xk xn ζx1 ···xk dμ−1 x1 · · · dμ−1 xk En,χ,ζ x.
X
1.12
i1
The purpose of this paper is to present a systemic study of some formulas of the
twisted q-extension of the generalized Euler numbers and polynomials of order k attached
to χ.
2. On the Twisted q-Extension of the Generalized Euler Polynomials
In this section, we assume that q ∈ Cp with |1 − q|p < 1 and ζ ∈ Tp . For d ∈ N with 2 d, let
χ be the Dirichlet’s character with conductor d. For h ∈ Z, k ∈ N, let us consider the twisted
h, q-extension of the generalized Euler numbers and polynomials of order k attached to χ.
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Advances in Difference Equations
We firstly consider the twisted q-extension of the generalized Euler polynomials of higher
order as follows:
∞
tn
En,χ,ζ,q x exyq t ζy χ y dμ−1 y
n!
X
n0
2.1
2
∞
χm−1m ζm emxq t .
m0
By 2.1, we see that
X
∞
n x y q χ y ζy dμ−1 y 2 χm−ζm emxq t
m0
d−1
2 χa−1a ζa 1
1−q
a0
n
2.2
n n
l0
ζla qlax
.
−1l
l
1 qld ζld
From the multivariate fermionic p-adic invariant integral on Zp , we can derive the twisted
q-extension of the generalized Euler polynomials of order k attached to χ as follows:
∞
n0
tn
k
En,χ,ζ,q x
n!
···
X
k
X
χxi ex1 ···xk xq t ζx1 ···xk dμ−1 x1 · · · dμ−1 xk .
2.3
i1
Thus, we have
k
En,χ,ζ,q x
···
k
X
X
d−1
2
k
j1 aj
χai −ζ
k
k
χai −ζ
a1 ,...,ak 0 i1
Let Fq,χ,ζ t, x k
i1
d−1
k
χxi x1 · · · xk xnq ζx1 ···xk dμ−1 x1 · · · dμ−1 xk i1
k
a1 ,...,ak 0
∞
n0
j1
aj
∞ mk−1
m0
k
k
n
nl −1l qlx j1 aj n
1 qld ζd
1 − q l0
2k
m
m
× −ζd x a1 · · · ak mdnq .
2.4
k
En,χ,ζ,q xtn /n! be the generating function for En,χ,ζ,q x. By 2.3,
Advances in Difference Equations
5
we easily see that
k
Fq,χ, t, x
d−1
2
k
a1 ,...,ak
k
∞ k
mk−1
χai −ζ j1 aj
× −ζdm exa1 ···ak mdq t
m
0
m0
i1
∞
2k
−ζn1 ···nk
k
n1 ,...,nk 0
2.5
χni en1 ···nk xq t .
i1
Therefore, we obtain the following theorem.
Theorem 2.1. For k ∈ N, n ≥ 0, one has
k
En,χ,ζ,q x
∞
2
k
n1 ···nk
−ζ
k
n1 ,...,nk 0
a1 ,...,ar 0
χni n1 · · · nk xnq
i1
k
d−1
k
n
nl −1l qlx j1 aj 2
χai −ζ j1 aj .
n
n
1 − q l0
1 qld ζd
i1
k
k
2.6
r
Let h ∈ Z, r ∈ N. Then we define the extension of En,χ,ζ,q x as follows:
∞
r
tn
h,r
En,χ,ζ,q x ···
q j1 h−jxj
n!
X
X
n0
k
χxi ex
r
j1
x j q t
× ζx1 ···xr dμ−1 x1 · · · dμ−1 xr .
i1
2.7
r
Then, En,χ,ζ,q x are called the nth generalized h, q-Euler polynomials of order r attached
r
r
to χ. In the special case x 0, En,χ,ζ,q En,χ,ζ,q 0 are called the nth generalized h, r-Euler
r
numbers of order r. By 1.7, we obtain the Witt’s formula for En,χ,ζ,q x as follows:
h,r
En,χ,ζ,q x
···
X
q
r
j1 h−jxj
X
j1
i1
r
d−1
a1 ,...,ar 0
⎤n
⎡
k
r
χxi ⎣x xj ⎦ ζx1 ···xr dμ−1 x1 · · · dμ−1 xr r
χai −ζ
j1 aj
q
i1
where a; qr 1 − a1 − aq · · · 1 − aqr−1 .
r
j1 aj h−j
q
r
n
nl −1l qlx j1 aj ×
,
n
1 − q l0 −qdh−rl ζd ; qd r
2r
2.8
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Advances in Difference Equations
Let nk q nq n − 1q · · · n − k 1q /kq ! nq !/kq !n − kq ! where kq ! kq k − 1q · · · 2q 1q . From 2.8, we note that
h,r
En,χ,ζ,q x
2r
1−q
×
2
r
n
2
i1
n ∞ m
r
n
mr−1
−ζd qdh−rm qldm
−1l qlx j1 aj l
m
qd
m0
l0
r−1
d−1
r
χai −ζ
j1
∞ mr−1
−ζd
m
m
r
j1 h−jaj
r
aj
j1
q
1
qdh−rm 1−q
−ζ
m
m0
× −ζ
q
qd
∞ mr−1
dnq
aj
i1
m0
r
r
r
r
χai −ζ j1 aj q j1 h−jaj
a1 ,...,ar 0
a1 ,...,ar 0
×
d−1
m
d
q
a1 ,...,ar 0
⎡
j1 h−jaj
r−1
d−1
dh−rm
qd
r
n
dmx kj1 aj /d
n 1 − q
⎣m x
d−1
j1
aj
d
2.9
χai i1
⎤n
⎦ .
qd
h,r
h,r
h,r
n
Let Fq,χ,ζ t, x ∞
n0 En,χ,ζ,q xt /n! be the generating function for En,χ,ζ,q x. From
2.8, we can easily derive
h,r
Fq,χ,ζ t, x
2r
⎞
⎛
r
q j1 h−jnj −ζ j1 nj ⎝ χ nj ⎠en1 ···nr xq t
∞
r
r
n1 ,...,nr 0
2
r
j1
∞ mr−1 −ζ
m
m0
r
× −ζ
j1
aj
q
d
m
q
j1 h−jaj
By 2.10, we obtain the following theorem.
r−1
a1 ,...,ar 0
q
r
d−1
dh−rm
emdx
r
j1
aj q t
.
i1
χai 2.10
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Theorem 2.2. For h ∈ Z, r ∈ N, one has
h,r
En,χ,ζ,q x
⎛
⎞
r
q j1 h−jnj −ζ j1 nj ⎝ χ nj ⎠n1 · · · nr xnq
∞
2r
r
r
n1 ,...,nr 0
2
r
j1
∞ mr−1 dnq
r
× −ζ
j1
−ζ
m
m0
aj
q
d
m
q
a1 ,...,ar 0
q
r
j1 h−jaj
r−1
χai d−1
dh−rm
m
r
x
j1
aj
i1
n
d
qd
⎛
⎞
r
r
n
r
r
nl −1l qlx j1 aj 2r
j1 aj
j1 h−jaj
⎝
⎠
χ nj
q
×
.
−ζ
n
1 − q l0 −qdh−rl ζd ; qd r
a1 ···ar 0
j1
d−1
2.11
Let h r. Then we see that
r,r
En,χ,ζ,q x
r
r−1
n
r
r
nl −1l ql j1 aj x
j1 h−jaj
i1 ai
χai −ζ
q
×
ld d d n
−q ζ ; q r
1 − q a1 ,...,ar 0 i1
l0
d−1
2r
2
r
∞ mr−1
dnq
m
m0
r
× −ζ
j1 aj
q
m
−ζ
j1 r−jaj
r
χai a1 ,...,ar 0
q
r
d−1
m
r
x
j1
2.12
i1
aj
n
.
d
qd
It is easy to see that
···
k
X
X
χxi q
r
j1 h−jxj xm
ζx1 ···xr dμ−1 x1 · · · dμ−1 xr i1
d−1
r
a1 ,...,ar 0
2r qmx
χai qmx
r
j1 h−jaj
r
−ζ
j1
aj
···
X
i1
d−1
a1 ,...,ar 0
×
r
j1 χ aj
q
r
−qdm−r ζd ; qd
j1 m−jaj
r
r
−ζ
j1
q
r
j1 m−jxj
dμ−1 x1 · · · dμ−1 xr X
aj
.
2.13
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Advances in Difference Equations
Thus, we have
2r qmx
d−1
a1 ,...,ar 0
q
r
j1 m−jaj
r
−ζ
j1
aj
−qdm−r ζd ; qd r
m r
···
x x1 · · · xr q q − 1 1 q− j1 jxj ζx1 ···xr
X
⎛
×⎝
r
j1 χ aj
X
r
⎞
χ xj ⎠dμ−1 x1 · · · dμ−1 xr j1
m m l
l0
l
q−1
⎛
⎝
···
X
X
2.14
⎞
r
χ xj ⎠
j1
r
× x x1 · · · xr lq q− j1 jxj ζx1 ···xr dμ−1 x1 · · · dμ−1 xr m l 0,r
m q − 1 El,χ,ζ,q x.
l
l0
By 2.14, we obtain the following theorem.
Theorem 2.3. For d, k ∈ N with 2 d, one has
r
r m−jaj
r
m q j1
2r qmx d−1
−ζ j1 aj a1 ,...,ar 0
j1 χ aj
l 0,r
m q − 1 El,χ,ζ,q x.
dm−r d d l
ζ ;q r
−q
l0
2.15
By 1.7, we easily see that
d−1
fxdμ−1 x 2 −1l fl.
fx ddμ−1 x X
X
2.16
l0
Thus,we have
r
r
···
qdh−1
x d x1 · · · xr nq q j1 r−jxj ζ j1 xj
X
X
⎞
⎛
r
× ⎝ χ xj ⎠dμ−1 x1 · · · dμ−1 xr j1
−
···
X
x x1 · · · X
xr nq q
r
j1 r−jxj
ζ
r
j1 xj
⎞
⎛
r
× ⎝ χ xj ⎠dμ−1 x1 · · · dμ−1 xr j1
⎛
⎞
r−1
d−1
2 χl−ζl
···
x l x2 · · · xr nq ⎝ χ xj1 ⎠
×q
l0
r−1
X
X
j1 xj1 h−1−j x2 x3 ···xr
ζ
j1
dμ−1 x2 · · · dμ−1 xr .
2.17
By 2.17, we obtain the following theorem.
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Theorem 2.4. For h ∈ Z, d ∈ N with 2 d, one has
d−1
h,r
h,r
h−1,r−1
qdh−1 En,χ,ζ,q x d En,χ,ζ,q x 2 χl−1l En,χ,ζ,q x l.
2.18
l0
It is easy to see that
h,r
h1,r
h,r
qx En,χ,ζ,q x q − 1 En1,χ,ζ,q En,χ,ζ,q x.
h,1
Let Fq,χ,ζ t, x ∞
n0
2.19
h,1
En,χ,ζ,q xtn /n!. Then we note that
∞
h,1
Fq,χ,ζ t, x 2 χnqh−1n −ζn enxq t .
2.20
n0
From 2.20, we can derive
h,1
En,χ,ζ,q x 2
∞
2
χmqh−1m −ζm m xnq 1−q
m0
n
n
d−1
nl −1l qlxa
χa−ζa
. 2.21
1 qld ζd
a0
l0
3. Further Remark
In this section, we assume that q ∈ C with |q| < 1. Let χ be the Dirichlet’s character with an
h,r
odd conductor d ∈ N. From the Mellin transformation of Fq,χ,ζ t, x in 2.10, we note that
1
Γs
∞
h,r
Fq,χ,ζ −t, xts−1 dt 2r
q
r
j1 h−jmj
−ζm1 ···mr
m1 · · · mr xsq
m1 ,...,mr 0
r
j1 χ mj
,
3.1
where h, s ∈ C, x /
0, −1, −2, . . . , and r ∈ N, ζ e2πi/d . By 3.1, we can define the Dirichlet’s
type multiple h, q-l-function as follows.
Definition 3.1. For s ∈ C, x ∈ R with x / 0, −1, −2, . . ., one defines the Dirichlet’s type multiple
h, q-l-function related to higher order h, q-Euler polynomials as
h,r lq
s, x
| χ 2r
∞
m1 ,··· ,mr 0
q
r
r
−ζm1 ···mr
i1 χmi ,
m1 · · · mr xsq
j1 h−jmj
where s, h ∈ C, x /
0, −1, −2, · · · , r ∈ N, and ζ e2πi/d .
3.2
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h,r
Note that lq
note that
s, x | χ is analytic continuation in whole complex s-plane. In 2.10, we
h,r
Fq,χ,ζ t, x
∞
2r
q
⎛
r
j1 h−jnj
n1 ,...,nr 0
∞
n0
h,r
En,χ,ζ,q x
n1 ···nr ⎝
−ζ
⎞
r
χ nj ⎠en1 ···nr xq t
j1
3.3
n
t
.
n!
By Laurent series and Cauchy residue theorem in 3.1 and 3.3, we obtain the
following theorem.
Theorem 3.2. Let χ be Dirichlet’s character with odd conductor d ∈ N, and let ζ e2πi/d . For
h, s ∈ C, x /
0, −1, −2, . . . , r ∈ N, and n ∈ Z , one has
h,r lq
h,r
−h, x | χ En,χ,ζ,q x.
3.4
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