Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2009, Article ID 639439, 7 pages
doi:10.1155/2009/639439
Research Article
Some Identities of the Frobenius-Euler Polynomials
Taekyun Kim1 and Byungje Lee2
1
2
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, South Korea
Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, South Korea
Correspondence should be addressed to Byungje Lee, bj lee@kw.ac.kr
Received 5 November 2008; Accepted 5 January 2009
Recommended by Ferhan Atici
By using the ordinary fermionic p-adic invariant integral on Zp , we derive some interesting
identities related to the Frobenius-Euler polynomials.
Copyright q 2009 T. Kim and B. Lee. 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
Let p be a fixed prime. Throughout this paper Zp , Qp , C, and Cp will, respectively, denote the
ring of p-adic rational integers, the field of p-adic rational numbers, the complex number
field, and the completion of algebraic closure of Qp . When one talks about q-extension, q is
variously considered as an indeterminate, a complex q ∈ C, or a p-adic number q ∈ Cp ; see
1–14. If q ∈ C, then we assume |q| < 1. If q ∈ Cp , then we assume |1 − q|p < 1. For x ∈ Qp ,
we use the notation xq 1 − qx /1 − q, and x−q 1 − −qx /1 q; see 15, 16. The
normalized valuation in Cp is denoted by | · |p with |p|p 1/p. We say that f is a uniformly
differentiable function at a point a ∈ Zp and denote this property by f ∈ UDZp , if the
difference quotients Ff x, y fx − fy/x − y have a limit l f a as x, y → a, a.
For f ∈ UDZp , let us start with the expression
1
pN q 0≤j<pN
qj fj fjµq j pN Zp ,
1.1
0≤j<pN
representing a q-analogue of Riemann sums for f; see 15, 16. The integral of f on Zp will
be defined as a limit n → ∞ of those sums, when it exists. The q-deformed bosonic p-adic
integral of the function f ∈ UDZp is defined by
Iq f Zp
1
N → ∞ dpN q
fxdµq x lim
0≤x<dpN
fxqx ,
1.2
2
Abstract and Applied Analysis
see 15. Thus, we note that
qIq f1 Iq f q − 1f0 q−1 f 0,
log q
1.3
where f1 x fx 1, f 0 df0/dx.
The fermionic p-adic invariant integral on Zp is defined as
I−1 f p
−1
1
fx−1x ,
N → ∞ pN −1 x0
N
Zp
fxdµ−1 x lim
1.4
see 15.
In this paper, we prove an identity of symmetry for the Frobenius-Euler polynomials.
Finally we investigate the several further interesting properties of the symmetry for the
fermionic p-adic invariant integral on Zp related to the Frobenius-Euler polynomials and
numbers.
2. Some Identities of the Frobenius-Euler Polynomials
Let u /
1 ∈ Cp or C be algebraic. Then the nth Frobenius-Euler numbers Hn u are defined
as
H0 u 1,
Hu 1n − uHn u 0,
if n ≥ 1,
2.1
with the usual convention about replacing H n u by Hn u.
The nth Frobenius-Euler polynomials Hn u, x are also defined as
Hn u, x n n
l0
l
xn−l Hl u.
2.2
From 1.4, we can easily derive
I−1 f1 I−1 f 2f0,
where f 1 x fx 1.
2.3
By continuing this process, we see that
n−1
I−1 fn −1n−1 I−1 f 2 −1n−1−l fl,
where f n x fx n.
2.4
l0
When n is an odd positive integer, we obtain
n−1
I−1 fn I−1 f 2 −1l fl.
l0
2.5
Abstract and Applied Analysis
3
If n ∈ N with n ≡ 0 mod 2, then we have
n−1
I−1 fn − I−1 f 2 −1l−1 fl.
2.6
l0
From 1.4 and 2.3, we derive
Zp
ext qx dµ−1 x −1
∞
2 1 − −q
2 tn
Hn −q−1 .
−1
2q et − −q
2q n0
n!
2.7
Thus, we note that
2
x q dµ−1 x Hn −q−1 ,
2
Zp
q
n x
Zp
y xn qy dµ−1 x 2
Hn −q−1 , x.
2q
2.8
Let n ∈ N with n ≡ 1 mod 2. Then we obtain
2q
n−1
−1l ql lm qn Hm −q−1 , n Hm −q−1 .
2.9
l0
For n ∈ N with n ≡ 0 mod 2, we have
qn Hm −q−1 , n − Hm −q−1 2q
n−1
−1l−1 ql lm .
2.10
l0
By substituting fx qx ext into 2.5, we can easily see that
Zp
qnx exnt dµ−1 x Zp
qx ext dµ−1 x 2
n−1
qn ent 1
2 −1l ql elt .
t
qe 1
l0
2.11
n
l k l
Let Sk,q n l0 −1 l q . Then Sk,q n is called the alternating sums of powers of
consecutive q-integers. From the definition of the fermionic p-adic invariant integral on Zp ,
we can derive
2 Zp qx ext dµ−1 x
.
qxn exnt dµ−1 x qx ext dµ−1 x nxt nx
e q dµ−1 x
Zp
Zp
Zp
2.12
By 2.12, we easily see that
Zp
qnx enxt dµ−1 x 2
qn ent
1
.
2.13
4
Abstract and Applied Analysis
Let w1 , w2 ∈ N be odd. By using double fermionic p-adic invariant integral on Zp , we
obtain
Zp Zp
ew1 x1 w2 x2 t qw1 x1 w2 x2 dµ−1 x1 dµ−1 x2 2 q w1 w2 e w1 w2 t 1
w wt
.
ew1 w2 xt qw1 w2 x dµ−1 x
q 1 e 1 1 q w2 e w2 t 1
Zp
2.14
Now we also consider the following fermionic p-adic invariant integral on Zp
associated with Frobenius-Euler polynomials:
Zp Zp
ew1 x1 w2 x2 w1 w2 xt qw1 x1 w2 x2 dµ−1 x1 dµ−1 x2 2ew1 w2 xt qw1 w2 ew1 w2 t 1
.
ew1 w2 xt qw1 w2 x dµ−1 x
q w1 e w1 t 1 q w2 e w2 t 1
Zp
2.15
From 2.15 and 2.12, we can derive
2 Zp qx ext dµ−1 x
ew1 xt qw1 x dµ−1 x
Zp
w
1 −1
−1l ql elt
2
l0
w −1
∞
1
tk
l l k
2
−1 q l
k!
k0
l0
2.16
∞
tk
2Sk,q w1 − 1 .
k!
k0
Let
M
w1 ,w2 t, x Zp Zp
qw1 x1 w2 x2 ew1 x1 w2 x2 w1 w2 xt dµ−1 x1 dµ−1 x2 .
ew1 w2 x3 t qw1 w2 x3 dµ−1 x3 Zp
2.17
By 2.15, 2.16, and 2.17, we see that
ew1 w2 xt qw1 w2 ew1 w2 t 1
Mw1 ,w2 t, x w w t
w w t
.
q 1e 1 1 q 2e 2 1
2.18
From 2.17 we derive
M
w1 ,w2 w2 x2 t w2 x2
q
dµ−1 x2 2 Zp e
1
w1 x1 w2 xt w1 x1
.
t, x e
q
dµ−1 x1 2 Zp
ew1 w2 xt qw1 w2 x dµ−1 x
Z
p
2.19
Abstract and Applied Analysis
5
By 2.16 and 2.19, we see that
M
w1 ,w2 t, x ∞
w1i i
1 −w1
t
H
,
w
x
−
q
i
2
1 qw1 i0
i!
∞
w2l l
Sl,qw2 w1 − 1 t
l!
l0
−w
∞
n
tn
n Hi − q 1 , w2 x
i
n−i
.
Sn−i,qw2 w1 − 1w1 w2
w
i
1q 1
n!
n0
i0
2.20
By the symmetry of p-adic invariant integral on Zp , we also see that
M
w1 ,w2 ∞
n
n Hi −q−w2 , w1 x
t, x n0
1 q w2
i
i0
Sn−i,qw1 w2 −
1w2i w1n−i
tn
,
n!
2.21
where Hn −q−1 , x are the nth Frobenius-Euler polynomials.
By comparing the coefficients on the both sides of 2.20 and 2.21, we obtain the
following theorem.
Theorem 2.1. For w1 , w2 , n ∈ N with n ≡ 1 mod 2, w1 ≡ 1 mod 2, w2 ≡ 1 mod 2, one has
n H − q−w1 , w x
n
i
2
1 q w1
i
i0
Sn−i,qw2 w1 − 1w1i w2n−i
n H − q−w2 , w x
n
i
1
1 q w2
i
i0
2.22
Sn−i,qw1 w2 − 1w2i w1n−i ,
where Hn q, x are the nth Frobenius-Euler polynomials.
If we take w2 1 in Theorem 2.1, then we have
n H − q−w1 , x
Hn − q−1 , w1 x
n
i
Sn−i,q w1 − 1w1i .
w1
i
1q
1
q
i0
2.23
From 2.11 and 2.12, we derive
Mw1 ,w2 t, x ew1 w2 xt
2
ew1 w2 xt
2
Zp
−1l qw2 l
∞
n0
e
w 1 x1 t w 1 x1
q
w
1 −1
Zp
−1
2
l0
dµ−1 x1 w
1 −1
l w2 l w2 lt
−1 q
2
e
l0
w
1 −1
l0
w2 x2 t w2 x2
q
dµ−1 x2 2 Zp e
w 1 x1 t w 1 x1
e
q
dµ−1 x1 ew1 w2 xt qw1 w2 x dµ−1 x
Zp
Zp
ex1 w2 xw2 /w1 ltw1 qx1 w1 dµ−1 x1 l Hn
− q−w1 , w2 x w2 /w1 l w2 l tn
.
q
1 q w1
n!
2.24
6
Abstract and Applied Analysis
From the symmetry of Mw1 ,w2 t, x, we note that
M
w1 ,w2 t, x ∞
n0
w
2 −1
−1
2
l0
l Hn
− q−w2 , w1 x w1 /w2 l w1 l tn
.
q
1 q w2
n!
2.25
By comparing the coefficients on the both sides of 2.24 and 2.25, we obtain the following
theorem.
Theorem 2.2. Let w1 , w2 ∈ N be odd, and let n ∈ Z with n ≡ 1 mod 2. Then, one has
w
1 −1
−1
l Hn
l0
−w2
2 −1
, w1 x w1 /w2 l w1 l
− q−w1 , w2 x w2 /w1 l w2 l w
l Hn − q
q −1
q .
1 q w1
1 q w2
l0
2.26
By setting w2 1 in Theorem 2.2, we get the multiplication theorem for the FrobeniusEuler polynomials as follows:
w −1
1
Hn − q−1 , w1 x
l
−1l ql Hn − q−w1 , x .
1q
w1
l0
2.27
Remark 2.3. By using the fermionic p-adic invariant q-integral on Zp , the symmetric properties
related to Frobenius-Euler polynomials are studied in 17. In this paper, we have studied
the symmetric properties of Frobenius-Euler polynomials, which are different from the
symmetric properties treated in a previous paper 17. To derive the symmetric properties
of Frobenius-Euler polynomials, we used the ordinary fermionic p-adic invariant integrals
on Zp in this paper.
Acknowledgment
The present research has been conducted by the research grant of the Kwangwoon University
in 2008.
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