Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 861797, 9 pages
doi:10.1155/2012/861797
Research Article
A Note on Some Identities of Frobenius-Euler
Numbers and Polynomials
J. Choi,1 D. S. Kim,2 T. Kim,3 and Y. H. Kim1
1
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Department of Mathematics, Sogang University, Seoul 121-742, Republic of Korea
3
Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
2
Correspondence should be addressed to D. S. Kim, dskim@sogang.ac.kr
and T. Kim, tkkim@kw.ac.kr
Received 28 November 2011; Accepted 9 January 2012
Academic Editor: Feng Qi
Copyright q 2012 J. Choi 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.
The purpose of this paper is to give some identities on the Frobenius-Euler numbers and
polynomials by using the fermionic p-adic q-integral equation on Zp .
1. Introduction
Let p be a fixed odd prime number. Throughout this paper, Zp , Qp , and Cp will denote the
ring of p-adic rational 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}.
The p-adic absolute value on Cp is normalized so that |p|p 1/p. Assume that q ∈ Cp with
|1 − q|p < 1.
Let f be a continuous function on Zp . Then the fermionic p-adic q-integral on Zp for f
is defined by Kim as follows:
I−q f −1
x
1 q p
fxdμ−q x lim
fx −q ,
N
p
N
→
∞
1 q x0
Zp
N
see 1.
1.1
From 1.1, we note that
n−1
qn I−q fn −1n I−q f 1 q
−1n−1−l flql ,
l0
1.2
2
International Journal of Mathematics and Mathematical Sciences
where n ∈ N and fn x fx n see 1. The ordinary Euler polynomials En x are defined
by
et
∞
tn
2
ext eExt En x ,
n!
1
n0
1.3
with the usual convention about replacing En x by En x see 1–10. In the special case,
x 0, En 0 En is called the nth Euler number.
As the extension of 1.3, the Frobenius-Euler polynomials are defined by
∞
tn
1 − q xt ,
e
H
q,
x
n
n!
et − q
n0
see 2.
1.4
In the special case, x 0, Hn q, 0 Hn q is called the nth Frobenius-Euler number.
By 1.3 and 1.4, we easily get Hn −1, x En x.
From 1.4, we note that
Hn q, x n
n
l0
l
n
Hl q xn−l H q x ,
see 2,
1.5
with the usual convention about replacing Hqn by Hn q.
In this paper, we consider the fermionic p-adic q-integral on Zp for the FrobeniusEuler numbers and polynomials. From these p-adic q-integrals on Zp , we derive some new
and interesting identities on the Frobenius-Euler numbers and polynomials.
2. Identities on the Frobenius-Euler Numbers
From 1.2 and 1.4, we can derive the following:
∞
tn
1 q−1 xt −1
.
−q
exyt dμ−q y t
e
H
,
x
n
n!
e q−1
Zp
n0
2.1
Thus, by 2.1, we get Witt’s formula for Hn −q−1 , x as follows:
n
x y dμ−q y Hn −q−1 , x ,
Zp
n ∈ Z .
2.2
By 1.5 and 2.1, we get
⎧
n
⎨1 q−1 ,
−1
−1
−1
H −q
1 q Hn −q
⎩0,
n
if n 0,
if n > 0,
with the usual convention about replacing H−q−1 by Hn −q−1 .
2.3
International Journal of Mathematics and Mathematical Sciences
3
From 1.5 and 2.3, we note that
H0 −q−1 1,
Hn −q−1 , 1 q−1 Hn −q−1 0,
if n > 0.
2.4
By 2.1 and 2.2, we get
y 1 − x dμ−q y −1n
Zp
n
Zp
n
y x dμ−q−1 y .
2.5
Therefore, by 2.5, we obtain the following lemma.
Lemma 2.1. For n ∈ Z , one has
Hn −q−1 , 1 − x −1n Hn −q, x .
2.6
From 2.3, we can derive the following:
2
q Hn
n
l
n −1 2
2
H −q
−q , 2 − q − q q
1 − q2 − q
l
l0
−1
n
l
n −1 q
q H −q
1 −q
l
l1
−q
n
n
l0
l
2.7
Hl −q−1
− 1 q δ0,n Hn −q−1 ,
where δk,n is the Kronecker symbol.
Therefore, by 2.7, we obtain the following theorem.
Theorem 2.2. For n ∈ Z , one has
Hn −q−1 , 2 1 q−1 − q−2 1 q δ0,n q−2 Hn −q−1 .
2.8
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International Journal of Mathematics and Mathematical Sciences
First we consider the fermionic p-adic q-integral on Zp for the nth Frobenius-Euler
polynomials as follows:
I1 Zp
Hn −q−1 , x dμ−q x
n
n
Hl −q
l
l0
−1
Zp
xn−l dμ−q x
n
n
l
l0
Hl −q−1 Hn−l −q−1 ,
2.9
where n ∈ Z .
On the other hand, by 2.5 and Lemma 2.1, we get
I1 Zp
Hn −q , x dμ−q x −1
−1
n
n
n
l
l0
Hn−l −q
n
n
−1n−l Hn−l −q
l
l0
−1
n
Zp
Hn −q, 1 − x dμ−q x
Zp
1 − xl dμ−q x
2.10
Zp
x − 1l dμ−q x
n
n
−1n−l Hn−l −q Hl −q−1 , −1 .
l
l0
From Lemma 2.1, Theorem 2.2, and 2.10, we note that
I1 n
n
l0
l
n
n
l0
−1n−l Hn−l −q Hl −q−1 , −1
−1n Hn−l −q Hl −q, 2
l
n
n
l0
l
−1n Hn−l −q 1 q − q2 1 q−1 δ0,l q2 Hl −q
−1n 1 q 1 q δ0,n − qHn −q − Hn −q q q2 −1n
n 2
−1 q
n
n
l0
l
Hn−l −q Hl −q .
Therefore, by 2.10 and 2.11, we obtain the following theorem.
2.11
International Journal of Mathematics and Mathematical Sciences
5
Theorem 2.3. For n ∈ Z , one has
n
n
Hl −q−1 Hn−l −q−1 −1n 1 q 1 q δ0,n − 2qHn −q
l
l0
−1n q2
n
n
l0
l
Hn−l
−q Hl −q .
In particular, for n ∈ N, one has
n
n
Hl −q−1 Hn−l −q−1 2−1n1 q 1 q Hn −q
l
l0
n
n
n 2
−1 q
Hn−l −q Hl −q .
l
l0
2.12
2.13
Let us consider the following fermionic p-adic q-integral on Zp for the product of
Bernoulli and Frobenius-Euler polynomials as follows:
Bm xHn −q−1 , x dμ−q x
I2 Zp
n
m m
n
k
k0 l0
l
Bm−k Hn−l −q−1
Zp
xkl dμ−q x
2.14
n
m m
n
k
k0 l0
l
Bm−k Hn−l −q−1 Hkl −q−1 .
It is known that Bn x −1n Bn 1 − x.
On the other hand, by Lemma 2.1, we get
mn
Bm 1 − xHn −q, 1 − x dμ−q x
I2 −1
Zp
−1
mn
m n
m
n
Bm−k Hn−l −q
k
l
m n
m
n
mn
k0 l0
−1
k0 l0
k
l
Zp
1 − xkl dμ−q x
Bm−k Hn−l −q
1 q − q2 1 q−1 δ0,kl q2 Hkl −q
−1mn 1 q Bm 1Hn −q, 1 − q2 q −1mn Bm Hn −q
q −1
2
mn
m n
m
n
k0 l0
k
l
Bm−k Hn−l −q Hkl −q .
Therefore, by 2.14 and 2.15, we obtain the following theorem.
2.15
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International Journal of Mathematics and Mathematical Sciences
Theorem 2.4. For m, n ∈ Z , one has
n
m m
n
k0 l0
k
l
Bm−k Hn−l −q−1 Hkl −q−1
−1mn 1 q Bm 1Hn −q, 1 − q2 q −1mn Bm Hn −q
q −1
2
mn
2.16
m n
m
n
k0 l0
k
l
Bm−k Hn−l −q Hkl −q .
In particular, for m ∈ N − {1}, n ∈ N, one has
n
m m
n
k0 l0
k
l
Bm−k Hn−l −q−1 Hkl −q−1
2−1mn1 q2 q Bm Hn −q
q −1
2
mn
m n
m
n
k0 l0
k
It is known that xn 1/n 1
fermionic p-adic q-integral on Zp :
2.17
Bm−k Hn−l −q Hkl −q .
l
n n1 l0
l
Bl x. Let us consider the following
n
n1
1 x dμ−q x Bl xdμ−q x.
n 1 l0
l
Zp
Zp
n
n
l
n1 l
1 Bl−k
xk dμ−q x
n 1 l0
l
Zp
k0 k
n
l
n1 l
1 Bl−k Hk −q−1 .
n 1 l0
l
k0 k
Therefore by 2.18, we obtain the following theorem.
2.18
International Journal of Mathematics and Mathematical Sciences
7
Theorem 2.5. For n ∈ Z , one has
Hn −q
−1
n
l
n1 l
1 Bl−k Hk −q−1 .
n 1 l0
l
k0 k
2.19
From 1.3, we can derive the following:
n−1 n
1
El x.
x En x 2 l0 l
2.20
n
Let us take the fermionic p-adic q-integral on Zp in 2.20 as follows:
n−1 n
1
x dμ−q x En xdμ−q x El xdμ−q x
2 l0 l
Zp
Zp
Zp
n
n
n
En−l Hl −q−1 l
l0
n−1 n l
l
1
2 l0
l
k0
k
2.21
El−k Hk −q−1 .
Therefore, by 2.21, we obtain the following theorem.
Theorem 2.6. For n ∈ N, one has
Hn −q
−1
n
n
l0
l
En−l Hl −q
−1
n−1 n l
l
1
El−k Hk −q−1 .
2 l0 l k0 k
2.22
By Theorems 2.5 and 2.6, we obtain the following corollary.
Corollary 2.7. For n ∈ N, one has
n
l
n1 l
1 Bl−k Hk −q−1
n 1 l0
l
k0 k
n
n
l0
l
En−l Hl −q−1
n−1 n l
l
1
El−k Hk −q−1 .
2 l0 l k0 k
By 1.3, we easily get En x −1n En 1 − x.
2.23
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International Journal of Mathematics and Mathematical Sciences
Thus, we have
Zp
xn dμ−q x
n−1 n
1
l
En 1 − xdμ−q x El 1 − xdμ−q x
−1
−1
2 l0 l
Zp
Zp
n
n
n
−1
En−l
1 − xl dμ−q x
l
Zp
l0
l
n−1 n
l
1
l
El−k
1 − xk dμ−q x
−1
2 l0 l
k
Zp
k0
n
n
n
l0
En−l −1
l
n
n
l0
l
n−l
2.24
n−1 n l
1
l
El−k −1l−k Hk −q−1 , −1
Hl −q , −1 2 l0 l k0 k
−1
n−1 n l
l
1
En−l −1 Hl −q, 2 El−k −1l Hk −q, 2 .
2 l0 l k0 k
n
Therefore, by 2.24, we obtain the following theorem.
Theorem 2.8. For n ∈ N, one has
Hn −q
−1
n
n
l0
l
En−l −1n Hl −q, 2
n−1 n l
l
1
El−k −1l Hk −q, 2 .
2 l0 l k0 k
2.25
Acknowledgment
The second author was supported by National Research Foundation of Korea Grant funded
by the Korean Government 2009-0072514.
References
1 T. Kim, “Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the
fermionic p-adic integral on Zp ,” Russian Journal of Mathematical Physics, vol. 16, no. 4, pp. 484–491,
2009.
2 L. Carlitz, “The product of two eulerian polynomials,” Mathematics Magazine, vol. 36, no. 1, pp. 37–41,
1963.
3 A. Bayad, “Modular properties of elliptic Bernoulli and Euler functions,” Advanced Studies in
Contemporary Mathematics, vol. 20, no. 3, pp. 389–401, 2010.
4 C.-P. Chen and L. Lin, “An inequality for the generalized-Euler-constant function,” Advanced Studies
in Contemporary Mathematics, vol. 17, no. 1, pp. 105–107, 2008.
5 T. Kim, J. Choi, and Y. H. Kim, “A note on Bernstein polynomials associated with q-Euler numbers
and polynomials,” International Journal of Mathematics and Analysis, vol. 5, no. 48, pp. 2413–2420, 2011.
International Journal of Mathematics and Mathematical Sciences
9
6 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks on q-Bernoulli numbers associated with Daehee
numbers,” Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.
7 S.-H. Rim, S. J. Lee, E. J. Moon, and J. H. Jin, “On the q-Genocchi numbers and polynomials associated
with q-zeta function,” Proceedings of the Jangjeon Mathematical Society, vol. 12, no. 3, pp. 261–267, 2009.
8 C. S. Ryoo, “A note on the weighted q-Euler numbers and polynomials,” Advanced Studies in
Contemporary Mathematics, vol. 21, pp. 47–54, 2011.
9 C. S. Ryoo, “On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots
of unity,” Proceedings of the Jangjeon Mathematical Society, vol. 13, no. 2, pp. 255–263, 2010.
10 Y. Simsek, “Generating functions of the twisted Bernoulli numbers and polynomials associated with
their interpolation functions,” Advanced Studies in Contemporary Mathematics, vol. 16, no. 2, pp. 251–
278, 2008.
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