Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 294715, 10 pages
doi:10.1155/2011/294715
Research Article
Some Identities on Bernstein Polynomials
Associated with q-Euler Polynomials
A. Bayad,1 T. Kim,2 B. Lee,3 and S.-H. Rim4
1
Département de Mathématiques, Université d’Evry Val d’Essonne, Bboulevard F. Mitterrand,
91025 Evry Cedex, France
2
Division of General Education-Mathematics, Kwang-Woon University, Seoul 139-701, Republic of Korea
3
Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
4
Department of Mathematics Education, Kyungpook National University,
Taegu 702-701, Republic of Korea
Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr
Received 11 January 2011; Accepted 11 February 2011
Academic Editor: John Rassias
Copyright q 2011 A. Bayad 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 investigate some interesting properties of the q-Euler polynomials. The purpose of this paper
is to give some relationships between Bernstein and q-Euler polynomials, which are derived by the
p-adic integral representation of the Bernstein polynomials associated with q-Euler polynomials.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper p, p , and p denote the ring of
p-adic integers, the field of p-adic numbers, and the field of p-adic completion of the algebraic
closure of p , respectively see 1–15. Let be the set of natural numbers and ∪ {0}.
The normalized p-adic absolute value is defined by |p|p 1/p. As an indeterminate, we
assume that q ∈ p with |1 − q|p < 1. Let UD p be the space of uniformly differentiable
function on p. For f ∈ UD p, the p-adic invariant integral on p is defined by
I−1 f fxdµ−1 x lim
p
N→∞
lim
fxµ−1 x pN
N
−1
p
x0
N
−1
p
N→∞
fx−1x ,
x0
p
1.1
2
Abstract and Applied Analysis
see 7–10. For n ∈ , we can derive the following integral equation from 1.1:
I−1 fn −1n
fxdµ−1 x 2
n−1
−1n−1−l fl,
1.2
l0
p
where fn x fx n see 7–11. As well-known definition, the Euler polynomials are
given by the generating function as follows:
et
∞
t
2
ext eExt En x ,
n!
1
n0
1.3
see 3, 5, 7–15, with usual convention about replacing En x by En x. In the special case
x 0, En 0 En are called the nth Euler numbers. From 1.3, we can derive the following
recurrence formula for Euler numbers:
E0 1,
E 1n En ⎧
⎨2 if n 0,
⎩0 if n > 0,
1.4
see 12, with usual convention about replacing En by En . By the definitions of Euler
numbers and polynomials, we get
n
n n−l
x El ,
En x E x l
l0
n
1.5
see 3, 5, 7–15. Let C0, 1 denote the set of continuous functions on 0, 1. For f ∈ C0, 1,
Bernstein introduced the following well-known linear positive operator in the field of real
numbers :
n
n n
n k
k
n−k
k
f |x x 1 − x
f
f
Bk,n x,
n
n
k
k0
k0
1.6
where nk nn − 1 · · · n − k 1/k! n!/k!n − k! see 1, 2, 7, 11, 12, 14. Here, n f | x
is called the Bernstein operator of order n for f. For k, n ∈ , the Bernstein polynomials of
degree n are defined by
n k
Bk,n x x 1 − xn−k ,
k
for x ∈ 0, 1.
1.7
In this paper, we study the properties of q-Euler numbers and polynomials. From these
properties, we investigate some identities on the q-Euler numbers and polynomials. Finally,
we give some relationships between Bernstein and q-Euler polynomials, which are derived
by the p-adic integral representation of the Bernstein polynomials associated with q-Euler
polynomials.
Abstract and Applied Analysis
3
2. q-Euler Numbers and Polynomials
In this section, we assume that q ∈ p with |1 − q|p < 1. Let fx qx ext . From 1.1 and 1.2,
we have
2
.
qet 1
2.1
∞
2
tn
Eq t
e
,
E
n,q
qet 1
n!
n0
2.2
fxdµ−1 x p
Now, we define the q-Euler numbers as follows:
with the usual convention about replacing Eqn by En,q .
By 2.2, we easily get
2
,
q1
E0,q
⎧
⎨2 if n 0,
n
q Eq 1 En,q ⎩0 if n > 0,
2.3
with usual convention about replacing Eqn by En,q .
We note that
∞
n
2
2
2
2 −1 t
·
H
,
−q
n
n!
qet 1 et q−1 1 q 1 q n0
2.4
where Hn −q−1 is the nth Frobenius-Euler numbers.
From 2.1, 2.2, and 2.4, we have
qx ext dµ−1 x En,q p
2
Hn −q−1 ,
1q
for n ∈
.
2.5
Now, we consider the q-Euler polynomials as follows:
∞
2
tn
xt
Eq xt
e
,
e
E
x
n,q
qet 1
n!
n0
2.6
with the usual convention Eqn x by En,q x.
From 1.2, 2.1, and 2.6, we get
qx exyt dµ−1 y p
∞
tn
2
xt
e
.
E
x
n,q
qet 1
n!
n0
2.7
4
Abstract and Applied Analysis
By comparing the coefficients on the both sides of 2.6 and 2.7, we get the following Witt’s
formula for the q-Euler polynomials as follows:
n
n n−l
q x y dµ−1 y En,q x x El,q .
l
p
l0
y
n
2.8
From 2.6 and 2.8, we can derive the following equation:
∞
2q
2
t
1−xt
−xt
e
En,q−1 x−1n .
e
t
−1
−t
qe 1
n!
1q e
n0
2.9
By 2.6 and 2.9, we obtain the following reflection symmetric property for the q-Euler
polynomials.
Theorem 2.1. For n ∈
,
one has
−1n En,q−1 x qEn,q 1 − x.
2.10
From 2.5, 2.6, 2.7, and 2.8, we can derive the following equation: for n ∈ ,
n
n
n En,q 2 Eq 1 1 El,q 1
l
l0
n
n
n
n
1
1
2
qEl,q 1 El,q
−
E0,q q l1 l
1 q q l1 l
n
2
1
2
El,q
−
1q q 1q
q l0 l
2.11
n
2
1
1
2
− qEn,q 1 2 En,q ,
q q2
q q
by using recurrence formula 2.3. Therefore, we obtain the following theorem.
Theorem 2.2. For n ∈ , one has
1
qEn,q 2 2 En,q .
q
2.12
Abstract and Applied Analysis
5
By using 2.5 and 2.8, we get
q−x 1 − xn dµ−1 x −1n
p
q−x x − 1n dµ−1 x
p
−1n En,q−1 −1 q
1
1
2 En,q 2 q
q
x 2n dµ−1 x q
p
xn qx dµ−1 x,
1
2
2 En,q
q q
2.13
for n > 0.
p
Therefore, we obtain the following theorem.
Theorem 2.3. For n ∈ , one has
q−x 1 − xn dµ−1 x 2 p
1
q
xn qx dµ−1 x.
2.14
p
By using Theorem 2.3, we will study for the p-adic integral representation on
Bernstein polynomials associated with q-Euler polynomials in Section 3.
p
of the
3. Bernstein Polynomials Associated with
q-Euler Numbers and Polynomials
Now, we take the p-adic integral on
Bk,n xq dµ−1 x x
p
p
n
p
for the Bernstein polynomials in 1.7 as follows:
k
xk 1 − xn−k qx dµ−1 x
n−k
n n−k
k
j0
j
n−k n n−k
k
j0
j
n−k n n−k
k
j0
j
n−k−j
−1
xn−j qx dµ−1 x
p
3.1
−1n−k−j En−j,q
−1j Ekj,q ,
where n, k ∈
.
By the definition of Bernstein polynomials, we see that
Bk,n x Bn−k,n 1 − x,
where n, k ∈
.
3.2
6
Abstract and Applied Analysis
Let n, k ∈
with n > k. Then, by 3.2, we get
qx Bk,n xdµ−1 x qx Bn−k,n 1 − xdµ−1 x
p
p
k k
k−j
−1
n − k j0 j
n
1 − xn−j qx dµ−1 x
p
k
n k
k−j
2q
−1
k j0 j
x
q dµ−1 x
n−j x
p
3.3
k
n k
−1k−j 2 qEn−j,q−1
k j0 j
⎧
⎪
2 qEn,q−1
⎪
⎪
⎪
⎨⎛ ⎞
⎛ ⎞
k
k
n
⎪
⎪⎝ ⎠q ⎝ ⎠−1k−j En−j,q−1
⎪
⎪
⎩ k
j
if k 0,
if k > 0.
j0
Thus, we obtain the following theorem.
Theorem 3.1. For n, k ∈
with n > k, one has
⎧
⎪
2q En,q
⎪
⎪
⎪
⎨⎛ ⎞ ⎛ ⎞
q1−x Bk,n xdµ−1 x k
⎪⎝n⎠⎝k⎠
⎪
p
⎪
−1k−j En−j,q
⎪
⎩ k
j
if k 0,
3.4
if k > 0.
j0
By 3.1 and Theorem 3.1, we get the following corollary.
Corollary 3.2. For n, k ∈
n−k
n−k
j0
j
with n > k, one has
−1j Ekj,q−1
⎧
1
⎪
⎪
2 En,q
⎪
⎪
⎪
q
⎨
⎛ ⎞
k
k
⎪
⎪
⎪ ⎝ ⎠−1k−j 1 En−j,q
⎪
⎪
⎩ j0 j
q
if k 0,
3.5
if k > 0.
Abstract and Applied Analysis
For m, n, k ∈
7
with m n > 2k. Then, we get
Bk,n xBk,m xq−x dµ−1 x
p
2k
n
m 2k
j2k
−1
k
k j0 j
q−x 1 − xnm−j dµ−1 x
p
2k
2k
n
m j2k
q
−1
k
k j0 j
x 2nm−j qx dµ−1 x
p
3.6
2k n
m 2k
1
2
2 Enm−j,q
−1j2k q
q q
k
k j0 j
⎧
1
⎪
⎪
⎪2 Enm,q
⎪
⎪
q
⎨
⎛ ⎞⎛ ⎞ ⎛ ⎞
2k
⎪
1
⎪⎝n⎠⎝m⎠⎝2k⎠
⎪
−1j2k Enm−j,q
⎪
⎪
⎩ k
q
j
k j0
if k 0,
if k > 0.
Therefore, we obtain the following theorem.
Theorem 3.3. For m, n, k ∈
with m n > 2k, one has
⎧
⎪
2q Enm,q
⎪
⎪
⎪
⎨⎛ ⎞⎛ ⎞ ⎛ ⎞
Bk,n xBk,m xq1−x dµ−1 x 2k
n
m 2k
⎪
⎪
⎝ ⎠⎝ ⎠ ⎝ ⎠−1j2k Enm−j,q
p
⎪
⎪
⎩
k
k j0 j
if k 0,
if k > 0.
3.7
By using binomial theorem, for m, n, k ∈
,
we get
3.8
Bk,n xBk,m xq1−x d µ−1 x
p
nm−2k
n m − 2k
n
m
j
−1
j
k
k
j0
xj2k q1−x dµ−1 x
p
n
m nm−2k
n m − 2k
q
−1j Ej2k,q−1 .
k
k
j
j0
3.9
By comparing the coefficients on the both sides of 3.8 and Theorem 3.3, we obtain the
following corollary.
8
Abstract and Applied Analysis
Corollary 3.4. Let m, n, k ∈
nm−2k
j0
with m n > 2k. Then, we get
n m − 2k
−1j Ej2k,q−1
j
For s ∈ , let n1 , n2 , . . . , ns , k ∈
⎧
1
⎪
⎪
2 Enm,q
⎪
⎪
⎪
q
⎨
⎛ ⎞
2k
2k
⎪
1
⎪
⎝ ⎠−1j2k Enm−j,q
⎪
⎪
⎪
⎩ q j0
j
if k 0,
3.10
if k > 0.
with n1 n2 · · · ns > sk. By induction, we get
Bk,n1 x · · · Bk,ns xq−x dµ−1 x
p
s
ni
i1
k
xsk 1 − xn1 ···ns −sk q−x dµ−1 x
p
s sk ni
sk
skj
−1
j
k
j0
i1
1 − xn1 ···ns −j q−x dµ−1 x
p
s
sk
ni
sk
skj
q
−1
j
k
j0
i1
x 2n1 ···ns −j qx dµ−1 x
3.11
p
s sk
ni
sk
1
2
2 En1 ···ns −j,q
−1skj q
q q
j
k
i1
j0
⎧
1
⎪
⎪
2 En1 ···ns ,q
⎪
⎪
⎪
q
⎨
⎛ ⎛ ⎞⎞
⎛ ⎞
s
sk
n
sk
⎪
i
1
⎪⎝ ⎝ ⎠⎠
⎝ ⎠−1skj En1 ···ns −j,q
⎪
⎪
⎪
⎩ i1 k
q j0 j
if k 0,
if k > 0.
Therefore, we obtain the following theorem.
Theorem 3.5. Let s ∈ . For n1 , n2 , . . . , ns , k ∈
with n1 n2 · · · ns > sk, one has
⎧
⎪
2q En1 n2 ···ns ,q
⎪
⎪
⎪
s
⎨⎛ ⎛ ⎞⎞ ⎛ ⎞
Bk,ni x q1−x dµ−1 x s
sk
ni
sk
⎪
⎪
⎝ ⎝ ⎠⎠ ⎝ ⎠−1skj En1 n2 ···ns −j,q
p
i1
⎪
⎪
⎩
k
j
i1
j0
if k 0,
if k > 0.
3.12
Abstract and Applied Analysis
For n1 , n2 , . . . , ns , k ∈
s
p
9
by binomial theorem, we get
Bk,ni x q−x dµ−1 x
i1
n1
k
n1
k
n ···n −sk
n1 · · · ns − sk
ns 1 s
···
−1j
j
k
j0
xjsk q−x dµ−1 x
3.13
p
n ···n −sk
n1 · · · ns − sk
ns 1 s
···
−1j Ejsk,q−1 .
k
j
j0
By using 3.13 and Theorem 3.5, we obtain the following corollary.
Corollary 3.6. Let s ∈ . For n1 , n2 , . . . , ns , k ∈
n1 ···n
s −sk
n1 · · · ns − sk
j0
j
−1j Ejsk,q−1
with n1 n2 · · · ns > sk, one has
⎧
1
⎪
⎪
2 En1 n2 ···ns ,q
⎪
⎪
⎪
q
⎨
⎛ ⎞
sk
sk
⎪
1
⎪
⎪
⎝ ⎠−1skj En1 n2 ···ns −j,q
⎪
⎪
⎩ q j0 j
if k 0,
if k > 0.
3.14
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