Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 864247, 12 pages
doi:10.1155/2010/864247
Research Article
On the Fermionic p-adic Integral Representation of
Bernstein Polynomials Associated
with Euler Numbers and Polynomials
T. Kim,1 J. Choi,1 Y. H. Kim,1 and C. S. Ryoo2
1
2
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea
Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea
Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr
Received 30 August 2010; Accepted 3 December 2010
Academic Editor: Paolo E. Ricci
Copyright q 2010 T. Kim 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 properties of several Bernstein type polynomials to
represent the fermionic p-adic integral on p. From these properties, we derive some interesting
identities on the Euler numbers and polynomials.
1. Introduction
Throughout this paper, let p be an odd prime number. The symbol, p, p , and p denote the
ring of p-adic integers, the field of p-adic rational numbers, the complex number field and the
completion of algebraic closure of p , respectively.
Let be the set of natural numbers and ∪ {0}. Let νp be the normalized
exponential valuation of p with |p|p p−νp p 1/p. Note that p {x | |x|p ≤ 1} lim← /pN p.
N
When one talks of q-extension, q is variously considered as an indeterminate, a
complex number q ∈ , or p-adic number q ∈ p . If q ∈ , we normally assume |q| < 1,
and if q ∈ p , we always assume |1 − q|p < 1.
We say that f is uniformly differentiable function at a point a ∈ p and write
f ∈ UD p, if the difference quotient Ff x, y fx − fy/x − y has a limit f a
as x, y → a, a. For f ∈ UD p, the fermionic p-adic q-integral on p is defined as
I−q f −1
x
1 q p
fxdμ−q x lim
fx −q ,
N
p
N →∞1 q
p
x0
N
1.1
2
Journal of Inequalities and Applications
see 1. In the special case q 1 in 1.1, the integral
I−1 f fxdμ−1 x,
1.2
p
is called the fermionic p-adic invariant integral on
p
see 2. From 1.2, we note
I−1 f1 −I−1 f 2f0,
1.3
where f1 x fx 1.
Moreover, for n ∈ , let fn x fx n. Then we note that
n−1
I−1 fn −1n I−1 f 2 −1n−1−l fl.
1.4
l0
It is well known that the Euler polynomials are defined by
∞
2
tn
xt
e
E
,
x
n
n!
et 1
n0
1.5
see 1–15. In the special case, x 0, and En 0 En are called the nth Euler numbers.
Let fx etx . Then, by 1.3, 1.4, and 1.5, we see that
exyt dμ−1 y p
∞
tn
2
xt
e
.
E
x
n
et 1
n!
n0
1.6
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
xk 1 − xn−k Bk,n x,
f
f
f :x n
n
k
k0
k0
1.7
where nk nn − 1 · · · n − k 1/k! n!/k!n − k! see 3, 4, 7, 10, 11, 14. Here, n f : x
is called the Bernstein operator of order n for f.
For k, n ∈ , the Bernstein polynomial of degree n is defined by
n k
Bk,n x x 1 − xn−k ,
k
for x ∈ 0, 1.
1.8
For example, B0,1 x 1 − x, B1,1 x x, B0,2 x 1 − x2 , B1,2 x 2x − 2x2 , B2,2 x x2 , . . .,
and Bk,n x 0 for n < k, Bk,n x Bn−k,n 1 − x.
In this paper, we study the properties of Bernstein polynomials in the p-adic number
field. For f ∈ UD p, we give some properties of several type Bernstein polynomials
Journal of Inequalities and Applications
3
p.
to represent the fermionic p-adic invariant integral on
some interesting identities on the Euler polynomials.
From those properties, we derive
2. Fermionic p-adic Integral Representation of Bernstein Polynomials
By 1.5 and 1.6, we see that
∞
2
tn
1−xt
e
.
E
−
x
1
n
et 1
n!
n0
2.1
We also have that
et
∞
2
2
−1n n
e1−xt e−xt En x
t .
−t
n!
1
1e
n0
2.2
From 2.1 and 2.2, we note that En 1 − x −1n En x. It is easy to show that
n
n
El 2 En ,
En 2 2 −
l
l0
for n > 0.
2.3
By 1.5, 1.6, 2.1, 2.2, and 2.3, we see that for n > 0,
1 − xn dμ−1 x −1n
p
x − 1n dμ−1 x p
2
x 2n dμ−1 x
p
2.4
x dμ−1 x.
n
p
Therefore, we obtain the following theorem.
Theorem 2.1. For n ∈ , one has
n
1 − x dμ−1 x 2 p
xn dμ−1 x.
p
Theorem 2.1 is important to derive our main result in this paper.
2.5
4
Journal of Inequalities and Applications
p for one Bernstein polynomial
Taking the fermionic p-adic integral on
in 1.8, we get
n
xk 1 − xn−k dμ−1 x
k
p
n−k
n n−k
n−k−j
xn−j dμ−1 x
−1
k j0
j
p
Bk,n xdμ−1 x p
n−k
n n−k
k
j
j0
n−k
n n−k
k
j
j0
2.6
−1n−k−j En−j
−1j Ekj .
Therefore, we obtain the following proposition.
Proposition 2.2. For k, n ∈
,
one is
Bk,n xdμ−1 x n−k
n n−k
p
k
j0
j
−1j Ekj .
2.7
It is known that Bk,n x Bn−k,n 1 − x. Thus, one has
Bk,n xdμ−1 x Bn−k,n 1 − xdμ−1 x
p
p
k
k
−1k−j
n − k j0 j
n
2.8
1 − x
n−j
dμ−1 x.
p
By 2.8 and Theorem 2.1, we see that for n > k,
k
k
n k−j
Bk,n xdμ−1 x 2
−1
k j0 j
p
x
n−j
dμ−1 x
p
k
k
n −1k−j 2 En−j
k j0 j
⎧
⎪
if k 0,
⎪
⎪2 En
⎪
⎨⎛ ⎞ ⎛ ⎞
k
k
n ⎪
⎝ ⎠ ⎝ ⎠−1k−j En−j
⎪
if k > 0.
⎪
⎪
⎩ k j0 j
From 2.9, we obtain the following theorem.
2.9
Journal of Inequalities and Applications
Theorem 2.3. For n, k ∈
5
with n > k, we have
⎧
⎪
2 En
⎪
⎪
⎪
⎨⎛ ⎞ ⎛ ⎞
k
Bk,n xdμ−1 x n k
⎪
⎝ ⎠ ⎝ ⎠−1k−j En−j
⎪
p
⎪
⎪
⎩ k j0 j
if k 0,
2.10
if k > 0.
By Proposition 2.2 and Theorem 2.3, we obtain the following corollary.
Corollary 2.4. For n, k ∈
with n > k, we have
n−k
−1j Ekj
j
j0
n−k
⎧
⎪
2 En
⎪
⎪
⎪
⎨ ⎛ ⎞
k
k
⎪
⎝ ⎠−1k−j En−j
⎪
⎪
⎪
⎩ j0 j
if k 0,
2.11
if k > 0.
For m, n, k ∈ with m n > 2k, fermionic p-adic invariant integral for multiplication
of two Bernstein polynomials on p can be given by the following relation:
n k
m
n−k
Bk,n xBk,m xdμ−1 x x 1 − x
xk 1 − xm−k dμ−1 x
k
k
p
p
n
m
k
k
x2k 1 − xnm−2k dμ−1 x
p
2k
n
m 2k
−1j2k
k
k j0 j
1 − xnm−j dμ−1 x
p
2k
2k
n
m j2k
2
−1
k
k j0 j
xnm−j dμ−1 x
p
⎧
⎪
2 Enm
⎪
⎪
⎪
⎨⎛ ⎞⎛ ⎞ ⎛ ⎞
2k
n
m 2k
⎪
⎪⎝ ⎠⎝ ⎠ ⎝ ⎠−1j2k Enm−j
⎪
⎪
⎩ k
k j0
j
if k 0,
if k > 0.
2.12
6
Journal of Inequalities and Applications
Therefore, we obtain the following theorem.
Theorem 2.5. For m, n, k ∈
with m n > 2k, one has
⎧
⎪
2 Enm
⎪
⎪
⎪
⎨⎛ ⎞⎛ ⎞ ⎛ ⎞
2k
Bk,n xBk,m xdμ−1 x n
m 2k
⎪
⎪⎝ ⎠⎝ ⎠ ⎝ ⎠−1j2k Enm−j
p
⎪
⎪
⎩ k
k j0
j
For m, n, k ∈
,
if k 0,
if k > 0.
2.13
xj2k dμ−1 x
2.14
one has
Bk,n xBk,m xdμ−1 x p
n
m
k
k
x2k 1 − xnm−2k dμ−1 x
p
n m − 2k
n
m nm−2k
−1j
j
k
k
j0
p
nm−2k
n
m
n m − 2k
−1j Ej2k .
k
k
j
j0
Thus, we obtain the following proposition.
Proposition 2.6. For m, n, k ∈
,
one has
n m − 2k
n
m nm−2k
Bk,n xBk,m xdμ−1 x −1j Ej2k .
j
k
k
p
j0
2.15
By Theorem 2.5 and Proposition 2.6, we obtain the following corollary.
Corollary 2.7. For m, n, k ∈
nm−2k
n m − 2k
j0
j
with m n > 2k, one has
−1j Ej2k
⎧
⎪
2 Enm
⎪
⎪
⎪
⎨ ⎛ ⎞
⎪ 2k ⎝2k⎠
⎪
⎪
−1j2k Enm−j
⎪
⎩
j
j0
if k 0,
2.16
if k > 0.
Journal of Inequalities and Applications
7
In the same manner, multiplication of three Bernstein polynomials can be given by the
following relation:
Bk,n xBk,m xBk,s xdμ−1 x
p
nms−3k n m s − 3k
n
m
s
k
k
k
j0
j
k
k
j0
xj3k dμ−1 x
−1
p
n
m
s nms−3k
n m s − 3k
k
j
j
2.17
−1j Ej3k ,
where m, n, s, k ∈ with m n s > 3k.
For m, n, s, k ∈ with m n s > 3k, by the symmetry of Bernstein polynomals, we
see that
Bk,n xBk,m xBk,s xdμ−1 x
p
3k
n
m
s 3k
3k−j
−1
k
k
k j0 j
1 − xnms−j dμ−1 x
p
3k
3k
n
m
s 3k−j
2
−1
k
k
k j0 j
x
nms−j
μ−1 x
2.18
p
⎧
⎪
2 Enms
⎪
⎪
⎪
⎨⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞
3k
n
m
s 3k
⎪
⎪⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠−13k−j Enms−j
⎪
⎪
⎩ k
k
k j0
j
if k 0,
if k > 0.
Therefore, we obtain the following theorem.
Theorem 2.8. For m, n, s, k ∈
with m n s > 3k, one has
Bk,n xBk,m xBk,s xdμ−1 x
p
⎧
⎪
2 Enms
⎪
⎪
⎪
⎨⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞
3k
n
m
s 3k
⎪
⎪⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠−13k−j Enms−j
⎪
⎪
⎩ k
k
k j0 j
By 2.17 and Theorem 2.8, we obtain the following corollary.
if k 0,
if k > 0.
2.19
8
Journal of Inequalities and Applications
Corollary 2.9. For m, n, s, k ∈
nms−3k
j0
with m n s > 3k, one has
n m s − 3k
−1j Ej3k
j
⎧
⎪
2 Enms
⎪
⎪
⎪
⎨ ⎛ ⎞
3k
3k
⎪
⎪ ⎝ ⎠−13k−j Enms−j
⎪
⎪
⎩ j0
j
if k 0,
2.20
if k > 0.
Using the above theorems and mathematical induction, we obtain the following
theorem.
Theorem 2.10. Let s ∈ . For n1 , n2 , . . . , ns , k ∈ with n1 n2 · · · ns > sk, the multiplication
of the sequence of Bernstein polynomials Bk,n1 x, . . . , Bk,ns x with different degrees under fermionic
p-adic invariant integral on p can be given as
⎧
⎪
⎪
⎪2 En1 n2 ···ns
⎪
s
⎨⎛ ⎛ ⎞⎞ ⎛ ⎞
s
sk
Bk,ni x dμ−1 x ni
sk
⎪
⎝ ⎝ ⎠⎠ ⎝ ⎠−1sk−j En n ···n −j
⎪
p
i1
⎪
1
2
s
⎪
⎩ i1 k
j
j0
if k 0,
if k > 0.
2.21
We also easily see that
s
p
Bk,ni x dμ−1 x i1
s n ···n −sk
1 s
ni
n1 · · · ns − sk
i1
k
j0
j
−1j Ejsk .
2.22
By Theorem 2.10 and 2.22, we obtain the following corollary.
Corollary 2.11. Let s ∈ . For n1 , n2 , . . . , ns , k ∈
n1 ···n
s −sk
n1 · · · ns − sk
j0
j
−1j Ejsk
with n1 n2 · · · ns > sk, one has
⎧
⎪
2 En1 n2 ···ns
⎪
⎪
⎪
⎨ ⎛ ⎞
sk
sk
⎪
⎪
⎝ ⎠−1sk−j En1 n2 ···ns −j
⎪
⎪
⎩
j
j0
if k 0,
if k > 0.
2.23
Journal of Inequalities and Applications
Let m1 , . . . , ms , n1 , . . . , ns , k ∈
ms
x, we easily get
definition of Bk,n
s
9
with m1 n1 · · · ms ns > m1 · · · ms k. By the
s
mi
Bk,ni x dμ−1 x
i1
p
mi k s mi
s
i1
ni
i1
k
k
−1
s
i1
mi −j
j0
1 − x
s
i1
ni mi −j
dμ−1 x
p
⎛ s
⎞
mi k s mi
s
i1
s
ni
⎜k mi ⎟
⎝ i1 ⎠−1k i1 mi −j 2 Esi1 ni mi −j
k
j0
i1
j
⎧
⎪
2 Em1 n1 ···ms ns
⎪
⎪
⎪
⎪
⎨⎛ ⎛ ⎞m ⎞ ⎛ ⎞
s
i
s
k
m
i
s
i1
s
⎪⎝⎝ni ⎠ ⎠ ⎜k mi ⎟
⎪
⎝ i1 ⎠−1k i1 mi −j Esi1 ni mi −j
⎪
⎪
⎪
⎩ i1 k
j0
j
2.24
if k 0,
if k > 0.
Therefore, we obtain the following theorem.
Theorem 2.12. Let s ∈ . For m1 , . . . , ms , n1 , . . . , ns , k ∈
ms k, one has
s
i1
p
with m1 n1 · · · ms ns > m1 · · · mi
Bk,n
x
i
dμ−1 x
⎧
⎪
⎪
⎪2 Em1 n1 ···ms ns
⎪
⎪
⎨ ⎛ ⎛ ⎞m ⎞ ⎛ ⎞
s
i
s
k s
i1 mi
s
n
m
k
i
i⎟
⎜
⎪
⎝ ⎝ ⎠ ⎠
⎪
⎝ i1 ⎠−1k i1 mi −j Esi1 ni mi −j
⎪
⎪
⎪
⎩ i1 k
j0
j
if k 0,
2.25
if k > 0.
By simple calculation, we easily get
s
mi
Bk,ni x dμ−1 x
p
i1
⎛ s
⎞
s
mi s ni mi −k s mi s
i1
i1 ⎜ ni mi − k mi ⎟
ni
⎝ i1
⎠−1j Ek si1 mi −j ,
i1
k
j0
i1
j
where m1 , . . . , ms , n1 , . . . , ns , k ∈
the following corollary.
for s ∈
2.26
. By Theorem 2.12 and 2.26, we obtain
10
Journal of Inequalities and Applications
Corollary 2.13. Let s ∈ . For m1 , . . . , ms , n1 , . . . , ns , k ∈
ms k, one has
s
i1
ni m
i −k
s
i1
with m1 n1 · · · ms ns > m1 · · · ⎛
mi
j0
⎞
s
s
⎜ ni mi − k mi ⎟
⎝ i1
⎠−1j Ek si1 mi −j
i1
j
⎧
⎪
2 Em1 n1 ···ms ns
⎪
⎪
⎪
⎪
⎨ ⎛ ⎞
s
s
k i1 mi
s
m
k
i
⎜
⎟
⎪
⎪
⎝ i1 ⎠−1k i1 mi −j Esi1 ni mi −j
⎪
⎪
⎪
⎩ j0
j
if k 0,
2.27
if k > 0.
The fermionic p-adic invariant integral of multiplication of n 1 Bernstein
polynomials, the nth degree Bernstein polynomials Bi,n x with i 0, 1, . . . , n and with
multiplicity m0 , m1 , . . . , mn on p, respectively, can be given by
n
p
i0
mi
Bi,n
x
dμ−1 x mi n
n
i0
i
n
ni mi
i1
n
n i0 mi
n
i1 imi
x
n
i1
imi
1 − xn
n
i0
mi − ni1 imi
dμ−1 x
p
2.28
Bni1 imi , n ni0 mi xdμ−1 x,
p
where m0 , m1 , . . . , mn ∈ with n ∈ .
Assume that nm0 nm1 · · · nmn > m1 2m2 · · · nmn . Then one has
n
p
i0
mi
Bi,n
x
dμ−1 x
⎧
⎪
⎪
2 Enm0 nm1 ···nmn
⎪
⎪
⎪
⎪
⎨
⎞
⎛ n
⎛ n ⎛ ⎞mi ⎞ni1 mi n
n
⎪
im
i⎟
⎪
⎜
⎪⎝ ⎝ ⎠ ⎠
i1 imi −j
⎪
En ni0 mi −ni1 imi
⎠−1
⎝
⎪
i1
⎪
⎩ i0 i
j0
j
Therefore, we obtain the following theorem.
if
n
imi 0,
i1
if
n
i1
imi > 0.
2.29
Journal of Inequalities and Applications
Theorem 2.14. Let n ∈
.
i For m0 , m1 , . . . , mn ∈
n
p
i0
with n
11
n
i0
mi >
n
i1
imi , one has
mi
Bi,n
x
dμ−1 x
⎧
⎪
⎪
2 Enm0 nm1 ···nmn
⎪
⎪
⎪
⎪
⎨
⎛ n
⎞
⎛ n ⎛ ⎞mi ⎞ni1 mi n
n
⎪
⎪
⎜ imi ⎟
⎪
i1 imi −j
⎝ ⎝ ⎠ ⎠
⎪
En ni0 mi −ni1 imi
⎝
⎠−1
⎪
i1
⎪
⎩ i0 i
j0
j
ii For m0 , m1 , . . . , mn ∈
n
p
i0
mi
Bi,n
x dμ−1 x ,
if
n
imi 0,
i1
if
2.30
n
imi > 0.
i1
one has
⎞
⎛ n
n
mi n n mi −n imi
n
i0 i1
n
m
−
im
n
i
i⎟
⎜
⎠−1j Eni1 imi j .
⎝ i0
i1
i
i0
j0
j
2.31
By Theorem 2.14, we obtain the following corollary.
Corollary 2.15. For n, m0 , m1 , . . . , mn ∈
n
n
i0
n
m
i − i1 imi
j0
with n
n
i0
mi >
n
i1
imi , one has
⎛
⎞
n
n
m
−
im
n
i
i⎟
⎜
⎝ i0
⎠−1j Eni1 imi j
i1
j
⎧
⎪
⎪
2 Enm0 nm1 ···nmn
⎪
⎪
⎪
⎪
⎨
⎛ n
⎞
ni1 mi n
⎪
⎪
⎜ imi ⎟
⎪
i1 imi −j
⎪
En ni0 mi −ni1 imi
⎝
⎠−1
⎪
i1
⎪
⎩ j0
j
if
n
imi 0,
2.32
i1
if
n
imi > 0.
i1
References
1 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299,
2002.
2 T. Kim, “Barnes-type multiple q-zeta functions and q-Euler polynomials,” Journal of Physics A, vol. 43,
no. 25, Article ID 255201, 11 pages, 2010.
3 M. Acikgoz and S. Araci, “A study on the integral of the product of several type Bernstein
polynomials,” IST Transaction of Applied Mathematics-Modelling and Simulation. In press.
4 S. Bernstein, “Démonstration du théorème de Weierstrass, fondée sur le calcul des probabilities,”
Communications of the Kharkov Mathematical Society, vol. 13, pp. 1–2, 1912.
12
Journal of Inequalities and Applications
5 T. Kim, J. Choi, and Y. H. Kim, “On extended carlitz’s type q-Euler numbers and polynomials,”
Advanced Studies in Contemporary Mathematics, vol. 20, no. 4, pp. 499–505, 2010.
6 N. K. Govil and V. Gupta, “Convergence of q-Meyer-König-Zeller-Durrmeyer operators,” Advanced
Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 97–108, 2009.
7 V. Gupta, T. Kim, J. Choi, and Y.-H. Kim, “Generating function for q-Bernstein, q-Meyer-König-Zeller
and q-beta basis,” Automation Computers Applied Mathematics, vol. 19, pp. 7–11, 2010.
8 T. Kim, “q-extension of the Euler formula and trigonometric functions,” Russian Journal of
Mathematical Physics, vol. 14, no. 3, pp. 275–278, 2007.
9 T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients,”
Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.
10 T. Kim, J. Choi, and Y.-H. Kim, “Some identities on the q-Bernstein polynomials, q-Stirling numbers
and q-Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 335–341,
2010.
11 T. Kim, L.-C. Jang, and H. Yi, “A note on the modified q-bernstein polynomials,” Discrete Dynamics in
Nature and Society, vol. 2010, Article ID 706483, 12 pages, 2010.
12 T. Kim, “Note on the Euler q-zeta functions,” Journal of Number Theory, vol. 129, no. 7, pp. 1798–1804,
2009.
13 V. Kurt, “A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of
the Apostol-Genocchi polynomials,” Applied Mathematical Sciences, vol. 3, no. 53–56, pp. 2757–2764,
2009.
14 I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order w-q-Genocchi numbers,”
Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.
15 L.-C. Jang, W.-J. Kim, and Y. Simsek, “A study on the p-adic integral representation on p associated
with Bernstein and Bernoulli polynomials,” Advances in Difference Equations, vol. 2010, Article ID
163217, 6 pages, 2010.