RESEARCH Open Access
Some identities on the weighted q-Euler
numbers and q-Bernstein polynomials
Taekyun Kim
1*
, Young-Hee Kim
1
and Cheon S Ryoo
2
* Correspondence:
1
Division of General Education-
Mathematics, Kwangwoon
University, Seoul 139-701, Korea
Full list of author information is
available at the end of the article
Abstract
Recently, Ryoo introduced the weighted q-Euler numbers and polynomials which are
a slightly different Kim’s weighted q-Euler numbers and polynomials(see C. S. Ryoo, A
note on the weighted q-Euler numbers and polynomials, 2011]). In this pa per, we
give some interesting new identities on the weighted q-Euler numbers related to the
q-Bernstein polynomials
2000 Mathematics Subject Classification - 11B68, 11S40, 11S80
Keywords: Euler numbers and polynomials, q-Euler numbers and polynomials,
weighted, q-Euler numbers and polynomials, Bernstein polynomials, q-Bernstein
polynomials
1. Introduction
Let p be a fixed odd prime number. Throughout t his paper ℤ
p
,
Q

ℂ, then one normally assumes |q| <1, and if q Î ℂ
p
, then one normally assumes |q-1|
p
<1. In this paper, the q-number is defined by
[x]
q
=
1 − q
x
1 −
q
,
.
(see [1-19])
Note that lim
q®1
[x]
q
= x (see [1-19]). Let f be a continuous function on ℤ
p
.Fora Î
N and k, n Î ℤ
+
, the weighted p-adic q-Bernstein operator of order n for f is defined
by Kim as follows:
B
(α)
n,q
(f |x)=

n

B
(α)
k,n
(x, q), .
(1)
see [4,9,19].
Kim et al. Journal of Inequalities and Applications 2011, 2011:64
/>© 2011 Kim et al; licensee Sp ringer. This is an Open Access artic le distributed under the terms of the C reative Commons Attr ibution
License ( which permits unrestricted use, distribution, and reproduction in any medium,
provided the original work is properly cited .
Here
B
(α)
k,n
(x, q)=

n
k

[x]
k
q
α
[1 − x]
n−
k
q
−α

x
=
0
f (x)(−q)
x
,
(2)
see [5-19].
For n Î N, by (2), we get
q
n

Z
p
f (x + n)dμ
−q
(x)=(−1)
n

Z
p
f (x)dμ
−q
(x)+[2]
q
n−1

l=0
(−1)
n−1−l

n,
q
(0) = E
(
α
)
n,
q
are called the n-th q-Euler numbers with
weight a (see [14]).
From (4), we note that
E
(α)
n,q
(x)=
[2]
q
(1 − q
α
)
n
n

l
=
0

n
l


(α)
l,q
,
(6)
see [17].
That is, (6) can be written as
E
(α)
n,q
(x)=(q
αx
E
(α)
q
+[x]
q
α
)
n
, n ∈ Z
+
.
(7)
with usual convention about replacing
(E
(
α
)
q
)

E
(α)
n,q
(x)
t
n
n!
.
(8)
By (5) and (8), we get
F
q
(t , x)=
∞

n=0

[2]
q
(1 − q
α
)
n
n

l=0

n
l


q
(t). Then we obtain the following difference
equation.
qF
q
(t ,1)+F
q
(t )=[2]
q
.
(10)
Therefore, by (8) and (10), we obtain the following proposition.
Proposition 1. For n Î ℤ
+
, we have
E
(α)
0,
q
=1, and qE
(α)
n,q
(1) + E
(α)
n,q
=0ifn > 0
.
By (6), we easily get the following corollary.
Corollary 2. For n Î ℤ
+

n,
q
.
From (9), we note that
F
q
−1
(t ,1− x)=F
q
(−q
α
t, x)
.
(11)
Therefore, by (11), we obtain the following lemma.
Lemma 3. Let n Î ℤ
+
. Then we have
E
(α)
n,
q
−1
(1 − x)=(−1)
n
q
αn
E
(α)
n,q

−
q
= −q
n

l=1

n
l

q
αl
E
(α)
l,q
− q
= −q
n

l=0

n
l

q
αl
E
(α)
l,q
= −qE

3. Weighted q-Euler numbers concerning q-Bernstein polynomials
In this section we assume that a Î ℤ
p
and q Î ℂ
p
with |1 -q|
p
<1.
From (2), (3) and (4), we note that
q

Z
p
[1 − x]
n
q
−α
dμ
−q
(x)=(−1)
n
q
αn+1

Z
p
[x − 1]
n
q
α

, we get
q

Z
p
[1 − x]
n
q
−α
dμ
−q
(x)=(−1)
n
q
αn+1
E
(α)
n,q
(−1) = qE
(α)
n,q
−1
(2)
= q
n

l
=
0


q
−1
+[2]
q
.
For x Î ℤ
p
,thep-adic q-Bernstein polynomials with weight a of degree n are given
by
B
(α)
k,n
(x, q)=

n
k

[x]
k
q
α
[1 − x]
n−k
q
−α
,wheren, k ∈ Z
+
,
(14)
see [9].

p
B
(α)
n−k,n
(1 − x, q
−1
)dμ
−q
(x)
=

n
k

k

l
=
0

k
l

(−1)
k+l

Z
p
[1 − x]
n−l

k
l

(−1)
k+l

q
2
E
(α)
n−l,q
−1
+[2]
q

=
⎧
⎪
⎨
⎪
⎩
q
2
E
(α)
n,q
−1
+[2]
q
,ifk =0,

(α)
k,n
(x, q)dμ
−q
(x)=

n
k


Z
p
[x]
k
q
α
[1 − x]
n−k
q
−α
dμ
−q
(x)
=

n
k

n−k


l

(−1)
l
E
(α)
l+k,q
.
(18)
Therefore, by comparing the coefficients on the both sides of (17) and (18), we
obtain the following theorem.
Theorem 7. For n, k Î ℤ
+
with n>k, we have
n−k

l=0
(−1)
l

n − k
l

E
(α)
l+k,q
=
⎧
⎪
⎨

, n
2
, k Î ℤ
+
with n
1
+ n
2
>2k. Then we see that

Z
p
B
(α)
k,n
1
(x, q)B
(α)
k,n
2
(x, q)dμ
−q
(x)
=

n
1
k

n

1
k

n
2
k

2k

l
=
0

2k
l

(−1)
l+2k

q
2
E
(α)
n
1
+n
2
−l,q
−1
+[2]

1
+n
2
−2k

l=0
(−1)
l

n
1
+ n
2
− 2k
l


Z
p
[x]
2k+l
q
α
dμ
−q
(x
)
=

n

(20)
Kim et al. Journal of Inequalities and Applications 2011, 2011:64
/>Page 5 of 7
By comparing the coefficients on the both sides of (19) and (20), we obtain the fol-
lowing theorem.
Theorem 8. Let n
1
, n
2
, k Î ℤ
+
with n
1
+ n
2
>2k. Then we have
n
1
+n
2
−2k

l=0
(−1)
l

n
1
+ n
2


2k
l

(−1)
2k+l
E
(α)
n
1
+n
2
−l,q
−1
,ifk > 0
.
Let s Î N with s ≥ 2. For n
1
, n
2
, , n
s
, k Î ℤ
+
with n
1
+ + n
s
>sk, we have


Z
p
[x]
sk
q
α
[1 − x]
n
1
+···+n
s
−sk
q
−α
dμ
−q
(x)
=

n
1
k

···

n
s
k

sk


n
s
k

sk

l
=
0

sk
l

(−1)
l+sk

q
2
E
(α)
n
1
+···+n
s
−l,q
−1
+[2]
q


n
s
k

n
1
+···+n
s
−sk

l=0
(−1)
l

n
1
+ ···+ n
s
− sk
l


Z
p
[x]
sk+l
q
α
dμ
−q

− sk
l

E
(α)
sk+l,q
.
(22)
Therefore, by (21) and (22), we obtain the following theorem.
Theorem 9. Let s Î N with s ≥ 2. For n
1
, n
2
, , n
s
, k Î ℤ
+
with n
1
+ + n
s
>sk,we
have
n
1
+···+n
s
−sk

l=0

q
,ifk =0,
q
2

sk
l=0

sk
l

(−1)
l+sk
E
(α)
n
1
+···+n
s
−l,q
−1
,ifk > 0
.
Acknowledgements
The authors would like to express their sincere gratitude to referee for his/her valuable comments.
Kim et al. Journal of Inequalities and Applications 2011, 2011:64
/>Page 6 of 7
Author details
1
Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Korea

11. Kim, T, Lee, B, Choi, J, Kim, YH, Rim, SH: On the q-Euler numbers and weighted q-Bernstein polynomials. Adv Stud
Contemp Math. 21,13–18 (2011)
12. Kim, T: New approach to q- Euler polynomials of higher order. Russ J Math phys. 17, 218–225 (2010). doi:10.1134/
S1061920810020068
13. Kim, T, Lee, B, Choi, J, Kim, YH: A new approach of q-Euler numbers and polynomials. Proc Jangjeon Math Soc. 14,7–14
(2011)
14. Kim, T, Choi, J, Kim, YH: q-Bernstein polynomials associated with q-Stirling numbers and Carlitz’s q-Bernoulli numbers,
Abstract and Applied Analysis. 2010, 11 (2010) Article ID 150975
15. Ozden, H, Simsek, Y: A new extension of q-Euler numbers and polynomials related to their interpolation functions. Appl
Math Lett. 21, 934–939 (2008). doi:10.1016/j.aml.2007.10.005
16. Ryoo, CS: On the generalized Barnes type multiple q-Euler polynomials twisted by ramified roots of unity. Proc
Jangjeon Math Soc. 13, 255–263 (2010)
17. Ryoo, CS: A note on the weighted q-Euler numbers and polynomials. Adv Stud Contemp Math. 21,47–54 (2011)
18. Rim, S-H, Jin, J-H, Moon, E-J, Lee, S-J: On multiple interpolation function of the q-Genocchi polynomials. J Inequal Appl
13 (2010). Art ID 351419
19. Simsek, Y, Acikgoz, M: A new generating function of (q-) Bernstein-type polynomials and their interpolation function.
Abstr Appl Anal 12 (2010). Art. ID 769095
doi:10.1186/1029-242X-2011-64
Cite this article as: Kim et al.: Some identities on the weighted q-Euler numbers and q-Bernstein polynomials.
Journal of Inequalities and Applications 2011 2011:64.
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Kim et al. Journal of Inequalities and Applications 2011, 2011:64