Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 952384, 15 pages
doi:10.1155/2010/952384
Research Article
On q-Euler Numbers Related to
the Modified q-Bernstein Polynomials
Min-Soo Kim,1 Daeyeoul Kim,2 and Taekyun Kim3
1
Department of Mathematics, KAIST, 373-1 Guseong-dong, Yuseong-gu,
Daejeon 305-701, Republic of Korea
2
National Institute for Mathematical Sciences, Doryong-dong, Yuseong-gu,
Daejeon 305-340, Republic of Korea
3
Division of General Education-Mathematics, Kwangwoon University,
Seoul, 139-701, Republic of Korea
Correspondence should be addressed to Taekyun Kim, tkkim@kw.ac.kr
Received 19 July 2010; Accepted 27 August 2010
Academic Editor: Douglas Robert Anderson
Copyright q 2010 Min-Soo 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.
We consider q-Euler numbers, polynomials, and q-Stirling numbers of first and second kinds.
Finally, we investigate some interesting properties of the modified q-Bernstein polynomials related
to q-Euler numbers and q-Stirling numbers by using fermionic p-adic integrals on Zp .
1. Introduction
Let C0, 1 be the set of continuous functions on 0, 1. The classical Bernstein polynomials of
degree n for f ∈ C0, 1 are defined by
n
k
Bk,n x,
f
Bn f n
k0
0 ≤ x ≤ 1,
1.1
where Bn f is called the Bernstein operator and
Bk,n x n k
x x − 1n−k
k
1.2
2
Abstract and Applied Analysis
are called the Bernstein basis polynomials or the Bernstein polynomials of degree n
see 1. Recently, Acikgoz and Araci have studied the generating function for Bernstein
polynomials see 2, 3. Their generating function for Bk,n x is given by
F k t, x ∞
tn
tk e1−xt xk Bk,n x ,
k!
n!
n0
1.3
where k 0, 1, . . . and x ∈ 0, 1. Note that
⎧⎛ ⎞
n
⎪
⎪
⎨⎝ ⎠xk 1 − xn−k ,
Bk,n x k
⎪
⎪
⎩
0,
if n ≥ k,
1.4
if n < k,
for n 0, 1, . . . see 2, 3.
Let p be an 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
the algebraic closure of Qp , respectively. Let vp be the normalized exponential valuation of Cp
with |p|p p−1 .
Throughout this paper, we use the following notation:
1 − qx
,
xq 1−q
x−q
x
1 − −q
1q
1.5
cf. 4–7. Let N be the natural numbers and Z N ∪ {0}. Let UDZp be the space of
uniformly differentiable function on Zp .
Let q ∈ Cp with |1 − q|p < p−1/p−1 and x ∈ Zp . Then q-Bernstein type operator for
f ∈ UDZp is defined by see 8, 9
n
n
k
k
n
Bk,n x, q ,
f
f
Bn,q f xkq 1 − xn−k
q
k
n
n
k0
k0
1.6
for k, n ∈ Z , where
Bk,n x, q n
xkq 1 − xn−k
q
k
1.7
is called the modified q-Bernstein polynomials of degree n. When we put q → 1 in 1.7,
→ 1 − xn−k , and we obtain the classical Bernstein polynomial, defined
xkq → xk , 1 − xn−k
q
by 1.2. We can deduce very easily from 1.7 that
Bk,n x, q 1 − xq Bk,n−1 x, q xq Bk−1,n−1 x, q
1.8
Abstract and Applied Analysis
3
see 8. For 0 ≤ k ≤ n, derivatives of the nth degree modified q-Bernstein polynomials are
polynomials of degree n − 1:
ln q
d
Bk,n x, q n qx Bk−1,n−1 x, q − q1−x Bk,n−1 x, q
dx
q−1
1.9
see 8.
The Bernstein polynomials can also be defined in many different ways. Thus, recently,
many applications of these polynomials have been looked for by many authors. In the recent
years, the q-Bernstein polynomials have been investigated and studied by many authors in
many different ways see 1, 8, 9 and references therein 10, 11. In 11, Phillips gave many
results concerning the q-integers and an account of the properties of q-Bernstein polynomials.
He gave many applications of these polynomials on approximation theory. In 2, 3, Acikgoz
and Araci have introduced several type Bernstein polynomials. The Acikgoz and Araci paper
to announced in the conference is actually motivated to write this paper. In 1, Simsek
and Acikgoz constructed a new generating function of the q-Bernstein type polynomials
and established elementary properties of this function. In 8, Kim et al. proposed the
modified q-Bernstein polynomials of degree n, which are different q-Bernstein polynomials of
Phillips. In 9, Kim et al. investigated some interesting properties of the modified q-Bernstein
polynomials of degree n related to q-Stirling numbers and Carlitz’s q-Bernoulli numbers.
In the present paper, we consider q-Euler numbers, polynomials, and q-Stirling
numbers of first and second kinds. We also investigate some interesting properties of the
modified q-Bernstein polynomials of degree n related to q-Euler numbers and q-Stirling
numbers by using fermionic p-adic integrals on Zp .
2. q-Euler Numbers and Polynomials Related to
the Fermionic p-Adic Integrals on Zp
For N ≥ 1, the fermionic q-extension µq of the p-adic Haar distribution µHaar ,
µ−q a p Zp
N
a
−q
N ,
p −q
2.1
is known as a measure on Zp , where a pN Zp {x ∈ Qp | |x − a|p ≤ p−N } cf. 4, 12.
We will write dµ−q x to remind ourselves that x is the variable of integration. Let UDZp be the space of uniformly differentiable function on Zp . Then µ−q yields the fermionic p-adic
q-integral of a function f ∈ UDZp :
I−q f fxdµ−q x lim
Zp
1q
N →∞1
N
qp
N
p
−1
x0
x
fx −q
2.2
4
Abstract and Applied Analysis
cf. 12–15. Many interesting properties of 2.2 were studied by many authors see 12, 13
and the references given there. Using 2.2, we have the fermionic p-adic invariant integral
on Zp as follows:
lim Iq f I−1 f fadµ−1 x.
q → −1
2.3
Zp
For n ∈ N, write fn x fx n. We have
n−1
I−1 fn −1n I−1 f 2
−1n−l−1 fl.
2.4
l0
This identity is obtained by Kim in 12 to derive interesting properties and relationships
involving q-Euler numbers and polynomials. For n ∈ Z , we note that
I−1 xnq
Zp
xnq dµ−1 x En,q ,
2.5
where En,q are the q-Euler numbers see 16. It is easy to see that E0,q 1. For n ∈ N, we
have
N
−1
p
n n n l
n l
q El,q q lim
xlq −1x
l
l
N →∞
x0
l0
l0
lim
N
−1
p
N →∞
lim
n
−1x qxq 1
x0
N
−1
p
N →∞
− lim
2.6
−1x x 1nq
x0
N →∞
N
−1
p
−1
x
xnq
n pN
x0
q
−En,q .
From this formula, we have the following recurrence formula:
n
qE 1 En,q 0
E0,q 1,
if n ∈ N,
2.7
with the usual convention of replacing El by El,q . By the simple calculation of the fermionic
p-adic invariant integral on Zp , we see that
2
En,q 1−q
n
n n
l0
l
−1l
1
,
1 ql
2.8
Abstract and Applied Analysis
5
where nl n!/l!n − l! nn − 1 · · · n − l 1/l!. Now, by introducing the following
equations:
xn1/q
n −nx
q q
xnq ,
q
−nx
m n m − 1
1−q
xm
q
m
∞
m0
2.9
into 2.5, we find that
∞
1−q
En,1/q qn
m0
m n m − 1
Enm,q .
m
2.10
This identity is a peculiarity of the p-adic q-Euler numbers, and the classical Euler numbers
do not seem to have a similar relation. Let Fq t be the generating function of the q-Euler
numbers. Then we obtain that
Fq t ∞
tn
n!
En,q
n0
∞
n0
2
1−q
2et/1−q
n
n
−1l
l0
1 tn
n
l 1 ql n!
2.11
∞
1 tk
−1k
.
k
1 qk k!
k0 1 − q
From 2.11, we note that
Fq t 2et/1−q
∞
−1n e−q
n
/1−qt
2
n0
∞
−1n enq t .
2.12
n0
It is well known that
n I−1 x y
Zp
n
x y dµ−1 y En,q x,
2.13
where En,q x are the q-Euler polynomials see 16. In the special case x 0, the numbers
En,q 0 En,q are referred to as the q-Euler numbers. Thus, we have
n n
k
n
kx
q
x y dµ−1 y y dµ−1 y
xn−k
q
k
Zp
Zp
k0
n n
k0
k
kx
xn−k
q q Ek,q
qx E xq
n
.
2.14
6
Abstract and Applied Analysis
It is easily verified, using 2.12 and 2.13, that the q-Euler polynomials En,q x satisfy the
following formula:
∞
En,q x
n0
tn
n!
Zp
∞
exyq t dµ−1 y
n0
2
2
∞
1−q
n
n
−1l
l0
lx n
t
n q
l 1 ql n!
2.15
−1n enxq t .
n0
Using formula 2.15, when q tends to 1, we can readily derive the Euler polynomials, En x,
namely,
∞
tn
2ext
exyt dµ−1 y t
En x
n!
e 1 n0
Zp
2.16
see 12. Note that En 0 En are referred to as the nth Euler numbers. Comparing the
coefficients of tn /n! on both sides of 2.15, we have
En,q x 2
∞
m
−1 m m0
xnq
lx
n q
.
−1
n
l 1 ql
1 − q l0
2
n
l
2.17
We refer to nq as a q-integer and note that nq is a continuous function of q. In an
obvious way we also define a q-factorial,
nq ! ⎧
⎨nq n − 1q · · · 1q , n ∈ N,
⎩1,
n 0,
2.18
and a q-analogue of binomial coefficient,
xq x − 1q · · · x − n 1q
xq !
x
n q x − nq !nq !
nq !
2.19
cf. 14, 16. Note that
xx − 1 · · · x − n 1
x
x
.
n
n
q→1
n!
q
lim
2.20
It readily follows from 2.19 that
n n n
1 − q q− 2 x
i n
q 2
−1ni qn−ix
n q
i
nq !
q
i0
2.21
Abstract and Applied Analysis
7
cf. 7, 16. It can be readily seen that
l l m
l q xq q − 1 1 q − 1 xm
q .
m
m0
lx
2.22
Thus, by 2.13 and 2.22, we have
n n
n−i q−1 j
x
n−i i n
i
2
dµ−1 x q
q − 1 Ej,q .
−1
n
n
i
j
Zp
q
q
nq !q 2 i0
j0
2.23
From now on, we use the following notation:
xq !
x − kq !
xnq q
− k 2
k
s1,q k, lxlq ,
k ∈ Z ,
n
l0
k
q 2 s2,q n, k
k0
2.24
xq !
x − kq !
,
n ∈ Z
see 7. From 2.24, and 2.22, we calculate the following consequence:
xnq n
q
k
2 s2,q n, k
k0
n
k
q 2 s2,q n, k
k0
×
n
m
1
1−q
k
k k
l0
l
l0
l
2 −1l qlx−k1
q
l l1−k
2
−1l
q
k k
l
q
q
q
m
q − 1 xm
q
k
2 s2,q n, k
k0
×
1−q
k
l l m0
1
1
1−q
2.25
k
k
m k
l
l l1−k
l
q 2
q−1
−1 xm
q .
l
m
q
lm
k
m0
Therefore, we obtain the following theorem.
Theorem 2.1. For n ∈ Z ,
En,q k k
n k0 m0 lm
q
k
2 s2,q n, k q
m−k k
l
l l1−k
q 2
−1
−1lk Em,q .
l q
m
2.26
8
Abstract and Applied Analysis
By 2.22 and simple calculation, we find that
n n m0
m
m
q − 1 Em,q qnx dµ−1 x
Zp
k−1
k k n
2
q−1 q
x − iq dµ−1 x
k q Zp i0
n
k0
k0
k
k n s1,q k, m
q−1
xm
q dµ−1 x
k q m0
Zp
n
n
m0
2.27
k n
q−1
s k, m Em,q .
k q 1,q
n
km
Therefore, we deduce the following theorem.
Theorem 2.2. For n ∈ Z ,
n n m0
n n
k n
m
s k, mEm,q .
q−1
q − 1 Em,q k q 1,q
m0 km
m
2.28
Corollary 2.3. For m, n ∈ Z with m ≤ n,
n
k n
m n s k, m.
q−1
q−1 m
k q 1,q
km
2.29
By 2.17 and Corollary 2.3, we obtain the following corollary.
Corollary 2.4. For n ∈ Z ,
2
En,q x 1−q
n
k−l n
qlx
s1,q k, l
.
−1l q − 1
k q
1 ql
kl
n
n l0
2.30
It is easy to see that
k
n
q i0 ili
k q l ···l n−k
0
2.31
k
cf. 7. From 2.31 and Corollary 2.4, we can also derive the following interesting formula
for q-Euler polynomials.
Theorem 2.5. For n ∈ Z ,
En,q x 2
n
n l0 kl l0 ···lk n−k
q
k
i0
ili
lx
1
k q
.
nl−k s1,q k, l−1
1 ql
1−q
2.32
Abstract and Applied Analysis
9
These polynomials are related to the many branches of Mathematics, for example,
combinatorics, number theory, and discrete probability distributions for finding higher-order
moments cf. 14–16. By substituting x 0 into the above, we have
En,q 2
n
n q
k
i0
ili
l0 kl l0 ···lk n−k
1
1
k
,
nl−k s1,q k, l−1
1 ql
1−q
2.33
where En,q is the q-Euler numbers.
3. q-Euler Numbers, q-Stirling Numbers, and q-Bernstein Polynomials
Related to the Fermionic p-Adic Integrals on Zp
First, we consider the q-extension of the generating function of Bernstein polynomials in 1.3.
For q ∈ Cp with |1 − q|p < p−1/p−1 , we obtain that
k
Fq t, x
tk e1−xq t xkq
xkq
k!
∞ n0
tnk
nk
1 − xnq
k
n k!
∞ tn
n
xkq 1 − xn−k
q
k
n!
nk
3.1
∞
tn
,
Bk,n x, q
n!
n0
which is the generating function of the modified q-Bernstein type polynomials see 9.
Indeed, this generating function is also treated by Simsek and Acikgoz see 1. Note that
k
limq → 1 Fq t, x F k t, x. It is easy to show that
1 −
xn−k
q
∞ n−k lm−1
n−k
m0 l0
m
l
m
q−1 .
−1lm ql xlm
q
3.2
From 1.6, 2.3, 2.15, and 3.2, we derive the following theorem.
Theorem 3.1. For k, n ∈ Z with n ≥ k,
Zp
∞ n−k Bk,n x, q
m
lm−1
n−k
dµ
y
−1lm ql q − 1 Elmk,q ,
−1
m
l
nk m0 l0
where En,q are the q-Euler numbers.
3.3
10
Abstract and Applied Analysis
It is possible to write xkq as a linear combination of the modified q-Bernstein
polynomials by using the degree evaluation formulae and mathematical induction. Therefore,
we obtain the following theorem.
Theorem 3.2 see 8, Theorem 7. For k, n ∈ Z , i ∈ N, and x ∈ 0, 1,
k
n
i
n Bk,n x, q
ki−1 i
n−i
.
xiq xq 1 − xq
3.4
Let i − 1 ≤ n. Then from 1.7, 3.2, and Theorem 3.2, we have
nk / ni xkq 1 − xn−k
q
n−k
1 1 − xq /xq
xn−i
q
n
xiq
ki−1
k i
∞ n
mn−k
∞
m0 ki−1
×
l0
k
nk l p − 1
mn−k
p
l
ni i
p0
3.5
p i−n−mkpl
n−im−1
.
−1lpm ql q − 1 xq
m
Using 2.13 and 3.5, we obtain the following theorem.
Theorem 3.3. For k, n ∈ Z and i ∈ N with i − 1 ≤ n,
Ei,q ∞ n
mn−k
∞
m0 ki−1
l0
p0
k
nk l p − 1
mn−k
p
l
ni i
p
n−im−1
×
−1lpm ql q − 1 Ei−n−mkpl,q .
m
3.6
The q-String numbers of the first kind is defined by
n n
1 kq z S1 n, k; q zk ,
3.7
k0
k1
and the q-String number of the second kind is also defined by
n n
−1 1 kq z
S2 n, k; q zk
k1
k0
see 9. Therefore, we deduce the following theorem.
3.8
Abstract and Applied Analysis
11
Theorem 3.4 see 9, Theorem 4. For k, n ∈ Z and i ∈ N,
n
k
i / ni Bk,n x, q
S1 k, l; q S2 k, i − k; q xlq .
n−i
k0 l0
xq 1 − xq
ki−1
k i
3.9
By Theorems 3.2 and 3.4 and the definition of fermionic p-adic integrals on Zp , we
obtain the following theorem.
Theorem 3.5. For k, n ∈ Z and i ∈ N,
Ei,q n
k i
n
ki−1 i
k
i Zp
Bk,n x, q
xq 1 − xq
n−i dµ−1 x
3.10
S1 k, l; q S2 k, i − k; q El,q ,
k0 l0
where Ei,q is the q-Euler numbers.
Let i − 1 ≤ n. It is easy to show that
n−i
xiq xq 1 − xq
n−i n−i
n−i−l
xli
q 1 − xq
l
l0
n−i n−i−l
n − in − i − l
q−mx
−1m qm xmil
q
l
m
l0 m0
n−i n−i−l
∞ n−i
n−i−l
ms−1
l0 m0 s0
l
m
s
3.11
s
.
−1m qm 1 − q xmils
q
From 3.11 and Theorem 3.2, we have the following theorem.
Theorem 3.6. For k, n ∈ Z and i ∈ N,
n
k i
n
i
ki−1
n−i n−i−l
∞ n−i
n−i−l
ms−1
Bk,n x, q dµ−1 x l
m
s
Zp
l0 m0 s0
m m
× −1 q
s
1 − q Emils,q ,
where Ei,q are the q-Euler numbers.
In the same manner, we can obtain the following theorem.
3.12
12
Abstract and Applied Analysis
Theorem 3.7. For k, n ∈ Z and i ∈ N,
n ∞ m
j
n
j −km−1
Bk,n x, q dµ−1 x −1j−km qj−k q − 1 Emj,q ,
k
j
m
Zp
jk m0
3.13
where Ei,q are the q-Euler numbers.
4. Further Remarks and Observations
The q-binomial formulas are known as
a; q
n
1 − a 1 − aq · · · 1 − aq
n−1
n
n
i0
i
∞
1
1
a; q n 1 − a 1 − aq · · · 1 − aqn−1
i0
q
i
2 −1i ai ,
q
ni−1
i
4.1
ai .
q
For h ∈ Z, n ∈ Z , and r ∈ N, we introduce the extended higher-order q-Euler polynomials as
follows 16:
h,r
En,q x
Zp
···
Zp
q
r
j1 h−jxj
x x1 · · · xr nq dµ−1 x1 · · · dµ−1 xr .
4.2
Then,
h,r
En,q x
2r
1−q
2r
1−q
n
n n
l0
n
l
n n
l0
l
qlx
−qh−1l ; q−1 r
−1l −1l qlx
−qh−rl ; q
4.3
.
r
Let us now define the extended higher-order Nörlund type q-Euler polynomials as follows
16:
h,−r
En,q
1
x 1−q
n
n n
l0
l
−1l Zp
···
Zp
qlx1 ···xr q
r
qlx
j1 h−jxj
dµ−1 x1 · · · dµ−1 xr .
4.4
Abstract and Applied Analysis
13
h,−r
h,−r
In the special case x 0, En,q En,q 0 are called the extended higher-order Nörlund type
q-Euler numbers. From 4.4, we note that
h,−r
En,q
1
x n
2r 1 − q
n n
l0
l
−1l qlx −qh−rl ; q
r
4.5
r
1 m
h−rm r
2
r
q q
m xnq .
m q
2 m0
A simple manipulation shows that
q
m
2
r
m
m
q 2 rq · · · r − m 1q
mq !
q
n−1 z − kq zn
n−1
k0
1−
kq
k0
z
n
m−1 1 rq − kq ,
mq ! k0
4.6
S1 n − 1, k; q −1k zn−k .
k0
Formula 4.5 implies the following lemma.
Lemma 4.1. For h ∈ Z, n ∈ Z , and r ∈ N,
h,−r
En,q
x r m
1
qh−rm S1 m − 1, k; q −1k rm−k
x mnq .
q
r
2 mq ! m0 k0
4.7
From 2.22, we can easily see that
1
x mnq 1−q
n
j n l
n
j
−1jl 1 − q qmj xlq .
j
l
j0 l0
4.8
Using 2.13 and 4.8, we obtain the following lemma.
Lemma 4.2. For m, n ∈ Z ,
1
En,q m 1−q
n
j n l
n
j
−1jl 1 − q qmj El,q .
j
l
j0 l0
4.9
By Lemma 4.2, and the definition of fermionic p-adic integrals on Zp , we obtain the
following theorem.
14
Abstract and Applied Analysis
Theorem 4.3. For h ∈ Z, n ∈ Z , and r ∈ N,
h,−r
Zp
En,q
r m
2−r qh−rm S1 m − 1, k; q −1k rm−k
En,q m
q
mq ! m0 k0
xdµ−1 x r m
1
qh−rm S1 m − 1, k; q −1k rm−k
q
r
2 mq ! m0 k0
1
×
1−q
n
j n n
j
j0 l0
j
l
4.10
l
−1jl 1 − q qmj El,q .
0,−r
Put h 0 in 4.4. We consider the following polynomials En,q x:
0,−r
En,q x
n
l0
Zp
···
Zp
1−q
−n
qlx1 ···xr q−
nl −1l qlx
r
j1
jxj
dµ−1 x1 · · · dµ−1 xr .
4.11
Then,
0,−r
En,q x r m
1 r
q 2 −rm m xnq .
r
m
2 m0
4.12
A simple calculation of the fermionic p-adic invariant integral on Zp shows that
Zp
0,−r
En,q xdµ−1 x r m
1 r
q 2 −rm En,q m.
r
2 m0 m
4.13
Using Theorem 4.3, we can also prove that
Zp
0,−r
En,q xdµ−1 x r m
2−r q−rm S1 m − 1, k; q −1k rm−k
En,q m.
q
mq ! m0 k0
4.14
Therefore, we obtain the following theorem.
Theorem 4.4. For m ∈ Z , r ∈ N with m ≤ r,
m
m
1 r
q 2 −rm q−rm S1 m − 1, k; q −1k rm−k
.
q
m
mq ! k0
4.15
Acknowledgments
M.-S. Kim was supported by the Basic Science Research Program through the National
Research Foundation of Korea NRF funded by the Ministry of Education, Science, and
Technology 2010-0001654. T. Kim was supported by the research grant of Kwangwoon
University in 2010.
Abstract and Applied Analysis
15
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