Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 123483, 9 pages
doi:10.1155/2011/123483
Research Article
Some Identities of the Twisted q-Genocchi
Numbers and Polynomials with Weight α and
q-Bernstein Polynomials with Weight α
H. Y. Lee, N. S. Jung, and C. S. Ryoo
Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea
Correspondence should be addressed to H. Y. Lee, normaliz1@naver.com
Received 7 July 2011; Accepted 22 August 2011
Academic Editor: John Rassias
Copyright q 2011 H. Y. Lee 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.
Recently mathematicians have studied some interesting relations between q-Genocchi numbers,
q-Euler numbers, polynomials, Bernstein polynomials, and q-Bernstein polynomials. In this paper,
we give some interesting identities of the twisted q-Genocchi numbers, polynomials, and qBernstein polynomials with weighted α.
1. Introduction
Throughout this paper, let p be a fixed odd prime number. The symbols Zp , Qp , and Cp
denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion
of algebraic closure of Qp . Let N be the set of natural numbers and let Z N ∪ {0}. As a
well-known definition, the p-adic absolute value is given by |x|p p−r , where x pr t/s with
t, p s, p t, s 1. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q ∈ C, or a p-adic number q ∈ Cp . In this paper we assume that
q ∈ Cp with |1 − q|p < 1.
We assume that UDZp is the space of the uniformly differentiable function on Zp . For
f ∈ UDZp , Kim defined the fermionic p-adic q-integral on Zp as follows:
I−q f −1
p
x
1
fxdμ−q x lim N fx −q .
N
→
∞
p −q x0
Zp
N
1.1
For n ∈ N, let fn x fx n be translation. As a well known equation, by 1.1, we have
n−1
n
q
fx ndμ−q x −1n I−q f 2q −1n−1−l ql fl,
1.2
Zp
l0
2
Abstract and Applied Analysis
compared 1–4. Throughout this paper we use the notation:
1 − qx
,
xq 1−q
x−q
x
1 − −q
,
1q
1.3
cf. 1–16. limq → 1 xq x for any x with |x|p ≤ 1 in the present p-adic case. To investigate
relation of the twisted q-Genocchi numbers and polynomials with weight α and the Bernstein
polynomials with weight α, we will use useful property for xqα as follows;
xqα 1 − 1 − xq−α ,
1 − xq−α 1 − xqα .
1.4
The twisted q-Genocchi numbers and polynomials with weight α are defined by the
generating function as follows, respectively:
α
Gn,q,w n
α
Gn,q,w x
α
Zp
n
Zp
φw xxn−1
qα dμ−q x,
n−1
φw y y x qα dμ−q y .
1.5
1.6
α
In the special case, x 0, Gn,q,w 0 Gn,q,w are called the nth twisted q-Genocchi numbers
with weight α see 9.
n
Let Cpn {w | wp 1} be the cyclic group of order pn and let
Tp lim Cpn n→∞
Cpn ,
n≥1
1.7
see 9, 12–15.
Kim defined the q-Bernstein polynomials with weight α of degree n as follows:
α
Bn,k x
n
k
xkqα 1 − xn−k
q−α ,
where x ∈ 0, 1, n, k ∈ Z ,
1.8
compare 4, 7.
In this paper, we investigate some properties for the twisted q-Genocchi numbers and
polynomials with weight α. By using these properties, we give some interesting identities on
the twisted q-Genocchi polynomials with weight α and q-Bernstein polynomials with weight
α.
Abstract and Applied Analysis
3
2. Some Identities on the Twisted q-Genocchi Polynomials with
Weight α and q-Bernstein Polynomials with Weight α
From 1.8, we can derive the following recurrence formula for the twisted q-Genocchi numbers with weight α:
α
α
G0,q,w 0,
α
qwGn,q,w 1 Gn,q,w ⎧
⎨2q ,
if n 1,
⎩0,
if n > 1,
⎧
⎨qα 2q ,
n
α
α
qw 1 qα Gq,w qα Gn,q,w ⎩0,
α
G0,q,w 0,
if n 1,
if n > 1,
α
α n1
qαx Gn1,q,w x xαq qαx Gq,w
α
2.1
2.2
2.3
α
with usual convention about replacing Gq,w n by Gn,q,w .
By 1.5, we easily get
α
Gn,q,w x
n2q
1
1 − qα
n−1 n−1 n − 1
l0
−1l qαxl
l
1
.
1 wqαl1
2.4
By 2.4, we obtain the theorem below.
Theorem 2.1. Let n ∈ Z . For w ∈ Tp , one has
α
α
Gn,q,w x −1n−1 w−1 qα1−n Gn,q−1 ,w−1 1 − x.
2.5
By 2.1, 2.2, and 2.3 we note that
α
α
Gn,q,w −qwGn,q,w 1
α
−nwqG1,q,w
w q
n
n
α
qαl Gl,q,w 1
l
l2
n
n αl α
α l
2 2−2α
−nwqG1,q,w w q
q 1 qα Gq,w
l
l2
α
α n
α
−nwqG1,q,w w2 q2−2α 2qα q2α Gq,w − nw2 q2 G1,q,w
α
2 2−α
α
α
−nwqG1,q,w w2 q2 Gn,q,w 2 − nw2 q2 G1,q,w .
Therefore, by 2.6, we obtain the theorem below.
2.6
4
Abstract and Applied Analysis
Theorem 2.2. For n ∈ N with n > 1, one has
α
n2q
α
Gn,q,w 2 w−2 q−2 Gn,q,w w−1 q−1
1 qw
n2q
1 qw
2.7
.
By 1.6 and Theorem 2.2,
α
Gn1,q,w 2
n1
Zp
n
φw y y 2 qα dμ−q y
1
n1
n 12q
1 qw
2q
1 qw
w−1 q−1
n 1w−1 q−1 2q
α
w−2 q−2
1 qw
Gn1,q,w
2.8
n1
α
2q
1 qw
w−2 q−2
Gn1,q,w
n1
.
Hence, we obtain the corollary below.
Corollary 2.3. For n ∈ N, one has
Zp
n
φw y y 2 qα dμ−q y 2q
1 qw
−1 −1
w q
2q
1 qw
α
−2 −2
w q
Gn1,q,w
n1
2.9
.
By fermionic integral on Zp ,Theorems 2.1 and 2.2, we note that
Zp
φw x1 − xnq−α dμ−q x −1n qαn
Zp
φw xx − 1nqα dμ−q x
α
−1n qαn
Gn1,q,w −1
n1
α
w
−1
Gn1,q−1 ,w−1 2
n1
⎛
w−1 ⎝
2q−1
1 q−1 w−1
2q
1 qw
wq
wq
2q
1 qw
Therefore, we have the theorem below.
2q−1
1 q−1 w−1
α
w 2 q2
Gn1,q−1 ,w−1
n1
⎞
⎠
α
wq
2
Gn1,q−1 ,w−1
n1
.
2.10
Abstract and Applied Analysis
5
Theorem 2.4. For n ∈ N with n > 1, one has
Zp
xnq−α dμ−q x
φw x1 −
2q
1 qw
wq
α
2q
1 qw
wq
2
Gn1,q−1 ,w−1
n1
.
2.11
By 1.4, Theorem 2.4, we take the fermionic p-adic invariant integral on Zp for one
q-Bernstein polynomials as follows:
Zp
φw xBn,k
n
φw x
x, q dμ−q x xkqα 1 − xn−k
q−α dμ−q x
k
Zp
n−k
n
φw xxkqα 1 − xqα
dμ−q x
k
Zp
n−k n − k
n k
α
−1l
l
l0
2.12
Gkl1,q,w
kl1
.
By symmetry of q-Bernstein polynomials with weight α of degree n, we get the following formula;
Zp
φw xBn,k x, q dμ−q x
Zp
φw x
Zp
φw x
n
k
xn−k
qα 1 − xq−α dμ−q x
k
n
n−k
dμ−q x
1 − xkq−α 1 − 1 − xq−α
k
n−k n − k
n k
l
l0
n−k n − k
n k
l
l0
2.13
−1
n−k−l
Zp
⎛
−1n−k−l ⎝
φw x1 − xn−l
q−α dμ−q x
2q
1 qw
wq
α
2q
1 qw
wq2
Gn−l1,q−1 ,w−1
n−l1
⎞
⎠.
Therefore, by 2.12 and 2.13, we have the theorem below.
Theorem 2.5. For n ∈ N with n > 1, one has
n−k n − k
l
l0
α
−1
l
Gkl1,q,w
kl1
n−k n − k
l0
l
⎛
−1n−k−l ⎝
2q
1 qw
wq
2q
1 qw
α
wq2
Gn−l1,q−1 ,w−1
n−l1
⎞
⎠.
2.14
6
Abstract and Applied Analysis
Also, we note that
Zp
φw xBn,k x, q dμ−q x
n−k n − k
n k
n
k
Zp
n
k
Zp
−1
l
l0
α
φw x1 −
l0
kl1
xn−k
q−α
l0
1 − 1 − xq−α
k
2.15
dμ−q x
l
−1k−l
Zp
⎛
k
k
n k
Gkl1,q,w
k
φw x1 − xn−k
q−α xqα dμ−q x
k
n k
k
l
−1k−l ⎝
l
φw x1 − xn−l
q−α dμ−q x
2q
1 qw
wq
2q
1 qw
α
wq2
Gn−l1,q−1 ,w−1
n−l1
⎞
⎠.
Therefore, we have the theorem below.
Theorem 2.6. For n, k ∈ Z with n > k 1, one has
Zp
φw xBk,n x, q dμ−q x
⎛
k
k
n k
l0
l
−1k−l ⎝
2q
1 qw
wq
2q
1 qw
α
wq2
Gn−l1,q−1 ,w−1
n−l1
⎞
2.16
⎠.
By 2.11 and Theorem 2.6, we have the theorem below.
Theorem 2.7. Let n, k ∈ Z with n > k 1. Then one has
n−k n − k
l
l0
α
−1
l0
Gkl1,q,w
kl1
⎛
k
k
l
l
−1k−l ⎝
2q
1 qw
wq
2q
1 qw
α
wq2
Gn−l1,q−1 ,w−1
n−l1
⎞
⎠.
2.17
Abstract and Applied Analysis
7
Let n1 , n2 , k ∈ Z with n1 n2 > 2k 1. Then we get
Zp
α α φw xBn1 ,k x, q Bn2 ,k x, q dμ−q x
2k
2k
n2 n1
k
k
l0
l
2k
n1
2k
n2 k
k
l0
l
−1
2k−l
Zp
⎛
−12k−l ⎝
1 n2 −l
φw x1 − xnq−α
dμ−q x
2q
1 qw
wq
α
2q
1 qw
wq2
Gn n −l1,q−1 ,w−1
1
2
n1 n2 − l 1
⎞
⎠.
2.18
Therefore, we obtain the theorem below.
Theorem 2.8. For n1 , n2 , k ∈ Z , one has
Zp
α α φw xBn1 ,k x, q Bn2 ,k x, q dμ−q x
⎛
2k
n1
2k
n2 k
k
l0
l
−12k−l ⎝
2q
1 qw
wq
α
2q
1 qw
wq2
⎧
α
⎪
G
2q
2q
⎪
−1
−1
⎪
2 n1 n2 −l1,q ,w
⎪
⎪
wq
wq
,
⎪
⎨ 1 qw
1 qw
n1 n2 − l 1
⎞
⎛
⎞
⎛
⎛
⎞
⎪
2k
⎪
Gn1 n2 −l1,q−1 ,w−1
⎪ 2 ⎝n1 ⎠⎝n2 ⎠⎝2k ⎠
⎪
⎪
wq
,
−12k−l
⎪
⎩
n1 n2 − l 1
k
k
l
Gn n −l1,q−1 ,w−1
1
2
n1 n2 − l 1
⎞
⎠
if k 0,
if k > 0,
l0
2.19
By simple calculation, we easily see that
Zp
α α φw xBn1 ,k x, q Bn2 ,k x, q dμ−q x
n n −2k
2
n2 1 n1
k
k
k
−1
−1
l
n1 n2 − 2k
l0
l
l0
n n −2k
2
n2 1 n1
k
l
n1 n2 − 2k
l
Zp
φw xx2kl
qα dμ−q x
α
G2kl1,q,w
2k l 1
,
where n1 , n2 , k ∈ Z .
Therefore, by 2.20 and Theorem 2.8, we obtain the theorem below.
2.20
8
Abstract and Applied Analysis
Theorem 2.9. Let n1 , n2 , k ∈ Z with n1 n2 > 2k 1. Then one has
2k
2k
l0
l
⎛
−12k−l ⎝
n1 n
2 −2k
2q
1 qw
wq
1 qw
n1 n2 − 2k
−1l
wq2
α
G2kl1,q,w
2k l 1
l
l0
α
2q
Gn n −l1,q−1 ,w−1
1
2
n1 n2 − l 1
φw x
Zp
s
i1
.
k
l0
l
s
sk
ni sk
k
i1
s
i1
ni m, then by the
α Bk,ni x, q dμ−q x
s
sk
ni sk
i1
⎠
2.21
For n1 , n2 , . . . , ns , k ∈ Z , n1 n2 · · · ns > sk 1, and let
symmetry of q-Bernstein polynomials with weight α, we see that
⎞
l0
l
−1sk−l
Zp
⎛
φw x1 − xm−l
q−α dμ−q x
2q
−1sk−l ⎝
1 qw
wq
2.22
α
2q
1 qw
wq2
Gm−l1,q−1 ,w−1
m−l1
⎞
⎠.
Therefore, we have the theorem below.
Theorem 2.10. For n1 , n2 , . . . , ns , k ∈ Z with n1 n2 · · · ns > sk 1, one has
Zp
φw x
s
α Bk,ni x, q dμ−q x
i1
s
sk
ni sk
k
i1
l0
l
⎛
−1
sk−l ⎝
2q
1 qw
α
2q
wq
1 qw
wq
2
Gm−l1,q−1 ,w−1
m−l1
⎞
2.23
⎠,
where n1 · · · ns m.
In the same manner as in 2.15, we can get the following relation:
Zp
φw x
s
α Bk,ni x, q dμ−q x
i1
s
ni
i1
k
Zp
s
ni m−sk
i1
k
l0
φw xxsk
qα
m−sk
−1
l
l0
−1
l
m − sk
l
m − sk
l
α
Gskl1,q,w
sk l 1
,
where n1 , n2 , . . . , ns , k ∈ Z with m n1 n2 · · · ns > sk 1.
−1l xlqα dμ−q x
2.24
Abstract and Applied Analysis
9
By Theorem 2.10 and 2.13, we have the following corollary.
Corollary 2.11. Let m ∈ N. For n1 , n2 , . . . , ns , k ∈ Z with n1 · · · ns > mk 1, one has
sk
sk
l0
l
⎛
−1
m−sk
l0
sk−l ⎝
2q
1 qw
−1l
m − sk
l
wq
2q
1 qw
sk l 1
wq
Gm−l1,q−1 ,w−1
m−l1
⎞
⎠
2.25
α
Gskl1,q,w
α
2
,
where n1 · · · ns m.
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