We first consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to χ.. The purpose of this paper is to present a s
Trang 1Volume 2011, Article ID 424809, 8 pages
doi:10.1155/2011/424809
Review Article
Polynomials of Higher-Order
C S Ryoo,1 T Kim,2 J Choi,2 and B Lee3
Seoul 139-701, Republic of Korea
Seoul 139-701, Republic of Korea
Correspondence should be addressed to T Kim,tkkim@kw.ac.kr
Received 10 August 2010; Accepted 18 February 2011
Academic Editor: Roderick Melnik
Copyrightq 2011 C S Ryoo 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 first consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to χ The purpose of this paper is to present a systemic study of some families of higher-order generalized q-Genocchi numbers and polynomials attached to χ by using the generating function of those numbers and polynomials.
1 Introduction
As a well known definition, the Genocchi polynomials are defined by
2t
e t 1
e xt e Gxt∞
n0
G n x t n
n! , |t| < π, 1.1
where we use the technical method’s notation by replacing G n x by G n x, symbolically,
see 1,2 In the special case x 0, Gn G n 0 are called the nth Genocchi numbers From the definition of Genocchi numbers, we note that G1 1, G3 G5 G7 · · · 0, and even coefficients are given by G2n 21 − 22n B 2n 2nE 2n−10 see 3, where Bnis a Bernoulli
number and E n x is an Euler polynomial The first few Genocchi numbers for 2, 4, 6, are
−1, 1, −3, 17, −155, 2073, The first few prime Genocchi numbers are given by G6 −3 and
G8 17 It is known that there are no other prime Genocchi numbers with n < 105 For a real
or complex parameter α, the higher-order Genocchi polynomials are defined by
2t
e t 1
α
e xt∞
n0
G α n x t n
Trang 2see 1,4 In the special case x 0, Gα n G α n 0 are called the nth Genocchi numbers
of order α From 1.1 and 1.2, we note that G n G1n For d ∈ N with d ≡ 1mod 2, let χ be the Dirichlet character with conductor d It is known that the generalized Genocchi polynomials attached to χ are defined by
2td−1
a0 χ a−1 a e at
e dt 1
e xt∞
n0
G n,χ x t n
see 1 In the special case x 0, Gn,χ G n,χ 0 are called the nth generalized Genocchi numbers attached to χ see 1,4 6
For a real or complex parameter α, the generalized higher-order Genocchi polynomials attached to χ are also defined by
2td−1
a0 χa−1 a e at
e dt 1
α
e xt∞
n0
G α n,χ x t n
see 7 In the special case x 0, Gα n,χ G α n,χ 0 are called the nth generalized Genocchi numbers attached to χ of order α see 1,4 9 From 1.3 and 1.4, we derive Gn,χ G1n,χ
Let us assume that q ∈ C with |q| < 1 as an indeterminate Then we, use the notation
x q 1− q x
The q-factorial is defined by
n q! n q n − 1 q· · · 2q1q , 1.6 and the Gaussian binomial coefficient is also defined by
n k
q
n − k n q!
q!kq! n q n − 1 q · · · n − k 1 q
see 5,10 Note that
lim
q → 1
n k
q
n k
n n − 1 · · · n − k 1
It is known that
n 1 k
q
n
k − 1
q
q k
n k
q
q n1−k
n
k − 1
q
n k
q
Trang 3see 5,10 The q-binomial formula are known that
x − y n
q x − y
x − qy
· · · x − q n−1 y
n
i0
n i
q
q
i 2
−1i
x n−i y i ,
1
x − y n
q
x − y
x − qy
· · ·x − q n−1 y ∞
l0
n l − 1 l
q
x n−l y l ,
1.10
see10,11
There is an unexpected connection with q-analysis and quantum groups, and thus with noncommutative geometry q-analysis is a sort of q-deformation of the ordinary analysis Spherical functions on quantum groups are q-special functions Recently, many authors have studied the q-extension in various areas see 1 15 Govil and Gupta 10 have introduced
a new type of q-integrated Meyer-K¨onig-Zeller-Durrmeyer operators, and their results are closely related to the study of q-Bernstein polynomials and q-Genocchi polynomials, which are treated in this paper In this paper, we first consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and polynomials attached to χ.
The purpose of this paper is to present a systemic study of some families of higher-order
generalized q-Genocchi numbers and polynomials attached to χ by using the generating
function of those numbers and polynomials
2 Generalized q-Genocchi Numbers and Polynomials
For r ∈ N, let us consider the q-extension of the generalized Genocchi polynomials of order r attached to χ as follows:
F q,χ r t, x 2 r t r
∞
m1, ,m r0
⎛
⎝r
j1
χ
m j
⎞⎠−1r
j1 m j e xm1···m rq t∞
n0
G r n,χ,q x t n
n! . 2.1
Note that
lim
q → 1 F q,χ r t, x
2td−1
a0 χa−1a
e at
e dt 1
r
By2.1 and 1.4, we can see that limq → 1 G r n,χ,q x G r n,χ x From 2.1, we note that
G r 0,χ,q x G r 1,χ,q x · · · G r r−1,χ,q x 0,
G r nr,χ,q x
nr r
r! 2r ∞
m1, ,m r0
⎛
⎝r
j1
χ
m j
⎞⎠−1r
j1 m j x m1 · · · m rn
q
2.3
Trang 4In the special case x 0, G r n,χ,q G r n,χ,q 0 are called the nth generalized q-Genocchi numbers
of order r attached to χ Therefore, we obtain the following theorem.
Theorem 2.1 For r ∈ N, one has
G r nr,χ,q
nr
r r! 2r
∞
m1, ,m r0
r
i1
χ m i
−1r j1 m j m1 · · · m rn
q 2.4
Note that
2r
∞
m1, ,m r0
r
i1
χ m i
−1r j1 m j m1 · · · m rn
q
2r
1− q n
n
l0
n l
−1l d−1
a1, ,a r0
⎛
⎝r
j1
χ
a j
⎞⎠−q l r
i1 a i
1 q ld r
2.5
Thus we obtain the following corollary
Corollary 2.2 For r ∈ N, we have
G r nr,χ,q
nr
2r
1− q n
n
l0
n l
−1l d−1
a1, ,a r0
⎛
⎝r
j1
χ
a j
⎞⎠−q l r
i1 a i
1 q ld r
2r∞
m0
m r − 1 m
−1m d−1
a1, ,a r0
−1r i1 a i
r
i1
χ a i
r
i1
a i md
n q
.
2.6
For h ∈ Z and r ∈ N, one also considers the extended higher-order generalized h, q-Genocchi polynomials as follows:
F q,χ h,r t, x 2 r t r
∞
m1, ,m r0
qr j1 h−jm j
r
i1
χ m i
−1r j1 m j e x
r
j1 m jq t
∞
n0
G h,r n,χ,q x t n
n! .
2.7
Trang 5From2.7, one notes that
G h,r 0,χ,q x G h,r 1,χ,q x · · · G h,r r−1,χ,q x 0,
G h,r nr,χ,q x
nr
∞
m1, ,m r0
qr j1 h−jm j
r
i1
χ m i
−1r j1 m j x m1 · · · m rn
q
2r
1− q n
n
l0
n l
q lx−1l d−1
a1, ,a r0
⎛
⎝r
j1
χ
a j
⎞⎠qr j1 h−ja j−1a1···a r q la1···a r
× ∞
m1, ,m r0
−1m1···m r q dm1···m r dr
j1 h−jm j
2r
1− q n
n
l0
n
l q lx−1ld−1
a1, ,a r0 r j1 χ
a j
qr j1 h−ja j
−q l r j1 a i
−q dh−rl ; q
r
,
2.8
where −x; q r 1 x1 xq · · · 1 xq r−1 .
Therefore, we obtain the following theorem
Theorem 2.3 For h ∈ Z, r ∈ N, one has
G h,r nr,χ,q x
nr
∞
m1, ,m r0
qr j1 h−jm j
r
i1
χ m i
−1r j1 m j x m1 · · · m rn
q
2r
1− q n
n
l0
n
l−q x ld−1
a1, ,a r0 r j1 χ
a j
qr j1 h−ja j
−q l r
j1 a i
−q dh−rl ; q
r
,
G h,r 0,χ,q x G h,r 1,χ,q x · · · G h,r r−1,χ,q x 0.
2.9
Note that
1
−q dh−rl ; q
r
1
1 q dh−rl ∞
m0
m r − 1 m
q
By2.10, one sees that
1
1− q n
n
l0
n
l−1l
q lxr i1 a i
−q dh−rl ; q
r
∞
m0
m r − 1 m
q
−1m
q dh−rm 1
1− q n
n
l0
n l
−1l
q lxr i1 a i dm
∞
m0
m r − 1 m
q
−1m q dh−rm
x
r
i1
a i dm
n
q
.
2.11
Trang 6By2.10 and 2.11, we obtain the following corollary.
Corollary 2.4 For h ∈ Z, r ∈ N, we have
G h,r nr,χ,q x
nr
r r!
2r∞
m0
m r − 1 m
q
−1m
q dh−rm
d−1
a1, ,a r0
⎛
⎝r
j1
χ
a j ⎞⎠qr
j1 h−ja j
x
r
i1
a i dm
n
q
2.12
By2.7, we can derive the following corollary
Corollary 2.5 For h ∈ Z, r, d ∈ N with d ≡ 1 mod 2, we have
q dh−1 G
h,r
nr,χ,q x d
nr r
r! G
h,r
nr,χ,q x
nr r
r! 2d−1
l0
χ l−1 l G h−1,r−1 nr−1,χ,q
nr−1 r−1
r − 1! ,
q x G
h1,r
nr,χ,q x
nr r
r! q − 1 G h,r nr1,χ,q x
nr1 r
r! G
h,r
nr,χ,q x
nr r
r! .
2.13
For h r inTheorem 2.3, we obtain the following corollary
Corollary 2.6 For r ∈ N, one has
G r,r nr,χ,q x 2r
1− q n
n
l0
n l
−q x l d−1
a1, ,a r0
⎛
⎝r
j1
χ
a j
⎞⎠qr j1 r−ja j la j−1a1···a r
−q dl ; q
r
2r∞
m0
m r − 1 m
q
−1m d−1
a1, ,a r0
⎛
⎝r
j1
χ
a j
⎞⎠qr j1 r−ja j
x
r
i1
a i dm
n
q
.
2.14
In particular,
G r,r nr,χ,q−1r − x
nr
r r! −1n q nr
2 G r,r nr,χ,q x
nr
Let x r in Corollary 2.6 Then one has
G r,r nr,χ,q−1
nr
r r! −1n
q nr
2 G r,r nr,χ,q r
nr
Trang 7Let w1, w2, , w r ∈ Q Then, one has defines Barnes’ type generalized q-Genocchi polynomials attached to χ as follows:
F q,χ r t, x | w1, w2, , wr 2r t r
∞
m1, ,m r0
r
i1
χ m i
−1m1···m r e xr j1 w j m jq t
∞
n0
G r n,χ,q x | w1, w2, , wrt n
n! .
2.17
By2.17, one sees that
G r nr,χ,q x | w1, , wr
nr
∞
m1, ,m r0
r
i1
χ m i
−1r j1 m j
⎡
⎣x r
j1
w j m j
⎤
⎦
n
q
. 2.18
It is easy to see that
2r
∞
m1, ,m r0
r
i1
χ m i
−1m1···m r
⎡
⎣x r
j1
w j m j
⎤
⎦
n
q
2r
1− q n
n
l0
n
l−q x ld−1
a1, ,a r0 r j1 χ
a j
−1r j1 a j q lr j1 w i a i
1 q dlw1
· · ·1 q dlw r
2.19
Therefore, we obtain the following theorem
Theorem 2.7 For r ∈ N, w1, w2, , wr∈ Q, one has
G r nr,χ,q x | xw1, w2, , wr
nr
∞
m1, ,m r0
r
i1
χ m i
−1r j1 m j x w1m1 · · · w r m rn
q
2r
1− q n
n
l0
n
l−q x ld−1
a1, ,a r0 r j1 χ
a j
−1r j1 a j q lr
i1 w i a i
1 q dlw1
· · ·1 q dlw r
2.20
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