Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2011, Article ID 424809, 8 pages
doi:10.1155/2011/424809
Review Article
On the Generalized q-Genocchi Numbers and
Polynomials of Higher-Order
C. S. Ryoo,1 T. Kim,2 J. Choi,2 and B. Lee3
1
Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea
Division of General Education-Mathematics, Kwangwoon University,
Seoul 139-701, Republic of Korea
3
Department of Wireless Communications Engineering, Kwangwoon University,
Seoul 139-701, Republic of Korea
2
Correspondence should be addressed to T. Kim, tkkim@kw.ac.kr
Received 10 August 2010; Accepted 18 February 2011
Academic Editor: Roderick Melnik
Copyright q 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
∞
tn
2t
xt
Gxt
, |t| < π,
e
e
G
x
n
n!
et 1
n0
1.1
where we use the technical method’s notation by replacing Gn x by Gn x, symbolically,
see 1, 2. In the special case x 0, Gn Gn 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 B2n 2nE2n−1 0 see 3, where Bn is a Bernoulli
number and En 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
α
∞
tn
2t
α
xt
e
G
x
1.2
n
n!
et 1
n0
2
Advances in Difference Equations
α
α
see 1, 4. In the special case x 0, Gn Gn 0 are called the nth Genocchi numbers
1
of order α. From 1.1 and 1.2, we note that Gn Gn . 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
2t
d−1
χa−1a eat
edt 1
a0
ext ∞
tn
Gn,χ x
n!
n0
1.3
see 1. In the special case x 0, Gn,χ Gn,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
2t
d−1
χa−1a eat
edt 1
α
a0
α
ext ∞
α
Gn,χ x
n0
tn
n!
1.4
α
see 7. In the special case x 0, Gn,χ Gn,χ 0 are called the nth generalized Genocchi
1
numbers attached to χ of order α see 1, 4–9. From 1.3 and 1.4, we derive Gn,χ Gn,χ .
Let us assume that q ∈ C with |q| < 1 as an indeterminate. Then we, use the notation
1 − qx
.
1−q
xq 1.5
The q-factorial is defined by
nq ! nq n − 1q · · · 2q 1q ,
1.6
and the Gaussian binomial coefficient is also defined by
n
k
q
nq !
n − kq !kq !
nq n − 1q · · · n − k 1q
1.7
kq !
see 5, 10. Note that
lim
n
q→1
k
n
k
q
nn − 1 · · · n − k 1
.
k!
1.8
It is known that
n1
k
q
n
k−1
q
q
k
n
k
q
q
n1−k
n
k−1
q
n
k
q
,
1.9
Advances in Difference Equations
3
see 5, 10. The q-binomial formula are known that
x−y
n
q
x−y
x − qy · · · x − q
n−1
y n
n
i0
i
i
q
2
−1i xn−i yi ,
q
∞
nl−1
1
n x − y x − qy · · · x − qn−1 y
x−y q
l
l0
1
1.10
x
n−l l
y,
q
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-König-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:
r
Fq,χ t, x
⎛
⎞
r
∞
r
tn
r
⎝ χ mj ⎠−1 j1 mj exm1 ···mr q t Gn,χ,q x .
2 t
n!
m1 ,...,mr 0
n0
j1
r r
∞
2.1
Note that
r
lim Fq,χ t, x
q→1
2t
d−1
r
χa−1a eat
edt 1
a0
r
ext .
2.2
r
By 2.1 and 1.4, we can see that limq → 1 Gn,χ,q x Gn,χ x. From 2.1, we note that
r
r
r
G0,χ,q x G1,χ,q x · · · Gr−1,χ,q x 0,
⎞
⎛
r
r
∞
r
Gnr,χ,q x
⎝ χ mj ⎠−1 j1 mj x m1 · · · mr n .
2r
nr q
r!
m1 ,...,mr 0
j1
r
2.3
4
Advances in Difference Equations
r
r
In the special case x 0, Gn,χ,q Gn,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
r
Gnr,χ,q
nr
r
2
r!
r
∞
r
m1 ,...,mr 0
χmi −1
r
j1
mj
m1 · · · mr nq .
2.4
i1
Note that
r
2
r
∞
m1 ,...,mr 0
χmi −1
i1
2r
1−q
n
j1
mj
m1 · · · mr nq
⎛
⎞ r
a
r
−ql i1 i
⎝ χ aj ⎠ −1
r .
1 qld
a1 ,...,ar 0
j1
n
n
l
l0
r
d−1
l
2.5
Thus we obtain the following corollary.
Corollary 2.2. For r ∈ N, we have
r
Gnr,χ,q
nr
r r!
2r
1−q
n
l0
l
⎛
⎞ r
a
r
−ql i1 i
l
⎝
⎠
χ aj
−1
r
1 qld
a1 ,...,ar 0
j1
n
n
d−1
n
r
d−1
r
∞
r
mr−1
m
a
r
i
2
χai ai md .
−1 i1
−1
m
m0
a1 ,...,ar 0
i1
i1
q
2.6
For h ∈ Z and r ∈ N, one also considers the extended higher-order generalized h, q-Genocchi
polynomials as follows:
h,r
Fq,χ t, x
2 t
r r
∞
q
r
j1 h−jmj
m1 ,...,mr 0
∞
tn
h,r
Gn,χ,q x .
n!
n0
r
r
x rj1 mj t
q
χmi −1 j1 mj e
i1
2.7
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5
From 2.7, one notes that
h,r
h,r
h,r
G0,χ,q x G1,χ,q x · · · Gr−1,χ,q x 0,
h,r
r
∞
r
r
Gnr,χ,q x
h−jm
r
j
2
q j1
χmi −1 j1 mj x m1 · · · mr nq
nr
r r!
m1 ,...,mr 0
i1
⎛
⎞
r
d−1
n
r
r
n
2
⎝ χ aj ⎠q j1 h−jaj −1a1 ···ar qla1 ···ar qlx −1l
n
1 − q l0 l
a1 ,...,ar 0
j1
∞
×
r
−1m1 ···mr qdm1 ···mr d
m1 ,...,mr 0
r
2
1−q
n
l
n
n l qlx −1
d−1
j1 h−jmj a1 ,...,ar 0
r
j1 χ aj
−qdh−rl ; q
l0
q
r
j1 h−jaj
−ql
rj1 ai
,
r
2.8
where −x; qr 1 x1 xq · · · 1 xqr−1 .
Therefore, we obtain the following theorem.
Theorem 2.3. For h ∈ Z, r ∈ N, one has
h,r
Gnr,χ,q x
nr
r
r!
∞
2
r
q
r
j1 h−jmj
m1 ,...,mr 0
2r
1−q
h,r
G0,χ,q x
n
r
l
n n l −qx
χmi −1
i1
d−1
a1 ,...,ar 0
j1
mj
r
j1 χ aj
··· h,r
Gr−1,χ,q x
x m1 · · · mr nq
−qdh−rl ; q
l0
h,r
G1,χ,q x
r
q
r
j1 h−jaj
−ql
rj1 ai
,
2.9
r
0.
Note that
1
−qdh−rl ; q
1
1 qdh−rl
r
∞
mr−1
m0
m
−1m qdh−rlm .
2.10
q
By 2.10, one sees that
r
n
nl −1l qlx i1 ai dh−rl n
;q r
−q
1 − q l0
∞
n
r
n
mr−1
1
m dh−rm
−1 q
−1l qlx i1 ai dm
n
m
1 − q l0 l
m0
q
1
∞
mr−1
m0
m
m dh−rm
−1 q
q
r
x ai dm
i1
n
.
q
2.11
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Advances in Difference Equations
By 2.10 and 2.11, we obtain the following corollary.
Corollary 2.4. For h ∈ Z, r ∈ N, we have
h,r
Gnr,χ,q x
nr
r r!
2
r
∞
mr−1
m
m0
d−1
m dh−rm
−1 q
⎛
⎞
n
r
r
r
h−ja
j
⎝ χ aj ⎠q j1
x ai dm
a1 ,...,ar 0
q
i1
j1
q
2.12
By 2.7, we can derive the following corollary.
Corollary 2.5. For h ∈ Z, r, d ∈ N with d ≡ 1 mod 2, we have
h−1,r−1
q
dh−1
h,r
h,r
d−1
Gnr−1,χ,q
Gnr,χ,q x d Gnr,χ,q x
nr 2 χl−1l nr−1 ,
nr r!
r!
r − 1!
l0
r
r
r−1
h,r
Gnr1,χ,q x
h1,r
Gnr,χ,q x q−1
qx nr r!
r
h,r
2.13
Gnr,χ,q x
nr .
nr1 r!
r!
r
r
For h r in Theorem 2.3, we obtain the following corollary.
Corollary 2.6. For r ∈ N, one has
2r
r,r
Gnr,χ,q x 1−q
n
n
n l0
−q
l
⎛
⎞ r
r
q j1 r−jaj laj −1a1 ···ar
⎝ χ aj ⎠
dl −q ; q r
a1 ,...,ar 0
j1
d−1
x l
⎛
⎞
n
r
d−1
r
∞
r
m
r
−
1
m
r
j1 r−jaj
⎝
⎠
q
x ai dm .
2
χ aj
−1
m
m0
a1 ,...,ar 0
i1
j1
q
q
2.14
In particular,
r,r
Gnr,χ,q−1 r − x
nr
r r!
r,r
G
x
n n 2r nr,χ,q
.
−1 q
nr
r!
r
2.15
Let x r in Corollary 2.6. Then one has
r,r
Gnr,χ,q−1
nr
r r!
r,r
G
r
n n 2r nr,χ,q
.
−1 q
nr
r!
r
2.16
Advances in Difference Equations
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Let w1 , w2 , . . . , wr ∈ Q . Then, one has defines Barnes’ type generalized q-Genocchi
polynomials attached to χ as follows:
r
Fq,χ t, x
∞
| w 1 , w 2 , . . . , wr 2 t
r r
r
m1 ,...,mr 0
χmi −1m1 ···mr ex
r
j1
w j m j q t
i1
2.17
∞
tn
r
Gn,χ,q x | w1 , w2 , . . . , wr .
n!
n0
By 2.17, one sees that
r
Gnr,χ,q x | w1 , . . . , wr nr
r r!
r
∞
2r
m1 ,...,mr 0
χmi −1
r
⎡
j1
mj ⎣
x
r
j1
i1
⎤n
w j mj ⎦ .
2.18
q
It is easy to see that
∞
2r
r
m1 ,...,mr 0
⎡
⎤n
r
x w j mj ⎦
χmi −1
m1 ···mr ⎣
j1
i1
q
r
r
l d−1
r
n j1 aj l
j1 wi ai
n l −qx
χ
a
q
−1
r
j
,...,a
0
a
j1
1
r
2
.
n
1 qdlw1 · · · 1 qdlwr
1 − q l0
2.19
Therefore, we obtain the following theorem.
Theorem 2.7. For r ∈ N, w1 , w2 , . . . , wr ∈ Q , one has
r
Gnr,χ,q x | xw1 , w2 , . . . , wr nr
r
r!
∞
2
r
r
m1 ,...,mr 0
χmi −1
r
j1
mj
x w1 m1 · · · wr mr nq
i1
r
r
l d−1
r
n j1 aj l
n l −qx
χ
a
q i1 wi ai
−1
j
,...,a
0
a
j1
1
r
2
.
n
1 qdlw1 · · · 1 qdlwr
1 − q l0
2.20
r
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