Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 316870, 8 pages
doi:10.1155/2010/316870
Research Article
On Multiple Generalized w-Genocchi Polynomials
and Their Applications
Lee-Chae Jang
Department of Mathematics and Computer Science, Konkuk University, Chungju,
Chungcheongbuk-do 380-701, Republic of Korea
Correspondence should be addressed to Lee-Chae Jang, leechae.jang@kku.ac.kr
Received 26 September 2010; Accepted 3 December 2010
Academic Editor: Ben T. Nohara
Copyright q 2010 Lee-Chae Jang. 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 define the multiple generalized w-Genocchi polynomials. By using fermionic p-adic invariant
integrals, we derive some identities on these generalized w-Genocchi polynomials, for example,
fermionic p-adic integral representation, Witt’s type formula, explicit formula, multiplication
formula, and recurrence formula for these w-Genocchi polynomials.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper, , p, p , , and p will,
respectively, denote the ring of integers, the ring of p-adic integers, the field of p-adic rational
numbers, the complex number field, and. the p-adic completion of the algebraic closure of
p . Let νp be the normalized exponential valuation of p with |p|p p−νp p p−1 .
The q-basic natural numbers are defined by
nq 1 − qn
1 q q2 · · · qn−1
1−q
1.1
for n ∈ , and the binomial coefficient is defined as
n
k
n!
nn − 1 · · · n − k 1
.
k!
n − k!k!
1.2
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Mathematical Problems in Engineering
The binomial formulas are well known that
n
n
1 − b −1i bi ,
i
i0
n
n
ni−1 i
1
b
1 − bn i0
i
1.3
see, 1, 2. When one talks of q-extension, q is variously considered as an indeterminate,
a complex number q ∈ , or a p-adic number q ∈ p . If q ∈ , one normally assumes that
|q| < 1. If q ∈ p , one normally assumes that |q − 1|p < 1. We use the notation
1 − qx
,
xq 1−q
x−q
x
1 − −q
,
1q
1.4
see 1–13 for all x ∈ p. Note that limq → 1 xq x for x ∈ p in presented p-adic case.
Let UD p be denoted by the set of uniformly differentiable function on p. For f ∈
UD p, an invariant p-adic q-integral on p is defined as
I−q f −1
x
1 q p
fxdμ−q x lim
fx −q .
N
p
n→∞1 q
p
x0
N
1.5
Thus, we have the following integral relation:
lim qI−q f1 I−q f 1 q f0,
q→1
1.6
where f1 x fx 1, and the fermionic p-adic invariant integral relation:
I−1 f lim I−q f q→1
fxdμ−1 x,
1.7
p
I−1 f1 I f 2f0.
1.8
Now, we recall that the definitions of w-Euler polynomials and w-Genocchi
polynomials are defined as
∞
2ext
tn
,
E
x
n,w
wet 1 n0
n!
∞
2text
tn
,
G
x
n,w
wet 1 n0
n!
t ∈ , w ∈ ,
1.9
with |1 − w|p < 1, respectively. In the special case x 0, En,w 0 En,w , and Gn,w 0 Gn,w
are called w-Euler numbers and w-Genocchi numbers see 2, 9.
Mathematical Problems in Engineering
3
In 13, Bayard and Simsek have studied multiple generalized Bernoulli polynomials
as follows:
r
aj t logwaj e t
wet aj − 1
j1
∞
t
r
Bn,w x; a1 , . . . ar n
n!
n0
,
π
π
t log|w| < min
,
···
a1
ap
1.10
where a1 , . . . , ar are strictly positive real numbers.
The purpose of this paper is to define another construction of multiple generalized wGenocchi polynomials and numbers, which are different from multiple generalized Bernoulli
polynomials and numbers in 13. By using fermionic p-adic invariant integrals, we derive
some identities on these generalized w-Genocchi polynomials, for example, fermionic p-adic
integral representation, Witt’s type formula, explicit formulas, multiplication formula, and
recurrence formula for these w-Genocchi polynomials.
2. Multiple Generalized w-Genocchi Polynomials and Numbers
Let r ∈ and a1 , . . . , ar be strictly positive real numbers. The multiple generalized wr
Genocchi polynomials Gn,w x; a1 , . . . , ar are defined as
r
j1
∞
tn
2tr
r
xt
,
e
G
,
.
.
.
a
a
x;
1
r
n,w
a
n!
wet j 1
n0
for t ∈ , w ∈ ,
2.1
r
where | log w t| ≤ min1≤j≤r {π/aj }. The values of Gn,w x; a1 , . . . , ar at x 0 are called the
multiple generalized w-Genocchi numbers: when r 1, w 1, and aj 0 j 1, . . . r, the
polynomials or numbers are called the ordinary Genocchi polynomials or numbers.
It is known that
wz etzx dμ−1 z t
p
wz1 z2 ···zr etz1 ···zr x dμ−1 z1 · · · dμ−1 zr ···
tr
p
∞
tn
2t
Gn,w x ,
t
we 1 n0
n!
p
2t
wet 1
r
ext ∞
n0
r
Gn,w x
tn
.
n!
2.2
In fact, let us take t ∈ ,
w ∈ , and we apply the above difference integral formula
1.8 for fz waz etaz , then we obtain
2
etx t
we a 1
waz etazx dμ−1 z.
p
2.3
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Mathematical Problems in Engineering
By 2.3, we easily see that
r
j1
2tr
ext tr
wet aj 1
wa1 z1 ···ar zr eta1 z1 ···ar zr x dμ−1 z1 · · · dμ−1 zr ···
p
p
∞ n0
wa1 z1 ···ar zr a1 z1 · · · ar zr xn
···
p
× dμ−1 z1 · · · dμ−1 zr r
2.4
p
tnr
,
n!
r
G0,w x; a1 , . . . , ar · · · Gr−1,w x; a1 , . . . , ar 0.
2.5
By 2.4 and 2.5, we obtain the following fermionic p-adic integral representation
formula for these numbers.
Theorem 2.1 p-adic integral representation. Let r ∈ and a1 , . . . , ar be strictly positive real
numbers. Then one has a fermionic p-adic invariant integral representation for the multiple generalized
r
w-Genocchi polynomials Gn,w x; a1 , . . . , ar as follows:
r
Gnr,w x; a1 , . . . , ar r! nr
r wa1 z1 ···ar zr a1 z1 · · · ar zr xn dμ−1 z1 · · · dμ−1 zr ···
p
p
2.6
for n ≥ r and
r
r
G0,w x; a1 , . . . , ar · · · Gr−1,w x; a1 . . . , ar 0.
2.7
We remark that if we set r 1 and a1 1, then we have the following equation:
r
1
Gnr,w x; 1 Gn1,w x
En,w x.
n1
1! n1
1
2.8
The generalized w-Genocchi polynomials are given by
∞
2t
tn
xt
,
e
G
a
x;
n,w
n!
wet a 1
n0
∞ waz etazx dμ−1 z waz az xn dμ−1 ztn .
p
n0
p
2.9
Mathematical Problems in Engineering
5
By comparing the coefficients on both sides in 2.9, we obtain the following identity on the
generalized w-Genocchi polynomials
Gn,w x; a
n!
waz az xn dμ−1 z.
2.10
p
Similarly, from 2.4, we can obtain the following Witt’s type formula for the multiple
generalized w-Genocchi polynomials.
Theorem 2.2 Witt’s type formula. Let r ∈
Then one has
r
Gn,w x; a1 , . . . , ar n!
and a1 , . . . , ar be strictly positive real numbers.
wa1 z1 ···ar zr a1 z1 · · · ar zr xn dμ−1 z1 · · · dμ−1 zr .
···
p
p
2.11
From 2.4, we can directly calculate the following:
r
Gn,w x; a1 , . . . , ar ···
wa1 z1 ···ar zr a1 z1 · · · ar zr xn × dμ−1 z1 · · · dμ−1 zr n!
p
p
n
n n−i
x
i
i0
2
n
n
i0
i
wa1 z1 ···ar zr × a1 z1 · · · ar zr i dμ−1 z1 · · · dμ−1 zr n!
···
p
p
r
n − i!xn−i Gi,w a1 , . . . , ar .
2.12
From 2.12, we get the following explicit formula.
Theorem 2.3 explicit formula. Let r ∈
one has
r
Gn,w x; a1 , . . . , ar and a1 , . . . , ar be strictly positive real numbers. Then
2
n
n
i0
i
r
n − i!xn−i Gi,w a1 , . . . , ar .
2.13
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Mathematical Problems in Engineering
Next we discuss the multiplication formula for the multiple generalized w-Genocchi
polynomials as follows:
r
Gn,w x; a1 , . . . , ar ···
wa1 z1 ···ar zr a1 z1 · · · ar zr xn dμ−1 z1 · · · dμ−1 zr n!
p
p
lim
N
mp
−1
wa1 z1 ···ar zr a1 z1 · · · ar zr xn −1z1 ···zr
N→∞
z1 ,...,zr 0
m−1
m
n
w
a1 t1 ···ar tr
t1 ···tr
−1
N
−1
p
lim
N →∞
t1 ,...,tr 0
−1my1 ···yr y1 ,...,yr 0
n
x a1 t1 · · · ar tr
a1 y1 · · · ar yr n!
× w m
m−1
mn
wa1 t1 ···ar tr −1t1 ···tr n!
···
wm a1 y1 ···ar yr
m a1 y1 ···ar yr
t1 ,...,tr 0
p
2.14
p
n
x a1 t1 · · · ar tr
a1 y1 · · · ar yr dμ−1 y1 · · · dμ−1 yr
m
m−1
x a1 t1 · · · ar tr
r
t1 ···tr
n
a1 t1 ···ar tr
m
; a1 , . . . , ar .
w
× Gn,wn
−1
m
t1 ,...,tr 0
×
Thus, we obtain the following multiplication formula for the multiple generalized wGenocchi polynomials.
Theorem 2.4 multiplication formula. Let r ∈ and a1 , . . . , ar be strictly positive real numbers.
For any m ∈ , one has
r
Gn,w mx; a1 , . . . , ar m
n
m−1
w
a1 t1 ···ar tr
t1 ···tr
×
−1
t1 ,...,tr 0
r
Gn,wn
x a1 t1 · · · ar tr
; a1 , . . . , ar .
m
Corollary 2.5. (1) If one sets w a1 · · · ar 1 and r, n ∈
r
formula for multiple Genocchi polynomials Gn x as follows:
r
Gn mx
r
where 2t/et 1 ext ∞
n0
r
m
n
m−1
r
Gn
t1 ,...,tr 0
Gn xtn /n!.
x
n
ti
i1
m
2.15
, then one obtains Raabe type
,
2.16
Mathematical Problems in Engineering
7
(2) If one sets w 1 and r, n ∈ , then one obtains Carlitz’s multiplication formula for the
r
multiple generalized Genocchi polynomials Gn x; a1 , . . . , ar as follows:
r
Gn mx; a1 , . . . , ar m
n
m−1
r
Gn
t1 ,...,tr 0
n
ti
x ai ; a1 , . . . , ar ,
m
i1
2.17
r
n
where 2tr / rj1 eaj t 1ext ∞
n0 Gn mx; a1 , . . . , ar t /n!.
Finally, we discuss the recurrence formula for the multiple generalized w-Genocchi
polynomials as follows. Let r ∈ and a1 , . . . , ar be strictly positive real numbers. For any
k 1, . . . , r, we can directly derive the following equation:
⎛
⎞
n
n
tn
k
r−k
⎝
Gj,w x | a1 , . . . , ak Gn−j,w ak1 , . . . , ar ⎠
n!
j
n0
j0
∞
⎛
⎞
∞
n
j
n−j
t
t
⎝ Gk x | a1 , . . . , ak Gr−k ak1 , . . . , ar ⎠
j,w
n−j,w
j!
n
−
j
!
n0
j0
∞
n0
k
Gm,w x
mln,m,l≥0
tm r−k
tl
| a1 , . . . , ak Gl,w ak1 , . . . , ar m!
l!
∞
r−k
t
tl
k
Gm,w x | a1 , . . . , ak Gl,w ak1 , . . . , ar m!
l!
m0
l0
⎛
⎞⎛
⎞
k
r
r
2tk
2tr−k ⎠ 2tr
xt ⎠⎝
xt
⎝
e
e
t a
t a
wet aj 1
j1 we j 1
j1
jk1 we j 1
∞
2.18
m
∞
tn
r
Gn mx | a1 , . . . , ar .
n!
n0
By comparing the coefficients on both sides in 2.18, we obtain the recurrence formula for
the multiple generalized w-Genocchi polynomials.
Theorem 2.6 recurrence formula. Let r ∈ and a1 , . . . , ar be strictly positive real numbers. For
any k 1, . . . , r, one has
r
Gn mx
| a1 , . . . , ar n
n
j0
j
k
r−k
Gj,w x | a1 , . . . , ak Gn−j,w ak1 , . . . , ar .
Acknowledgment
This paper was supported by Konkuk University in 2011.
2.19
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Mathematical Problems in Engineering
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