Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 860280, 14 pages
doi:10.1155/2010/860280
Research Article
Some Identities on the q-Genocchi Polynomials
of Higher-Order and q-Stirling Numbers by the
Fermionic p-Adic Integral on Zp
Seog-Hoon Rim, Jeong-Hee Jin, Eun-Jung Moon,
and Sun-Jung Lee
Department of Mathematics, Kyungpook National University, Tagegu 702-701, Republic of Korea
Correspondence should be addressed to Seog-Hoon Rim, shrim@knu.ac.kr
Received 25 September 2010; Accepted 8 November 2010
Academic Editor: H. Srivastava
Copyright q 2010 Seog-Hoon Rim 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.
A systemic study of some families of q-Genocchi numbers and families of polynomials of Nörlund
type is presented by using the multivariate fermionic p-adic integral on Zp. The study of these
higher-order q-Genocchi numbers and polynomials yields an interesting q-analog of identities for
Stirling numbers.
1. Introduction
Let p be a fixed odd prime number. Throughout this paper, Zp , Qp , C, and Cp denote the ring
of p-adic rational integers, the field of p-adic rational numbers, the complex number field,
and the completion of the algebraic closure of Qp , respectively. Let N be the set of natural
numbers and Z N ∪ {0}. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p.
When one talks of q-extension, q is variously considered as an indeterminate, a
complex q ∈ C, or a p-adic number q ∈ Cp . If q ∈ C, then one normally assumes |q| < 1.
If q ∈ Cp , then we assume |q − 1|p < 1. In this paper, we use the following notation:
1 − qx
,
xq 1−q
x−q
see 1–10. Hence limq → 1 xq x for all x ∈ Zp .
x
1 − −q
,
1q
1.1
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The q-factorial is defined as nq ! nq n − 1q · · · 2q 1q , and the Gaussian binomial
coefficient is defined by the standard rule
n
k
q
nq !
n − kq !kq !
nq n − 1q · · · n − k 1q !
kq !
1.2
,
see 7, 9. Note that limq → 1 nk q nk n!/n − k!k! nn − 1 · · · n − k 1/k!. It readily
follows from 1.2 that
n1
k
n
k−1
q
q
k
n
q
k
q
n
n−k1
q
k−1
q
n
k
,
1.3
q
see 4, 7.
The q-binomial formulas are known,
b; q
n
i
n
n
n−1
1 − b 1 − bq · · · 1 − bq
q 2 −1i bi ,
i q
i0
∞
ni−1
1
1
bi .
b; q n 1 − b 1 − bq · · · 1 − bqn−1
i
i0
1.4
q
We say that f : Zp → Cp is uniformly differentiable function at a point a ∈ Zp , and
we write f ∈ UDZp , if the difference quotients Φf : Zp × Zp → Cp such that Φf x, y fx − fy/x − y have a limit f a as x, y → a, a. For f ∈ UDZp , the q-deformed
fermionic p-adic integral is defined as
I−q f −1
p
x
1
fxdμ−q x lim N fx −q ,
N →∞ p
Zp
−q x0
N
1.5
see 7, 9. Note that
I−1 f lim I−q f q→1
Zp
fxdμ−1 x.
1.6
For n ∈ N, write fn x fx n. Then, we have
n−1
I−1 fn −1n I−1 f 2 −1n−1−l fl.
l0
1.7
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3
Using 1.7, we can readily derive the Genocchi polynomials, Gn x, namely,
t
Zp
exyt dμ−1 y ∞
tn
2t xt ,
e
G
x
n
n!
et 1
n0
1.8
see 1–27. Note that Gn 0 Gn are referred to as the nth Genocchi numbers. Let us now
introduce the Genocchi polynomials of Nörlund type as follows:
r
t
Zp
···
Zp
e
xx1 ···xr t
dμ−1 x1 · · · dμ−1 xr 2t
t
e 1
r
ext ∞
r
Gn x
n0
tn
,
n!
1.9
r times
et 1
2t
r
ext ∞
tn
−r
Gn x ,
n!
n0
−r
−r
r
1.10
r
see 7, 9. In the special case x 0, Gn 0 Gn , and Gn 0 Gn are referred to as the
Genocchi numbers of Nörlund type. Let Ehx hx 1 be the shift operator. Then, the
q-difference operator Δq is defined as
Δnq n E − qi−1 I ,
where Ihx hx,
1.11
i1
see 4, 7, 9. It follows from 1.11 that
fx x
n≥0
n
Δnq f0,
1.12
q
k
where Δnq f0 nk0 nk q q 2 fn − k see 5, 6, 10. The q-Stirling number of the second
kind as defined by Carlitz is given by
− k k
j
k n
q 2 j 2
S2 n, k; q k − j q,
−1 q
kq ! j0
j q
1.13
see 7, 10. By 1.12 and 1.13, we see that
− k q 2 k n
S2 n, k; q Δ 0 ,
kq ! q
1.14
see 6, 10.
In this paper, the q-extensions of 1.9 are considered in several ways. Using these qextensions, we derive some interesting identities and relations for Genocchi polynomials and
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numbers of Nörlund type. The purpose of this paper is to present a systemic study of some
families q-Genocchi numbers and polynomials of Nörlund type by using the multivariate
fermionic p-adic integral on Zp .
2. q-Extensions of Genocchi Numbers and
Polynomials of Nörlund Type
In this section, we assume that q ∈ Cp with |1 − q|p < 1. We first consider the q-extensions of
1.8 given by the rule
∞
tn
Gn,q x t
exyq t dμ−1 y
n!
Zp
n0
n
∞
∞
n
l −1l qlx tn
2t 2t
−1m emxq t .
n
l
n!
1
q
1
−
q
n0
m0
l0
2.1
Thus, we obtain the following lemma.
Lemma 2.1. If n ≥ 0, then
∞
n
Gn1,q x
nl −1l qlx
2
2 −1m m xnq .
n
n1
1 ql
1 − q l0
m0
2.2
By 1.14,
xnq
n
x
k0
k
k
kq !S2 k, n − k; q q 2
q
k − n−k n
2
q 2
k
Δn−k
xq x − 1q · · · x − k 1q
q 0
−
k
n
q!
k0
k
−
n
q 2
k0
n−k 2
n − kq !
1
k
Δn−k
q 0 1−q
k
k
k
l0
l
q
2.3
l
2 −1l qlx−k1 .
q
Thus, we have
k k
l
n q 2 S k, n − k; q k
l Gn1,q m Gm1,q 1 − k
2
l
l
2
,
q
q−1
−1
k
n1
m1
l
1−q
m0 m
k0
l0
q
and we obtain the following theorem.
2.4
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5
Theorem 2.2. If n ≥ 0, then
k k
l
n q 2 S k, n − k; q k
l Gn1,q m Gm1,q 1 − k
2
l
l
,
q 2 −1
q−1
k
n1
m1
l
1−q
m0 m
k0
l0
2.5
q
where Gn,q Gn,q 0 stand for the nth Genocchi numbers.
r
r
r
Consider a q-extensoin of 1.9 such that G0,q x G1,q x · · · Gr−1,q x 0 and
r
Gnr,q x
r! nr
r Zp
···
r
2
1−q
r
Let Fq t, x ∞
n0
x x1 · · · xr nq dμ−1 x1 · · · dμ−1 xr Zp
n
n
n
l
l0
−1l qlx
1
1 ql
r
∞
mr−1
r
2
−1m m xnq .
m
m0
2.6
r
Gn,q xtn /n!. Then,
r
Fq t, x
∞
mr−1
2 t
−1m emxq t .
m
m0
2.7
r r
r
r
In the special case x 0, the numbers Gn,q 0 Gn,q are referred to as q-extension of
the Genocchi numbers of order r. In the sense of the q-extension in 1.10, consider the qextension of Genocchi polynomials of Nörlund type given by
r
Gq t, x
−r
−r
Fq t, x
−r
∞
r
r
tn
1 −r
emxq t Gn,q x .
r r
2 t m0 m
n!
n0
−r
r
By 2.8, G0,q x G1,q x · · · Gr−1,q x 0 and r! nr Gn−r,q x 1/2r Therefore, we obtain the following theorem.
2.8
r
r
m0 m m
xnq .
Theorem 2.3. For r ∈ N, and, n ≥ 0, write
r r
2t
∞
mr−1
m0
m
−1m emxq t ∞
tn
r
Gn,q x .
n!
n0
2.9
Then,
r
r
n
∞
n
mr−1
Gnr,q x
1
2r
l lx
r
2
nr −1 q
−1m m xnq ,
n
l
1
q
1
−
q
l
m
r! r
m0
l0
r
n
r
n
r
n
1
1
−r
l lx
l
r!
Gn−r,q x r
−1 q 1 q
m xnq .
n
r
2 m0 m
2 1 − q l0 l
r
2.10
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International Journal of Mathematics and Mathematical Sciences
−r
−r
The numbers Gn,q 0 Gn,q are referred to as the q-extension of Genocchi numbers
of Nörlund type. For h ∈ Z and r ∈ N, introduce the extended higher-order q-Genocchi
polynomials as follows:
h,r
Gnr,q x
r! nr
r Zp
···
q
Zp
r
j1 h−jxj
x x1 · · · xr nq dμ−1 x1 · · · dμ−1 xr .
2.11
Then,
h,r
n
n
nl −1l qlx
nl −1l qlx
2r
n
r! nr
1 − q l0 −qh−1l ; q−1 r
1 − q l0 −qh−rl ; q r
r ∞
mr−1
r
2
−1m qh−rm x mnq .
m
m0
q
Gnr,q x
h,r
Let Fq
t, x ∞
n0
2r
n
2.12
h,r
Gn,q xtn /n!. Then, we can readily see that
h,r
Fq t, x
2 t
r r
∞
mr−1
m0
m
−1m qh−rm exmq t .
2.13
q
Therefore, we obtain the following theorem.
Theorem 2.4. For h ∈ Z and n ≥ 0, let
r r
2t
∞
mr−1
m0
m
q
−1m qh−rm exmq t ∞
tn
h,r
Gn,q x .
n!
n0
2.14
Then,
n
∞
mr−1
nl −1l qlx
r
2
−1m qh−rm x mnq .
n
r! nr
1 − q l0 −qh−rl ; q r
m
r m0
h,r
Gnr,q x
2r
2.15
q
Let us now define the extended higher-order Nörlund type q-Genocchi polynomials
as follows:
n
n
nl −1l qlx
1
h,−r
r
.
r!
Gn−r,q x n
1 − q l0 Z · · · Z qlx1 ···xr q j1 h−jxj dμ−1 x1 · · · dμ−1 xr r
p
p
2.16
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By 2.16,
r!
n
r
h,−r
Gn−r,q x
n
n
−1l qlx −qh−rl ; q
n
r
2r 1 − q l0 l
r
r
m
1
r
q 2 qh−rm m xnq .
2 m0 m
1
2.17
q
h,−r
Let Fq
t, x ∞
h,−r
n0
Gn,q
xtn /n!. Then, we have
h,−r
Fq
t, x
r
r
m
1 q 2 qh−rm emxq t ,
r r
2 t m0 m
2.18
q
h,−r
where, G0,q
theorem.
h,−r
x G1,q
h,−r
x · · · Gr−1,q x 0. Therefore, we obtain the following
Theorem 2.5. For h ∈ Z, n ≥ 0, and r ∈ N, write
r
∞
r
1 tn
h,−r
m
2 qh−rm emxq t q
Gn,q x .
r
r
2 t m0 m
n!
n0
2.19
q
Then,
r!
h,−r
where, G0,q
n
r
h,−r
Gn−r,q x
h,−r
x G1,q
n
n
−1l qlx −qh−rl ; q
n
r
2r 1 − q l0 l
r
r
m
1
r
q 2 qh−rm m xnq ,
2 m0 m
q
1
2.20
h,−r
x · · · Gr−1,q x 0.
For h r,
n
r,r
n
∞
−1l qlx
mr−1
Gnr,q x
2r
l
r
2
l nr −1m x mnq ,
n
;
q
−q
1
−
q
m
r! r
m0
l0
r
q
r!
n
r
r,−r
Gn−r,q x
1
n
2r 1 − q
n
n
l0
l
l lx
−1 q
−q ; q
l
r
2.21
m
r
r
1
r
q 2 m xnq .
2 m0 m
q
2.22
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International Journal of Mathematics and Mathematical Sciences
It can readily be seen that
qmx 2r
m−r −q ; q r
Zp
Zp
···
···
Zp
q
r
j1 m−jxj mx
m r
x x1 · · · xr q q − 1 1 q− j1 jxj dμ−1 x1 · · · dμ−1 xr Zp
m
m l
q−1
l
l0
Zp
m
m ···
Zp
x x1 · · · xr lq q−
r
j1
jxj
dμ−1 x1 · · · dμ−1 xr 0,r
l Glr,q x
q−1
.
r! lr
r
l
l0
dμ−1 x1 · · · dμ−1 xr By 2.23, qmx 2r /−qm−r ; qr m
m
l0 l q
2.23
0,r
− 1l Glr,q x/r!
I−1 f1 I−1 f 2f0,
lr r
. As is known,
where f1 x fx 1.
2.24
It follows from 2.24 that
qh−1
Zp
···
Zp
−
x 1 x1 · · · xr nq q
Zp
j1 h−jxj
···
Zp
2
r
x x1 · · · xr nq q
r
j1 h−jxj
Zp
···
Zp
dμ−1 x1 · · · dμ−1 xr x x2 · · · xr nq q
r−1
dμ−1 x1 · · · dμ−1 xr j1 h−1−jxj1
2.25
dμ−1 x2 · · · dμ−1 xr .
By 2.25,
h,r
qh−1
Gnr,q x 1
nr
h,r
Gnr,q x
nr
h−1,r−1
2Gnr−1,q x.
2.26
A simple manipulation shows that
q
x
Zp
···
Zp
x x1 · · · xr nq q
q−1
Zp
h1,r
j1 h−j1xj
dμ−1 x1 · · · dμ−1 xr Zp
···
···
r
Zp
Zp
x x1 · · · xr n1
q q
x x1 · · · xr nq q
r
r
j1 h−jxj
h,r
j1 h−jxj
dμ−1 x1 · · · dμ−1 xr 2.27
dμ−1 x1 · · · dμ−1 xr .
h,r
By 2.27, qx Gnr,q x/n 1 q − 1Gnr1,q x/n r 1 Gnr,q x/n 1.
International Journal of Mathematics and Mathematical Sciences
9
Therefore, we obtain the following proposition.
Proposition 2.6. For h ∈ Z, r ∈ N and n ≥ 0, the following equations
h,r
Gnr,q x 1
qh−1
nr
h1,r
q
G
x
x nr,q
n1
h,r
Gnr,q x
nr
h−1,r−1
2Gnr−1,q x,
h,r
2.28
h,r
Gnr1,q x Gnr,q x
q−1
nr1
n1
hold. Moreover, qmx 2r /−qm−r ; qr m
m
l0 l q
0,r
− 1l Glr,q x/r!
lr r
.
By 2.21,
r,r
Gnr,q−1 r − x
r! nr
r n
nl −1l q−lr−x
−l −1 n
−q ; q r
1 − q−1 l0
2r
r,r
n
G
x
nl −1l qx
2
n n 2r nr,q
.
q
−1
n
nr
r! r 1 − q l0 −ql ; q r
r
−1n qn 2 2.29
r
Hence,
Zp
···
Zp
r − x x1 · · · xr nq−1 q−
n n 2r −1 q
r
j1 r−jxj
dμ−1 x1 · · · dμ−1 xr Zp
···
Zp
x x1 · · · r
r,r
xr nq q
2.30
r
j1 r−jxj
dμ−1 x1 · · · dμ−1 xr .
r,r
For h r, Gnr,q−1 0 −1n qn 2 Gnr,q r. It also follows from 2.26 that
r,r
q
r−1
Gnr,q x 1
nr
r,r
Gnr,q x
nr
r−1,r−1
2Gnr−1,q x.
2.31
The Stirling numbers of the first kind are defined as
n n
1 kq z S1 n, k; q zk ,
2.32
k0
k1
see6, 9,
q
m
2
r
m
q
m
q 2 rq · · · r − m 1q
mq !
m−1 1 rq − kq .
mq ! k0
2.33
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International Journal of Mathematics and Mathematical Sciences
It can readily be seen that
n−1 z − kq z
k0
n
n−1
1−
kq
z
k0
n
S1 n − 1, k; q −1k zn−k .
2.34
k0
By 2.33 and 2.34,
m−1 m
S1 m − 1, k; q −1k rm−k
.
rq − kq q
2.35
k0
k0
Formulas 2.22 and 2.35 imply the following assertion.
Proposition 2.7. For r ∈ N and n ∈ Z ,
r!
n
r
r,−r
Gn−r,q x r m
1
S1 m − 1, k; q −1k rm−k
m xnq .
q
r
2 mq ! m0 k0
2.36
The generalized Genocchi numbers and polynomials of Nörlund type are defined by
∞
tn
2r tr
r
xt
,
e
G
,
.
.
.
,
w
|
w
x
1
r
n
n!
ew1 t 1ew2 t 1 · · · ewr t 1
n0
r
2.37
r
and Gn w1 , . . . , wr Gn 0 | w1 , . . . , wr . We can now also define a q-extension of 2.37 as
follows. For w1 , . . . , wr ∈ Zp and δ1 , . . . , δr ∈ Z, write
r
Gnr,q x | w1 , . . . , wr ; δ1 , . . . , δr r! nr
r Zp
···
r
Zp
x1 w1 · · · xr wr xnq dμ−qδ1 x1 · · · dμ−qδr xr ,
2.38
r
and Gnr,q w1 , . . . , wr ; δ1 , . . . , δr Gnr,q 0 | w1 , . . . , wr ; δ1 , . . . , δr . Thus,
r
Gnr,q x | w1 , . . . , wr ; δ1 , . . . , δr r! nr
r n
2qδ1 · · · 2qδr nl −1l qlx
.
n
δ1 lw1 · · · 1 qδr lwr
1−q
l0 1 q
2.39
Another q-extension of Nörlund type generalized Genocchi numbers and polynomials is also
of interest, namely,
∗r
Gnr,q x | w1 , . . . , wr ; δ1 , . . . , δr Zp
r! nr
r ···
x1 w1 · · · xr wr xnq qδ1 x1 ···δr xr dμ−1 x1 · · · dμ−1 xr ,
Zp
2.40
International Journal of Mathematics and Mathematical Sciences
∗r
11
∗r
and Gnr,q w1 , . . . , wr ; δ1 , . . . , δr Gnr,q 0 | w1 , . . . , wr ; δ1 , . . . , δr . By 2.40,
∗r
Gnr,q x | w1 , . . . , wr ; δ1 , . . . , δr r! nr
r 2r
1−q
n
n
nl −1l qlx
.
1 qδ1 lw1 · · · 1 qδr lwr
2.41
l0
3. Further Remarks
0,r
0,−r
n
For h 0, consider the following polynomials Gnr,q x/r! nr
r and r! r Gnr,q x:
r!
n
r
0,r
Gnr,q x
nr r! r
···
Zp
Zp
1
0,−r
Gn−r,q x 1−q
x x1 · · · xr nq q−
n
n
r
j1
jxj
dμ−1 x1 · · · dμ−1 xr ,
3.1
n
l0 Zp
···
Zp
−1l qlx
l
qlx1 ···xr q−
r
j1
jxj
dμ−1 x1 · · · dμ−1 xr .
Then,
0,r
Gnr,q x
r!
r!
n
r
0,r
Let Fq
r
2
nr 1−q
r
0,−r
Gn−r,q x
t, x l0
1
∞
n
n
n
2r 1 − q
n
∞
−1l qlx
mr−1
r
q−rm −1m x mnq
l−r 2
−q ; q r
m
m0
l
q
n
n
l
l0
−1 q
0,r
n0
l lx
−q
0,−r
Gn,q xtn /n! and let Fq
0,r
Fq t, x
2 t
r r
l−r
;q
r
t, x ∞
mr−1
m0
m
m
r
r
1
r
q 2 q−rm m xnq .
2 m0 m
q
∞
3.2
0,−r
n0
Gn,q xtn /n!. Then,
q−rm −1m exmq t ,
q
3.3
m
r
r
1 0,−r
Fq
q 2 q−rm emxq t .
t, x r r
2 t m0 m
q
Consider the following polynomials:
h,1
Gn1,q x
n1
Zp
2
qx1 h−1 x x1 nq dμ−1 x1 1−q
n
n
nl −1l qlx
l0
1 qlh−1
.
3.4
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International Journal of Mathematics and Mathematical Sciences
A simple calculation of the fermionic p-adic invariant integral on Zp show that
q
x x1 nq qx1 h−1 dμ−1 x1 x
Zp
q−1
Zp
x x1 h−2
dμ−1 x1 x1 n1
q q
h,1
3.5
Zp
h−1,1
x x1 nq qx1 h−2 dμ−1 x1 .
h−1,1
By 3.5, qx Gn1,q x q − 1Gn2,q x/2n 2 Gn1,q x. It can readily be proved that
Zp
x x1 nq qx1 h−1 dμ−1 x1 j0
h,1
By 3.6, Gn1,q x/n 1 that
Zp
n
n
x x1 n n j0
j
j
j
Zp
x1 q qx1 h−1 dμ−1 x1 .
3.6
h,1
n−j
xq qjx Gj1,q /j 1. Using 2.24, we can also prove
1nq qx1 1h−1 dμ−1 x1 h,1
n−j
xq qjx
Zp
x x1 nq qx1 h−1 dμ−1 x1 2xnq .
h,1
3.7
h,1
Thus, qh−1 Gn1,q x/n 1 Gn1,q x/n 1 2xnq . For x 0, we have qh−1 Gn1,q 1/
h,1
n 1 Gn1,q /n 1 2δn,0 , where δn,0 is the Kronecker delta.
h,1
It is easy to see that G1,q Zp qx1 h−1 dμ−1 x1 2/1 qh−1 2/2qh−1 . By 3.4,
h,1
Gn1,q−1 1 − x
n1
Zp
1 − x x1 nq−1 q−x1 h−1 dμ−1 x1 2
−1n qnh−1 1−q
n
n
nl −1l qlx
l0
3.8
1 qlh−1
h,1
−1n qnh−1
h,1
Gn1,q x
n1
.
h,1
In particular, if x 1, then Gn1,q−1 0/n 1 −1n qnh−1 Gn1,q 1/n 1 h,1
−1n−1 qn Gn1,q /n 1 for n ≥ 1.
Recently, Kim has studied p-adic fermionic integral on Zp connected with the problems
of mathematical physics see 6, 10, 11, and our result are closely related to his results. In
the future, we will try to study p-adic stochastic problems associated with our theorems.
For example, p-adic q-Bernstein polynomials seem to be closely related to our results see
6, 14, 20.
International Journal of Mathematics and Mathematical Sciences
13
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14
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