By using p-adic q-Vokenborn integral 6, Kim 2, 7 9, 14–18 constructed many kind of generating functions of the q-Euler numbers and polynomials and their interpolation functions.. 1.21 Fr
Trang 1Volume 2010, Article ID 351419, 13 pages
doi:10.1155/2010/351419
Research Article
On Multiple Interpolation Functions of
1 Department of Mathematics Education, Kyungpook National University, Tagegu 702-701, South Korea
2 Department of Mathematics, Kyungpook National University, Tagegu 702-701, South Korea
Correspondence should be addressed to Seog-Hoon Rim,shrim@knu.ac.kr
Received 14 December 2009; Accepted 29 March 2010
Academic Editor: Wing-Sum Cheung
Copyrightq 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
Recently, many mathematicians have studied various kinds of the q-analogue of Genocchi
numbers and polynomials In the workNew approach to q-Euler, Genocchi numbers and their
interpolation functions, “Advanced Studies in Contemporary Mathematics, vol 18, no 2, pp 105–
112, 2009.”, Kim defined new generating functions of q-Genocchi, q-Euler polynomials, and their interpolation functions In this paper, we give another definition of the multiple Hurwitz type q-zeta function This function interpolates q-Genocchi polynomials at negative integers Finally, we
also give some identities related to these polynomials
1 Introduction
Let p be a fixed odd prime number Throughout this paper Z p, Qp, C, and Cpdenote 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 ofQp, respectively LetN be the set of natural numbers andZ N ∪ {0} Let v pbe the normalized exponential valuation ofCpwith|p| p
p −v p p 1/p see 1
When one talks of q-extension, q is variously considered as an indeterminate, a complex q ∈ C or a p-adic number q ∈ C p If q ∈ C, then one normally assumes |q| < 1.
If q ∈ C p, then we assume that|q − 1| p < 1 In this paper, we use the following notation:
x x : q
1− q x
1− q , x −q 1−
−qx
see 2,3 Hence limq → 1 x x for all x ∈ Z p
Trang 2We say that f : Z p → Cpis uniformly differentiable function at a point a ∈ Zpand we
write f ∈ UDZ p if the difference quotients Φf :Zp× Zp → Cpsuch that
Φf
x, y
f x − f
y
have a limit fa as x, y → a, a For f ∈ UDZ p , the q-deformed fermionic p-adic
integral is defined as
I −q
f
Zp
f xdμ −q x lim
N → ∞
1
p N
−q
pN−1
x0
f x−qx 1.3
see 4 6 Note that
I−1
f
lim
q → 1 I −q
f
Zp
f xdμ−1x 1.4
see 7 9 Let f1x be the translation with f1x fx 1 Then we have the following
integral equation:
I−1
f1
I−1f
see 10–12
The ordinary Genocchi numbers and polynomials are defined by the generating functions as, respectively,
F t 2t
e t 1
∞
n0
Gn t n n! , |t| < π,
F t, x 2t
e t 1e xt
∞
n0
Gn x t n
n! , |t| < π.
1.6
Observe that G n 0 G nsee 10,11,13
These numbers and polynomials are interpolated by the Genocchi zeta function and Hurwitz-type Genocchi zeta function, respectively,
ζG s 2∞
n1
−1n
n s , s ∈ C,
ζG s, x 2∞
n0
−1n
n x s , s ∈ C, 0 < x ≤ 1.
1.7
Thus we note that Genocchi zeta functions are entire functions in the whole complex s-plane
see 14–16
Trang 3Various kinds of the q-analogue of the Genocchi numbers and polynomials, recently,
have been studied by many mathematicians In this paper, we use Kim’s14–16 methods
By using p-adic q-Vokenborn integral 6, Kim 2, 7 9, 14–18 constructed many kind
of generating functions of the q-Euler numbers and polynomials and their interpolation
functions He also gave many applications of these numbers and functions He14 defined
q-extension Genocchi polynomials of higher order He gave many applications and interesting identities We give some of them in what follows
Let q ∈ C with |q| < 1 The q-Genocchi numbers G n,q and polynomials G n,q x are
defined by Kim of the generating functions as, respectively,
t
Zp
e xt dμ −q x ∞
n0 Gn,q t n n! , |t| < π
t
Zp
e xyt dμ −q
y
∞
n0
Gn,q x t n
n! , |t| < π
1.8
see 1,8 11,13,14,17 By using the Taylor expansion of e xt,
∞
n0
Zp
x n dμ −q x t n
n! ∞
n0 Gn,q t n−1 n! G 0,q∞
n0
Gn1,q
n 1
t n n! . 1.9
By comparing the coefficient of both sides of tn /n! in the above,
G 0,q 0,
Gn1,q
n 1
Zp
x n
dμ −q x 2
1− qn
n
l0
n l
−1l 1
1 q l1
1.10
From the above, we can easily derive that
∞
n0 Gn,q t n n! ∞
n0
t
Zp
x n dμ −q x
t n n!
∞
n0
t 2
1− qn
n
l0
n l
−1l 1
1 q l1
t n n!
2t∞
m0
−1m q m e mt
1.11
Thus we have, following that,
Fq t 2t∞
m0
−1m
q m e mt∞
n0 Gn,q t n
Trang 4Using similar method to the above, we can find that
G 0,q x 0,
Gn1,q x
n 1
Zp
x yn
dμ −q
y
2
1− qn
n
l0
n l
−1l
q lx 1
1 q l1 1.13 Thus we can easily derive that
Fq t, x 2t∞
m0
−1m
q m e mxt∞
n0
Gn,q x t n
Observe that F q t F q t, 0 Hence we have G n,q 0 G n,q If q → 1 into 1.14, then we
easily obtain Ft, x in 1.6
Let q ∈ C with |q| < 1, r ∈ N, and n ≥ 0 We now define as the generating functions of higher order q-extension Genocchi numbers G r n,q and polynomials G r n,q x, respectively,
F q r t t r
Zp
· · ·
Zp
r times
e x1···x r t dμ −q x1 · · · dμ −q x r ∞
n0
G r n,q t n n! ,
F q r t, x t r
Zp
· · ·
Zp
r times
e xx1···x r t dμ −q x1 · · · dμ −q x r ∞
n0
G r n,q x t n
n! .
1.15
Then we have
∞
n0
Zp
· · ·
Zp
x1 · · · x rn dμ −q x1 · · · dμ −q x r
t n n!
∞
n0
G r n,q t n−r n! r−1
n0
G r n,q t n−r n! ∞
n0
G r nr,q
nr
r
r!
t n n! ,
1.16
wherenr
r
n r!/n!r!.
By comparing the coefficient of both sides of tn /n! in the above, we can derive that
G r 0,q G r 1,q · · · G r r−1,q 0,
G r nr,q
nr
r
r!
Zp
· · ·
Zp
x1 · · · x rn dμ −q x1 · · · dμ −q x r
2r
1− qn
n
l0
n l
−1l 1
1 q l1r
1.17
Trang 5Therefore we obtain
F q r t 2 r
t r
∞
m0
−1m
q m
m r − 1 m
e mt∞
n0
G r n,q t n n! . 1.18 Using similar method to the above, we can also derive that
G r 0,q x G r 1,q x · · · G r r−1,q x 0,
G r n,q x
nr
r
r! 2r
1− qn
n
l0
n l
−1l q lx 1
1 q l1r 1.19 Thus we can easily obtain the following theorem
Theorem 1.1 For r ∈ N and n ≥ 0, one has
F q r t, x 2 r
t r
∞
m0
−1m
q m
m r − 1 m
e mxt∞
n0
G r n,q x t n
n! . 1.20
It is noted that if r 1, then 1.20 reduces to 1.14
Remark 1.2 In1.20, we easily see that
lim
q → 1 F q r t, x 2 r t r∞
m0
−1m
m r − 1 m
e mxt
2r t r e tx
∞
m0
m r − 1 m
−e tm
2r t r e tx
1 e tr
F r t, x.
1.21
From the above, we obtain generating function of the Genocchi numbers of higher order That is
F r t, x 2r t r e tx
1 e tr ∞
n0
G r n x t n
Thus we have
lim
q → 1 G r n,q x G r n x. 1.23
Trang 6Hence we have
F r t, x
2t
e t 1
2t
e t 1
· · ·
2t
e t 1
rtimes
e tx
2r t r e tx
∞
n1 0
−1n1e n1t∞
n2 0
−1n2e n2t· · ·∞
n r0
−1n r e n r t
2r t r e tx
∞
n1,n2, ,n r0
−1n1n2···n r e n1n2···n r t
∞
n0
G r n x t n
n! .
1.24
In14, Kim defined new generating functions of q-Genocchi, q-Euler polynomials and
their interpolation functions In this paper, we give another definition of the multiple Hurwitz
type q-zeta function This function interpolates q-Genocchi polynomials at negative integers.
Finally, we also give some identities related to these polynomials
Polynomials and Numbers
In this section, we study modified generating functions of the higher order q-Genocchi
numbers and polynomials We obtain some relations related to these numbers and
polynomials Therefore we define generating function of modified higher order q-Genocchi polynomials and numbers, which are denoted by G r n,q x and G r n,q, respectively, in1.15 We give relations between these numbers and polynomials
We modify1.20 as follows:
Fr q t, x F r q
q −x t, x
where F q r t, x is defined in 1.20 From the above we find that
Fr q t, x ∞
n0
q −nrx G r n,q x t n
After some elementary calculations, we obtain
Fr q t, x q −rxexp
xq −x t
F q r t, 2.3 where
F q r t 2 r
t r
∞
m0
−1m
q m
r m − 1 m
e mt∞
n0
G r n,q t n n! . 2.4
Trang 7From the above, we can define the modified higher order q-Genocchi polynomials ε r n,q x as
follows
Fr q t, x ∞
n0
ε r n,q x t n
Then we have
ε r n,q x q −nrx G r n,q x. 2.6
By using Cauchy product in2.3, we arrive at following theorem
Theorem 2.1 For r ∈ N and n ≥ 0, one has
ε n,q r x q −nrxn
j0
n j
q jx x n−j G r j,q 2.7
By using2.7, we easily obtain the following result
Corollary 2.2 For r ∈ N, and n ≥ 0, one has
ε r n,q x q −nrx∞
m0
n
j0
n−j
l0
n
j, l, n − j − l
n − j m − 1 m
−1l
q mxjl G r j,q 2.8
We now give some identity related to the Genocchi polynomials and numbers of higher order
Substituting x 0 into 1.24, we find that
G r n 2r t r
∞
n1,n2, ,n r0
∞
j1,j2, ,j r0
j1j2···j r n
n
j1, j2, , jr
−1n1n2···n r
r
k0
n j k
k 2.9
By1.24 and 2.8, we arrive at the following theorem
Theorem 2.3 For r ∈ N and n ≥ 0, one has
G r n n
j0
n j
−x n−j
G r j x. 2.10
By using1.24, we easily arrive at the following result
Corollary 2.4 For r, v ∈ N and n ≥ 0, one has
G r x G v
yn
n
j0
n j
x n−j G rv j
y
where G r x n is replace by G r n x.
Trang 83 Interpolation Function of Higher Order q-Genocchi Polynomials
Recently, higher order Bernoulli polynomials, Euler polynomials, and Genocchi polynomials have been studied by many mathematicians Especially, in this paper, we study higher order Genocchi polynomials which constructed by Kim 15 and see also the references cited in each of the these earlier works
In14, by using the fermionic p-adic invariant integral on Z p , the set of p-adic integers, Kim gave a new construction of q-Genocchi numbers, Euler numbers of higher order By using q-Genocchi, Euler numbers of higher order, he investigated the interesting relationship between w-q-Euler polynomials and w-q-Genocchi polynomials He also defined the multiple
w-q-zeta functions which interpolate q-Genocchi, Euler numbers of higher order.
By using similar method to that in the papers given by Kim14, in this section, we give
interpolation function of the generating functions of higher order q-Genocchi polynomials.
From1.20, we easily see that
∞
k0
G r k,q x t k
k! ∞
k0
2r r!
k r r
∞
m0
−1m
q m
m r − 1 m
m x k t kr
k r! . 3.1
From the above we have
G r kr,q x 2 r r!
k r r
∞
m0
−1m q m
m r − 1 m
m x k ,
G r 0,q x G r 1,q x · · · G r r−1,q x 0.
3.2
Hence we have obtain the following theorem
Theorem 3.1 Let r, k ∈ Z Then one has
G r kr,q x 2 r
r!
k r r
∞
m0
−1m
q m
m r − 1 m
m x k
Let us define interpolation function of the G r kr,q x as follows.
Definition 3.2 Let q, s ∈ C with |q| < 1 and 0 < x ≤ 1 Then we define
ζ r q s, x 2 r∞
n0
n r − 1 n
−1n q n
n x s 3.4
We call ζ r q s, x are the multiple Hurwitz type q-zeta function.
Trang 9Remark 3.3 It holds that
lim
q → 1 ζ r q s, x 2 r∞
n0
n r − 1 n
−1n
n x s 3.5 From1.24, we easily see that
ζ r s, x 2 r ∞
n1,n2, ,n r0
−1n1n2···n r
r
j1 nj xs , 3.6
where s ∈ C.
The functions in 3.5 and 3.6 interpolate the same numbers at negative integers
That is, these functions interpolate higher order q-Genocchi numbers at negative integers So,
by3.5, we modify 3.6 in sense of q-analogue.
In3.5 and 3.6, setting r 1, we have
ζ1s, x 2∞
n0
−1n
n x s ζ G s, x, 3.7
where ζ G s, x denotes Hurwitz type Genocchi zeta function, which interpolates classical
Genocchi polynomials at negative integers
Substituting s −k, k ∈ Zinto3.4 Then we have
ζ r q −k, x 2 r∞
n0
r n − 1 n
−1n
q n n x k
Setting3.3 into the above, we easily arrive at the following result
Theorem 3.4 Let r, k ∈ Z Then one has
ζ r q −n, x G
r
nr,q x
r!nr
r
In this section, by using generating function of the higher order q-Genocchi polynomials,
which is defined by1.20, we obtain the following identities
Trang 10By using1.20, we find that
G r kr,q x
r!kr
r
2r∞
m0
−1m
q m
m r − 1 m
m q m xk
2r∞
m0
−1m q m
m r − 1 m
k
j0
k j
m j q mk−j x k−j
2r∞
m0
−1m q m
m r − 1 m
k
j0
k j
1− q mj
1− qj q mk−j x k−j
2r∞
m0
−1m
q m
m r − 1 m
k
j0
k j
j
a0
j a
−1a q mak−j x k−j
1− qj
2rk
j0
j
a0
k j
j a
−1a x k−j
1− qj
∞
m0
−1m
q m
m r − 1 m
q mak−j
2rk
j0
j
a0
k j
j a
−1a x k−j
1− qj
1 q ak−j1r
2rk
j0
j
a0
−1a
k
a, j − a, k − j
x k−j
1− qj
1 q ak−j1r
4.1
Thus we have the following theorem
Theorem 4.1 Let q ∈ C with |q| < 1 Let r be a positive integer Then one has
G r kr,q x
r!kr
r
2rk
j0
j
a0
−1a
k
a, j − a, k − j
x k−j
1− qj
1 q ak−j1r 4.2
By using1.20, we have
F r q t, x 2 r
t r
∞
m0
−1m
q m
m r − 1 m
e mxt
2r
t r
∞
m0
∞
n0
−1m
q m
m r − 1 m
1− q mx
1− q
n
t n n!
2r t r
∞
m0
∞
n0
−1m q m
m r − 1 m
1
1− qn
n
j0
n j
−q mxj t n
n!
2r t r
∞
n0
n
j0
n j
−1j q jx
1− qn
∞
m0
m r − 1 m
−1m q j1m t
n n! .
4.3
Trang 11Thus we have
∞
n0
G r n,q x t n
n! ∞
n0
2r
t r n
j0
n j
−1j
q jx
1 q j1−r
1− q−n t n
n! . 4.4
By comparing the coefficients tn /n! of both sides in the above, we arrive at the following
theorem
Theorem 4.2 Let q ∈ C with |q| < 1 Let r be a positive integer Then one has
G r nr,q x
r!nr
r
2rn
j0
n j
−1j
q jx
1 q j1−r
1− q−n 4.5
By using1.20, we have
∞
n0
G r n,q x t n
n!
∞
n0
G y n,q x t n
n!
2ry
t ry
∞
n0
−1n
q n
n r − 1 n
e nxt
∞
n0
−1n
q n
n y − 1 n
e nxt
4.6
By using Cauchy product into the above, we obtain
∞
n0
n
j0
n
j
G r j,q xG r n−j,qy t n
n!
2ry
t ry
∞
n0
n
j0
n j
−1j
q j
j r − 1 j
−1n−j
q n−j
n − j y − 1
n − j
e jxt e n−jxt
4.7 From the above, we have
∞
m0
⎛
⎝m
j0
m
j
G r j,q xG r m−j,qy⎞⎠ t m
m!
∞
m0
⎛
⎝2ry
t ry
∞
n0
n
j0
−1n
q n
j r − 1 j
n − j y − 1
n − j
j x
n − j xm⎞⎠ t m
m! .
4.8
By comparing the coefficients of both sides of tm /m! in the above, we have the following
theorem
... generating functions of q-Genocchi, q-Euler polynomials andtheir interpolation functions In this paper, we give another definition of the multiple Hurwitz
type q-zeta function This... using the fermionic p-adic invariant integral on Z p , the set of p-adic integers, Kim gave a new construction of q-Genocchi numbers, Euler numbers of higher order By using q-Genocchi, ... section, we study modified generating functions of the higher order q-Genocchi< /i>
numbers and polynomials We obtain some relations related to these numbers and
polynomials Therefore