constructed generating functions of higher-order twistedh, q-extension of Euler polynomials and numbers, by using p-adic, q-deformed fermionic integral onZp.. Cenkci, “The p-adic general
Trang 1Volume 2008, Article ID 371295, 9 pages
doi:10.1155/2008/371295
Research Article
Polynomials of Higher Order
Taekyun Kim, 1 Lee-Chae Jang, 2 and Cheon-Seoung Ryoo 3
1 The School of Electrical Engineering and Computer Science (EECS), Kyungpook National University, Taegu 702-701, South Korea
2 Department of Mathematics and Computer Science, KonKuk University,
Chungju 143-701, South Korea
3 Department of Mathematics, Hannam University, Daejeon 306-791, South Korea
Correspondence should be addressed to Cheon-Seoung Ryoo, ryoocs@hnu.kr
Received 1 November 2007; Accepted 22 December 2007
Recommended by Paolo Emilio Ricci
In 2007, Ozden et al constructed generating functions of higher-order twistedh, q-extension of Euler polynomials and numbers, by using p-adic, q-deformed fermionic integral onZp By apply-ing their generatapply-ing functions, they derived the complete sums of products of the twistedh, q-extension of Euler polynomials and numbers In this paper, we consider the new q-q-extension of
Eu-ler numbers and polynomials to be different which is treated by Ozden et al From our q-EuEu-ler
num-bers and polynomials, we derive some interesting identities and we construct q-Euler zeta functions which interpolate the new q-Euler numbers and polynomials at a negative integer Furthermore, we study Barnes-type q-Euler zeta functions Finally, we will derive the new formula for “sums of prod-ucts of q-Euler numbers and polynomials” by using fermionic p-adic, q-integral onZp.
Copyright q 2008 Taekyun Kim 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.
1 Introduction and notations
one talks of q-extension, q is considered in many ways such as an indeterminate, a complex
we use the following notation:
Trang 2x q x : q 1− q x
cf 1 5,22
integer and let p be a fixed prime number For any positive integer N, we set
N
dp NZ
,
X∗
0<a<d p
a,p1
,
,
1.2
μ q
a dp NZp
dp N q
1.3
q-integral onZpas follows:
Zp
N→∞
1
p N q
p N−1
x0
see1 23
Higher-order twisted Bernoulli and Euler numbers and polynomials are studied by
q-deformed fermionic integral onZp By applying their generating functions, they derived the
to be different which is treated by Ozden et al From our q-Euler numbers and polynomials,
we derive some interesting identities and we construct q-Euler zeta functions which interpo-late the new q-Euler numbers and polynomials at a negative integer Furthermore, we study Barnes-type q-Euler zeta functions Finally, we will derive the new formula for “sums of
2.q-extension of Euler numbers
polynomials as follows:
qe t 1e xt
∞
n0
E n,q x
n! t
Trang 3Note that
lim
e t 1e xt
∞
n0
E n x
n! t
∞
n0
E n,q x
n! t
n 2q
qe t 1e xt 2q
∞
n0
E k,q x 2 q ∞
n0
in-teger as follows
ζ q s, x 2 q ∞
n0
−1n q n
ζ q −k, x E k,q x.
Theorem 2.1 For k ∈ Z,
E k,q x 2 q ∞
n0
Let F q 0, t F q t Then
2q n−1
k0
−1k q k e kt 2q
1 qe t − 2q−1n q n e nt
1 qe t
F q t − −1 n q n F q n, t.
2.7
∞
k0
2q n−1
l0
−1l q l l k
t k k! ∞
k0
E k,q− −1n q n E k,q n t k
Trang 4Theorem 2.2 Let n ∈ N, k ∈ Z If n ≡ 0 mod 2, then
E k,q − q n E k,q n 2 q n−1
l0
E k,q q n E k,q n 2 q n−1
l0
For w1, w2, , w r ∈ C, consider the multiple q-Euler polynomials of Barnes-type as follows:
F q r
w1, w2, , w r | x, t 2
r
q e xt
qe w1t 1qe w2t 1· · ·qe w r t 1
n0
E n,q
w1, , w r | x t n
n! , where max1≤i≤r |w i t log q| < π.
2.11
q w1 , w2, , w r | x, t is analytic function in the
2r q
∞
n1, ,n r0
−qr1n i er1n i w i xt ∞
n0
E n,q
w1, , w r | x t n
2r q
∞
n1, ,n r0
−qr1n i
r
i1
n i w i x
E k,q
w1, , w r | x. 2.13
ζ r,q
w1, w2, , w r | s, x 2r
q
∞
n1, ,n r0
−1n1···n r q n1···n r
Theorem 2.3 For k ∈ Z, w1, w2, , w r ∈ C,
ζ r,q
w1, w2, , w r | −k, x E k,q
w1, w2, , w r | x. 2.15
generalized Euler numbers attached to χ as follows:
F χ,q t 2q
f−1
a0−1a q a χ ae at
q f e ft 1
∞
n0
E n,χ,q t
n
Trang 5where | log q t| < π/f The numbers E n,χ,q will be called the generalized q-Euler numbers
F χ,q t 2q
f−1
a0−1a q a χ ae at
q f e ft 1
2q
f−1
a0
−1a q a χ a ∞
n0
q nf−1n e anft
2q ∞
n0
f−1
a0
−1a nf q a nf χ a nfe anft
2q ∞
n0
−1n q n χ ne nt ∞
n0
E n,χ,q t n
n! .
2.17
Thus,
E k,χ,q d k
dt k F χ,q t
t0 2q ∞
n1
Therefore, we can define the Dirichlet-type l-function which interpolates at negative integer as
follows
l q s, χ 2 q ∞
n1
−1n q n χ n
Theorem 2.4 For k ∈ Z,
qe t 1e xt
∞
n0
E n,q t n
n!
∞
l0
x l
l! t
l
m0
m
n0
E n,q m n
x m −n
t m m! .
2.21
By2.21 it is shown that
E n,q x n
m0
E m,q n m
Trang 6
For f (=odd) ∈ N, note that
∞
n0
E n,q x t n
n! 2q
qe t 1e xt
q f e ft 1
f−1
a0
−1a q a e ax/fft
22q
q f
f−1
a0
−1a q a 2q f e ax/fft
q f e ft 1
2q f
f−1
a0
−1a q a
∞
n0
E n,q f
a x
f
f n t n n! .
2.23
Thus, we have the distribution relation for q-Euler polynomials as follows.
Theorem 2.5 For f (=odd) ∈ N,
E n,q x f
n2q
2q f
f−1
a0
−1a q a E n,q f
a x
f
For k, n ∈ N with n ≡ 0 (mod 2), it is easy to see that
2q n−1
l0
−1l−1q l l k q n E k,q n − E k,q
q n k
m0
k m
n k −m E m,q − E k,q
q n k−1
m0
k m
E m,q n k −mq n− 1E k,q
2.25
Therefore, we obtain the following
Theorem 2.6 For k, n ∈ N with n ≡ 0 (mod 2),
2q n−1
l0
−1l−1q l l k q n k−1
m0
k m
3 Witt-type formulae onZp inp-adic number field
I q g
Zp
N→∞
1
p N q
p N−1
x0
Trang 7The fermionic p-adic, q-integral is also defined as
I −qg
Zp
N→∞
2q
1 q p N
p N−1
x0
see 4
I q
e tx
Zp
∞
n0
Zp
x n dμ −q x t n
n! 2q
qe t 1
∞
n0
E n,q t n
By comparing the coefficient on both sides, we see that
Zp
By the same method, we see that
Zp
e xyt dμ −q y 2q
qe t 1e xt
∞
n0
E n,q x t n
Hence, we have the formula of Witt’s type for q-Euler polynomial as follows:
Zp
q n I −q
g n
−1n−1I −qg 2 q n−1
l0
If n is odd positive integer, then we have
q n I −q
g n
I−qg 2 q n−1
l0
I −q
χ xe xt
X
χ xe tx dμ −q x
a0−1a q a χ ae at
q f e ft 1
n0
E n,χ,q t n
n! .
3.12
Thus, we have the Witt formula for generalized q-Euler numbers attached to χ as follows:
X
Trang 84 Higher-orderq-Euler numbers and polynomials
q-Euler numbers and polynomials and sums of products of q-Euler numbers First, we try to
Zp
· · ·
Zp
r times
e a1x1a2x2···a r x r t e xt dμ −q
x1
· · · dμ −qx r
r q
qe a1t 1qe a2t 1· · ·qe a r t 1e xt ,
4.1
where a1 , a2, , a r ∈ Zp
2r q
qe a1t 1qe a2t 1· · ·qe a r t 1e xt
∞
n0
E n,q
a1, a2, , a r | x t n
In the special casea1 , a2, , a r 1, 1, , 1, we write
E n,q
a1, , a r
r times
E n,q
a1, a2, , a r | x
Zp
· · ·
Zp
r times
a1x1 · · · a r x r xnr
j1
dμ −q
x j
It is easy to check that
E n,q
a1, a2, , a r | x n
l0
n l
x n −l E l,q
a1, a2, , a r
where E n,q a1 , a2, , a r E n,q a1 , a2, , a r | 0 Multinomial theorem is well known as fol-lows:
r
j1
x j
l1, ,l r≥0
l1···l r n
n
l1, , l r
a1
x l a
where
n
l1, , l r
E r n,q x n
m0
l1, ,l r≥0
l ···l m
n m
m
l1, , l r
x n −m
r
j1
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Trang 91 M Cenkci, “The p-adic generalized twisted h,q -Euler- l-function and its applications,” Advanced... Ryoo, T Kim, and L.-C Jang, “Some relationships between the analogs of Euler numbers and< /small>
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