In this paper we investigate some symmetric property of the twisted
q
Euler zeta functions and twisted
q
Euler polynomials.
AMS Mathematics Subject Classification : 11B68, 11S40, 11S80.
1. Introduction
The Euler polynomials and numbers possess many interesting properties in many areas of mathematics and physics. Many mathematicians have studied in the area of the
q
extension of the Euler numbers and polynomials (see
[3

10]
).
Recently, Y. Hu studied several identities of symmetry for Carlitz's
q
Bernoulli numbers and polynomials in complex field (see
[2]
). D. Kim
et al.
[3]
derived some identities of symmetry for Carlitz's
q
Euler numbers and polynomials in complex field. J.Y. Kang and C.S. Ryoo investigated some identities of symmetry for
q
Genocchi polynomials (see
[1]
). In
[8]
, we obtained some identities of symmetry for Carlitz's twisted
q
Euler polynomials associated with
p
adic
q
integral on ℤ
_{p}
. In this paper, we establish some interesting symmetric identities for twisted
q
Euler zeta functions and twisted
q
Euler polynomials in complex field. If we take
ε
= 1 in all equations of this article, then
[3]
are the special case of our results. Throughout this paper we use the following notations. By ℕ we denote the set of natural numbers, ℤ denotes the ring of rational integers, ℚ denotes the field of rational numbers, ℂ denotes the set of complex numbers, and Z
_{+}
= ℕ ∪ {0}: We use the following notation:
Note that lim
_{q→1}
[
x
] =
x
. We assume that
q
∈ ℂ with 
q
 < 1. Let
ε
be the
p^{N}
th root of unity. Then the twisted
q
Euler polynomials
E_{n,q,ε}
are defined by the generating function to be
When
x
= 0,
E_{n,q,ε}
=
E_{n,q,ε}
(0) are called the twisted
q
Euler numbers. By (1.1) and Cauchy product, we have
with the usual convention about replacing (
E_{q,ε}
)
^{n}
by
E_{n,q,ε}
.
By using (1.1), we note that
By (1.3), we are now ready to define the Hurwitz type of the twisted
q
Euler zeta functions.
Definition 1.1.
Let
s
∈ ℂ and
x
∈ ℝ with
x
≠ 0,−1,−2,.... We define
Note that
ζ_{q,ζ}
(
s
,
x
) is a meromorphic function on ℂ. A relation between
ζ_{q,ε}
(
s
,
x
) and
E_{k,q,ε}
(
x
) is given by the following theorem.
Theorem 1.2.
For k
∈ ℕ, we get
Observe that
ζ_{q,ε}
(−
k
,
x
) function interpolates
E_{k,q,ε}
(
x
) polynomials at nonnegative integers.
2. Symmetric property of twistedqEuler zeta functions
In this section, by using the similar method of
[1
,
2
,
3]
, expect for obvious modifications, we investigate some symmetric identities for twisted
q
Euler polynomials and twisted
q
Euler zeta functions. Let
w
_{1}
,
w
_{2}
∈ ℕ with
w
_{1}
≡ 1 (mod 2),
w
_{2}
≡ 1 (mod 2).
Theorem 2.1.
For w
_{1}
,
w
_{2}
∈ ℕ with
w
_{1}
≡ 1 (mod 2),
w
_{2}
≡ 1 (mod 2),
we have
Proof
. Observe that [
xy
]
_{q}
= [
x
]
_{qy}
[
y
]
_{q}
for any
x
,
y
∈ ℂ. In Definition 1.1, we derive next result by substitute
for
x
in and replace
q
and
ε
by
q
^{w2}
and
ε
^{w2}
, respectively.
Since for any nonnegative integer
n
and odd positive integer
w
_{1}
, there exist unique nonnegative integer
r
,
j
such that
m
=
w
_{1}
r
+
j
with 0 ≤
j
≤
w
_{1}
−1. So, the equation (2.1) can be written as
In similarly, we can see that
Using the method in (2.2), we obtain
From (2.2) and (2.4), we have
Next, we derive the symmetric results by using definition and theorem of the twisted
q
Euler polynomials.
Theorem 2.2.
Let i
,
j and n be nonnegative integers. For w
_{1}
,
w
_{2}
∈ ℕ
with w
_{1}
≡ 1 (mod 2),
w
_{2}
≡ 1 (mod 2)
, we have
Proof
. By substitute
for
x
in Theorem 1.2 and replace
q
and
ε
by
q
^{w2}
and
ε
^{w2}
, respectively, we derive
Since for any nonnegative integer
m
and odd positive integer
w
_{1}
, there exist unique nonnegative integer
r
,
j
such that
m
=
w
_{1}
r
+
j
with 0 ≤
j
≤
w
_{1}
− 1.
Hence, the equation (2.6) is written as
In similar, we have
and
It follows from the above equation that
From (2.8) and (2.9), the proof of the Theorem 2.2 is completed. □
By (1.2) and Theorem 2.2, we have the following theorem.
Theorem 2.3.
Let i
,
j and n be nonnegative integers. For w
_{1}
,
w
_{2}
∈ ℕ
with w
_{1}
≡ 1 (mod 2),
w
_{2}
≡ 1 (mod 2)
, we have
Proof
. After some calculations, we obtain
and
From (2.11), (2.12) and Theorem 2.2, we obtain that
Hence, we have above theorem.
By Theorem 2.3, we obtain the interesting symmetric identity for twisted
q
Euler numbers in complex field.
Corollary 2.4.
For w
_{1}
,
w
_{2}
∈ ℕ
with w
_{1}
≡ 1 (mod 2),
w
_{2}
≡ 1 (mod 2)
, we have
BIO
N.S. Jung received Ph.D. degree from Hannam University. Her research interests are analytic number theory and padic functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306791, Korea.
email: jns4235@nate.com
C.S. Ryoo received Ph.D. degree from Kyushu University. His research interests focus on the numerical verification method, scientific computing and padic functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306791, Korea.
email: ryoocs@hnu.kr
Kang J.Y.
,
Ryoo C.S.
(2014)
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8
9 
16
He Yuan
(2013)
Symmetric identities for Carlitz's qBernoulli numbers and polynomials
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Kim D.
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(2014)
Symmetric Identities of the qEuler Polynomials
Adv. Studies Theor. Phys.
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1149 
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On the Barnes type multiple qEuler polynomials twisted by ramified roots of unity
Proc. Jangjeon Math. Soc. 13
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Ryoo C.S.
(2011)
A note on the weighted qEuler numbers and polynomials
Advan. Stud. Contemp. Math.
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47 
54
Ryoo C.S.
(2015)
Some Identities of Symmetry for Carlitz's Twisted qEuler Polynomials Associated with pAdic qIntegral on Zp
Int. Journal of Math. Analysis
9
1747 
1753
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(2014)
Analytic Continuation of Euler Polynomials and the Euler Zeta Function
Discrete Dynamics in Nature and Society
Article ID 568129
2014
DOI : 10.1155/2014/568129
Ryoo C.S.
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A Note on the Reflection Symmetries of the Genocchi polynomials
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1404