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

11]
). Recently, Y. Hu investigated several identities of symmetry for Carlitz's
q
Bernoulli numbers and polynomials in complex field (see
[3]
). D. Kim
et al.
[4]
derived some identities of symmetry for Carlitz's
q
Euler numbers and polynomials in complex field. J. Y. Kang and ℂ. S. Ryoo studied some identities of symmetry for
q
Genocchi polynomials (see
[2]
). In
[1]
, we obtained some identities of symmetry for Carlitz's twisted
q
Euler zeta function in complex field. In this paper, we establish some interesting symmetric identities for generalized twisted
q
Euler zeta functions and generalized ] twisted
q
Euler polynomials in complex field. If we take
X
= 1 in all equations of this article, then
[1]
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
r
be a positive integer, and let
ε
be the
r
th root of unity. Let
X
be Dirichlet's character with conductor
d
∈ ℕ with
d
≡ 1(mod2). Then the generalized twisted
q
Euler polynomials associated with associated with
X
,
E_{n,X,q,ε}
, are defined by the following generating function
and their values at
x
= 0 are called the generalized twisted
q
Euler numbers and denoted
E_{n,X,q,ε}
.
By (1.1) and Cauchy product, we obtain
with the usual convention about replacing (
E_{X,q,ε}
)
^{n}
by
E_{n,X,q,ε}
.
By using (1.1), we note that
By (1.3), we are now ready to define the Hurwitz type of the generalized twisted
q
Euler zeta functions.
Definition 1.1.
Let
s
∈ ℂ and
x
∈ ℝ with
x
≠ 0,−1,−2,.... We define
Note that
ζ_{X,q,ε}
(
s, x
) is a meromorphic function on ℂ. Relation between
ζ_{X,q,ε}
(
s, x
) and
E_{k,X,q,ε}
(
x
) is given by the following theorem.
Theorem 1.2.
For k
∈ ℕ,
we get
Observe that
ζ_{X,q,ε}
(
−k, x
) function interpolates
E_{k,X,q,ε}
(
x
) polynomials at nonnegative integers. If
X
= 1, then
ζ_{X,q,ε}
(
s, x
) =
ζ_{q,ε}
(
s, x
) (see
[1]
).
2. Symmetric property of generalized twistedqEuler zeta functions
In this section, by using the similar method of
[1
,
2
,
3
,
4
,
9]
, expect for obvious modifications, we give some symmetric identities for generalized twisted
q
Euler polynomials and generalized 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.
Let X be Dirichlet's character with conductor d
∈ ℕ
with d
≡ 1
(mod 2) and ε be the rth root of unity. For w
_{1}
,w
_{2}
∈ ℕ
with w
_{1}
≡ 1 (mod 2)
, w
_{2}
≡ 1 (mod 2)
, we obtain
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 obtain
Using the method in (2.2), we obtain
By (2.2) and (2.4), we obtain
Next, we obtain the symmetric results by using definition and theorem of the generalized twisted
q
Euler polynomials.
Theorem 2.2.
Let X be Dirichlet's character with conductor d
∈ ℕ
with d
≡ 1
(mod 2) and ε be the rth root of unity. For w
_{1}
,
w
_{2}
∈ ℕ
with w
_{1}
≡ 1 (mod 2)
, w
_{2}
≡ 1 (mod 2)
, i, j and n be nonnegative integer, we obtain
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 obtain
and
It follows from the above equation that
From (2.7), (2.8), (2.9) and (2.10), 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 have
and
By (2.11), (2.12) and Theorem 2. 2, we obtain that
Hence, we have above theorem. □
By Theorem 2.3, we have the interesting symmetric identity for generalized 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
Jung N.S.
,
Ryoo C.S.
(2015)
Symmetric identities for twisted qEuler zeta functions
J. Appl. Math. & Informatics
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Kang J.Y.
,
Ryoo C.S.
(2014)
On Symmetric Property for qGenocchi Polynomials and Zeta Function
Int. Journal of Math. Analysis
8
9 
16
He Yuan
(2013)
Symmetric identities for Carlitz's qBernoulli numbers and polynomials
Advances in Difference Equations
246
Kim D.
,
Kim T.
,
Lee S.H.
,
Seo J.J.
(2014)
Symmetric Identities of the qEuler Polynomials
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Ryoo C.S.
(2010)
On the Barnes type multiple qEuler polynomials twisted by ramified roots of unity
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Ryoo C.S.
(2011)
A note on the weighted qEuler numbers and polynomials
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Ryoo C.S.
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Some Identities of Symmetry for Carlitz's Twisted qEuler Polynomials Associated with pAdic qIntegral on ℤp
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Ryoo C.S.
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Analytic Continuation of Euler Polynomials and the Euler Zeta Function
Discrete Dynamics in Nature and Society
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Ryoo C.S.
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A Note on the Reflection Symmetries of the Genocchi polynomials
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