Download SYMMETRIC PROPERTIES FOR GENERALIZED TWISTED q

J. Appl. Math. & Informatics Vol. 34(2016), No. 1 - 2, pp. 107 - 114
http://dx.doi.org/10.14317/jami.2016.107
SYMMETRIC PROPERTIES FOR GENERALIZED TWISTED
q-EULER ZETA FUNCTIONS AND q-EULER POLYNOMIALS
N.S. JUNG AND C.S. RYOO∗
Abstract. 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.
Key words and phrases : generalized twisted q-Euler numbers and polynomials, generalized twisted q-Euler zeta function, symmetric property, power
sum.
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 [111]). 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 χ = 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:
1 − qx
[x]q =
(see [1, 2, 3, 4]).
1−q
∗
Received September 20, 2015. Revised November 4, 2015.
Corresponding author.
c 2016 Korean SIGCAM and KSCAM.
⃝
107
Accepted November 11, 2015.
108
N.S. Jung and C.S. Ryoo
Note that limq→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 χ be Dirichlet’s character
with conductor d ∈ N with d ≡ 1(mod2). Then the generalized twisted qEuler polynomials associated with associated with χ, En,χ,q,ε , are defined by
the following generating function
Fχ,q,ε (t, x) = [2]q
∞
∑
(−1)n q n εn χ(n)e[x+n]q t
=
n=0
∞
∑
En,χ,q,ε (x)
n=0
tn
n!
(1.1)
and their values at x = 0 are called the generalized twisted q-Euler numbers and
denoted En,χ,q,ε .
By (1.1) and Cauchy product, we obtain
n ( )
∑
n lx n−l
En,χ,q,ε (x) =
q [x]q El,χ,q,ε ,
(1.2)
l
l=0
with the usual convention about replacing (Eχ,q,ε )n by En,χ,q,ε .
By using (1.1), we note that
∞
∑
dk
F
(t,
x)
=
[2]
χ(n)(−1)n εn q n [n + x]kq , (k ∈ N).
χ,q,ε
q
dtk
t=0
n=0
(1.3)
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
ζχ,q,ε (s, x) = [2]q
∞
∑
(−1)n χ(n)εn q n
.
[n + x]sq
n=1
(1.4)
Note that ζχ,q,ε (s, x) is a meromorphic function on C. Relation between
ζχ,q,ε (s, x) and Ek,χ,q,ε (x) is given by the following theorem.
Theorem 1.2. For k ∈ N, we get
ζχ,q,ε (−k, x) = Ek,χ,q,ε (x).
(1.5)
Observe that ζχ,q,ε (−k, x) function interpolates Ek,χ,q,ε (x) polynomials at
non-negative integers. If χ = 1, then ζχ,q,ε (s, x) = ζq,ε (s, x) (see [1]).
2. Symmetric property of generalized twisted q-Euler 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 w1 , w2 ∈ N with
w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2).
Symmetric identities for generalized twisted q-Euler zeta functions
109
Theorem 2.1. Let χ be Dirichlet’s character with conductor d ∈ N with d ≡
1(mod 2) and ε be the r-th root of unity. For w1 , w2 ∈ N with w1 ≡ 1 (mod 2),
w2 ≡ 1 (mod 2), we obtain
)
(
w1
i
s, w1 x +
w2
i=0
)
(
w∑
1 d−1
w1
s
j
w
j
w
j
2
2
=
[2]qw2 [w2 ]q (−1) χ(j)ε q ζχ,qw1 ,εw1 s, w2 x +
j .
w2
j=0
w∑
2 d−1
[2]qw1 [w1 ]sq (−1)i χ(i)εw1 i q w1 i ζχ,qw2 ,εw2
Proof. Observe that [xy]q = [x]qy [y]q for any x, y ∈ C. In Definition 1.1, we
w2
1
derive next result by substitute w1 x + w
w2 i for x in and replace q and ε by q
w2
and ε , respectively.
ζχ,qw2 ,εw2 (s, w1 x +
∞
∑
w1
(−1)n χ(n)εw2 n q w2 n
i) = [2]qw2
w1 s
i]qw2
w2
[n + w1 x + w
2
n=0
= [2]
q w2
[w2 ]sq
∞
∑
(−1)n χ(n)εw2 n q w2 n
.
[w1 w2 x + w1 i + w2 n]sq
n=0
(2.1)
Since for any non-negative integer n and odd positive integer w1 , there exist
unique non-negative integer r, j such that m = w1 r + j with 0 ≤ j ≤ w1 − 1. So,
the equation (2.1) can be written as
w1
i)
w2
∞
∑
(−1)w1 dr+j χ(w1 dr + j)εw2 (w1 dr+j) q w2 (w1 dr+j)
ζχ,qw2 ,εw2 (s, w1 x +
= [2]qw2 [w2 ]sq
[w1 w2 dr + w1 w2 x + w1 i + w2 j]sq
w1 dr+j=0
0≤j≤w1 d−1
= [2]qw2 [w2 ]sq
w∑
∞
1 d−1 ∑
j=0 r=0
(2.2)
(−1)j χ(j)εw2 (w1 dr+j) q w2 (w1 dr+j)
.
[w1 w2 (dr + x) + w1 i + w2 j]sq
In similarly, we obtain
ζχ,qw1 ,εw1 (s, w2 x +
∞
∑
(−1)n χ(n)εw1 n q w1 n
w2
j) = [2]qw1
w2 s
w1
[n + w2 x + w
j]qw1
1
n=0
=
[2]qw1 [w1 ]sq
∞
∑
(−1)n χ(n)εw1 n q w1 n
.
[w1 w2 x + w1 n + w2 j]sq
n=0
(2.3)
110
N.S. Jung and C.S. Ryoo
Using the method in (2.2), we obtain
w2
ζχ,qw1 ,ζ w1 (s, w2 x +
j)
w1
∞
∑
(−1)w2 dr+i χ(w2 dr + i)εw1 (w2 dr+i) q w1 (w2 dr+i)
= [2]qw1 [w1 ]sq
[w1 w2 dr + w1 w2 x + w1 i + w2 j]sq
w2 dr+i=0
0≤i≤w2 d−1
= [2]qw1 [w1 ]sq
w∑
∞
2 d−1 ∑
i=0
r=0
(2.4)
(−1)i χ(i)εw1 (w2 dr+i) q w1 (w2 dr+i)
.
[w1 w2 (dr + x) + w1 i + w2 j]sq
By (2.2) and (2.4), we obtain
)
w1
s, w1 x +
i
w2
i=0
(
)
w∑
1 d−1
w2
j
w2 j w2 j
s
w
w
w
=
[2]q 2 [w2 ]q (−1) χ(j)ε q ζχ,q 1 ,ε 1 s, w2 x +
j .
w1
j=0
w∑
2 d−1
(
[2]qw1 [w1 ]sq (−1)i χ(i)εw1 i q w1 i ζχ,qw2 ,εw2
(2.5)
Next, we obtain the symmetric results by using definition and theorem of the
generalized twisted q-Euler polynomials.
Theorem 2.2. Let χ be Dirichlet’s character with conductor d ∈ N with d ≡
1(mod 2) and ε be the r-th root of unity. For w1 , w2 ∈ N with w1 ≡ 1 (mod 2),
w2 ≡ 1 (mod 2), i, j and n be non-negative integer, we obtain
(
)
w2 d−1
w1
[2]qw1 ∑
i
w1 i w1 i
w
w
(−1)
χ(i)ε
q
E
w
x
+
i
2
2
n,χ,q ,ε
1
[w1 ]nq i=0
w2
(
)
w1 d−1
w2
[2]qw2 ∑
j
w2 j w2 j
w1 ,εw1
(−1)
χ(j)ε
q
E
w
x
+
=
j
.
n,χ,q
2
[w2 ]nq j=0
w1
Proof. By substitute w1 x + ww12i for x in Theorem 1.2 and replace q and ε by q w2
and εw2 , respectively, we derive
(
)
w1
w
w
En,χ,q 2 ,ε 2 w1 x +
i
w2
[
]n
∞
∑
w1
i+m
= [2]qw2
(−1)m χ(m)εw2 m q w2 m w1 x +
(2.6)
w2
q w2
m=0
=
∞
[2]qw2 ∑
(−1)m χ(m)εw2 m q w2 m [w1 w2 x + w1 i + w2 m]nq .
[w2 ]nq m=0
Since for any non-negative integer m and odd positive integer w1 , there exist
unique non-negative integer r, j such that m = w1 r + j with 0 ≤ j ≤ w1 − 1.
Symmetric identities for generalized twisted q-Euler zeta functions
111
Hence, the equation (2.6) is written as
E
n,χ,q w2 ,εw2
=
[2]qw2
[w2 ]nq
(
)
w1
w1 x +
i
w2
∞
∑
(−1)w1 dr+j χ(w1 dr + j)εw2 (w1 dr+j) q w2 (w1 dr+j)
w1 dr+j=0
0≤j≤w1 d−1
× [w1 w2 x + w1 i + w2 (w1 dr + j)]nq
=
(2.7)
w1 d−1 ∑
∞
[2]qw2 ∑
(−1)w1 dr+j χ(j)εw2 (w1 dr+j) q w2 (w1 dr+j)
[w2 ]nq i=0 r=0
× [w1 w2 (x + dr) + w1 i + w2 j]nq .
In similar, we obtain
E
(
)
w2
w2 x +
j
w1
]n
[
∞
∑
w2
j+m
(−1)m χ(m)εw1 m q w1 m w2 x +
w1
q w1
m=0
n,χ,q w1 ,ζ w1
= [2]qw1
=
(2.8)
∞
[2]qw1 ∑
(−1)m χ(m)εw1 m q w1 m [w1 w2 x + w2 j + w1 m]nq ,
[w1 ]nq m=0
and
E
n,χ,q w1 ,εw1
=
[2]qw1
[w1 ]nq
(
)
w2
w2 x +
j
w1
∞
∑
(−1)w2 dr+i χ(w2 dr + i)εw1 (w2 dr+i) q w1 (w2 dr+i)
w2 dr+i=0
0≤i≤w2 d−1
× [w1 w2 x + w2 j + w1 (w2 dr + i)]nq
=
w2 d−1 ∑
∞
[2]qw1 ∑
(−1)w2 dr+i χ(i)εw1 (w2 dr+i) q w1 (w2 dr+i)
[w1 ]nq i=0 r=0
× [w1 w2 (x + dr) + w1 i + w2 j]nq .
(2.9)
112
N.S. Jung and C.S. Ryoo
It follows from the above equation that
(
)
w2 d−1
[2]qw1 ∑
w1
i
w1 i w1 i
w2 ,εw2
(−1)
χ(i)ε
q
E
w
x
+
i
n,χ,q
1
[w1 ]nq i=0
w2
=
w1 d−1 w∑
∞
2 d−1 ∑
[2]qw1 [2]qw2 ∑
(−1)i+j χ(i)χ(j)εw1 w2 dr+w1 i+w2 j
[w1 ]nq [w2 ]nq j=0 i=0 r=0
(2.10)
× q w1 w2 dr+w1 i+w2 j [w1 w2 (x + dr) + w1 i + w2 j)]nq
(
)
w1 d−1
[2]qw2 ∑
w2
j
w2 j w2 j
w
w
=
(−1)
χ(j)ε
q
E
w
x
+
j
.
1
1
n,χ,q ,ε
2
[w2 ]nq j=0
w1
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 non-negative integers. For w1 , w2 ∈ N with
w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), we have
[2]qw1
n ( )
∑
n
k=0
k
[w1 ]kq [w2 ]n−k
En−k,χ,qw2 ,εw2 (w1 x)
q
×
w∑
2 d−1
(−1)i χ(i)εw1 i q (1+n−k)w1 i [i]kqw1
i=0
= [2]qw2
n ( )
∑
n
[w1 ]n−k
[w2 ]kq En−k,χ,qw1 ,εw1 (w2 x)
q
k
k=0
×
w∑
1 d−1
(−1)j χ(j)εw2 j q (1+n−k)w2 j [j]kqw2 .
j=0
Proof. After some calculations, we have
)
(
w2 d−1
[2]qw1 ∑
w1
i
w1 i w1 i
w
w
i
(−1)
χ(i)ε
q
E
w
x
+
n,χ,q 2 ,ε 2
1
[w1 ]nq i=0
w2
]
(
)
[
n
k
[2]qw1 ∑ n
w1
=
En−k,χ,qw2 ,εw2 (w1 x)
[w1 ]nq
w2 qw2
k
k=0
×
w∑
2 d−1
(−1)i χ(i)εw1 i q (1+n−k)w1 i [i]kqw1 ,
i=0
(2.11)
Symmetric identities for generalized twisted q-Euler zeta functions
113
and
(
)
w1 d−1
[2]qw2 ∑
w2
j
w2 j w2 j
(−1) χ(j)ε q En,χ,qw2 ,εw2 w2 x +
j
[w2 ]nq j=0
w1
]k
n ( )[
[2]qw2 ∑ n
w1
=
En−k,χ,qw1 ,εw1 (w2 x)
[w2 ]nq
k
w2 qw1
(2.12)
k=0
×
w∑
1 d−1
(−1)j χ(j)εw2 j q (1+n−k)w2 j [j]kqw2 .
j=0
By (2.11), (2.12) and Theorem 2.2, we obtain that
n ( )
∑
n
1
[2]qw1
En−k,χ,qw2 ,εw2 (w1 x)
n−k
k [w1 ]q [w2 ]kq
k=0
×
= [2]qw2
(−1)i χ(i)εw1 i q (1+n−k)w1 i [i]kqw1
i=0
n ( )
∑
n
k=0
w∑
2 d−1
1
En−k,χ,qw1 ,εw1 (w2 x)
k [w1 ]kq [w2 ]qn−k
×
w∑
1 d−1
(−1)j χ(j)εw2 j q (1+n−k)w2 j [j]kqw2 .
j=0
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 w1 , w2 ∈ N with w1 ≡ 1 (mod 2), w2 ≡ 1 (mod 2), we have
n ( )
∑
n
[2]qw1
[w1 ]kq [w2 ]n−k
En−k,χ,qw2 ,εw2
q
k
k=0
×
w∑
2 d−1
(−1)i χ(i)εw1 i q (1+n−k)w1 i [i]kqw1
i=0
= [2]qw2
n ( )
∑
n
[w1 ]qn−k [w2 ]kq En−k,χ,qw1 ,εw1
k
k=0
×
w∑
1 d−1
j=0
(−1)j χ(j)εw2 j q (1+n−k)w2 j [j]kqw2 .
114
N.S. Jung and C.S. Ryoo
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N.S. Jung received Ph.D. degree from Hannam University. Her research interests are analytic number theory and p-adic functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306-791, Korea.
e-mail: [email protected]
C.S. Ryoo received Ph.D. degree from Kyushu University. His research interests focus on
the numerical verification method, scientific computing and p-adic functional analysis.
Department of Mathematics, Hannam University, Daejeon, 306-791, Korea.
e-mail: [email protected]