Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 86052, 22 pages
doi:10.1155/2007/86052
Research Article
Some Relationships between the Analogs of
Euler Numbers and Polynomials
C. S. Ryoo, T. Kim, and Lee-Chae Jang
Received 5 June 2007; Revised 28 July 2007; Accepted 26 August 2007
Recommended by Narendra K. Govil
We construct new twisted Euler polynomials and numbers. We also study the generating
functions of the twisted Euler numbers and polynomials associated with their interpolation functions. Next we construct twisted Euler zeta function, twisted Hurwitz zeta function, twisted Dirichlet l-Euler numbers and twisted Euler polynomials at non-positive
integers, respectively. Furthermore, we find distribution relations of generalized twisted
Euler numbers and polynomials. By numerical experiments, we demonstrate a remarkably regular structure of the complex roots of the twisted q-Euler polynomials. Finally,
we give a table for the solutions of the twisted q-Euler polynomials.
Copyright © 2007 C. S. Ryoo 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
Throughout this paper, we use the following notations. By Z p we denote the ring of p-adic
rational integers, Q denotes the field of rational numbers, Q p denotes the field of p-adic
rational numbers, C denotes the complex numbers field, and C p denotes the completion
of algebraic closure of Q p . Let ν p be the normalized exponential valuation of C p with
| p| p = p−ν p (p) = p−1 . When one talks of q-extension, q is considered in many ways such
as an indeterminate, a complex number q ∈ C, or p-adic number q ∈ C p . If q ∈ C, one
normally assumes that |q| < 1. If q ∈ C p , we normally assume that |q − 1| p < p−1/(p−1) so
that qx = exp(x log q), for |x| p ≤ 1.
[x]q = [x : q] =
1 − qx
1−q
cf. [1–18] .
(1.1)
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Journal of Inequalities and Applications
Hence, limq→1 [x] = x for any x with |x| p ≤ 1 in the present p-adic case. Let d be a fixed
integer and let p be a fixed prime number. For any positive integer N, we set
Z
X = lim
,
←− d pN Z
N
X∗ =
(a + d pZ p ),
(1.2)
0<a<d p
(a,p)=1
a + d pN Z p = x ∈ X | x ≡ a modd pN
,
where a ∈ Z lies in 0 ≤ a < d pN . For any positive integer N,
μq a + d p N Z p = qa
d pN q
(1.3)
is known to be a distribution on X (cf. [1–18]).
For
g ∈ UD Z p = g | g : Z p → C p is a uniformly differentiable function ,
(1.4)
the p-adic q-integral was defined by [1, 2, 6–18]
Iq (g) =
Zp
g(x)dμq (x) = lim N →∞
1
g(x)qx .
N
p 0≤x<pN
(1.5)
Note that
I1 (g) = lim Iq (g) =
q→1
Zp
1
N →∞ pN
g(x)dμ1 (x) = lim
g(x)
(1.6)
0≤x<pN
(see [1, 2, 6–18]). For q ∈ [0,1], certain q-deformed bosonic operators may be introduced
which generalize the undeformed bosonic ones (corresponding q = 1); see [1, 2, 6–18].
For g ∈ UD(Z p )
Zp
g(x)dμ1 (x) =
X
g(x)dμ1 (x)
(1.7)
(see [6–18] for details).
We assume that q ∈ C with |1 − q| p < 1. Using definition, we note that I1 (g1 ) = I1 (g) +
g (x), where g1 (x) = g(x + 1).
Let
Tp =
m≥1
m
C pm = lim C pm ,
m→∞
(1.8)
where C pm = {w | w p = 1} is the cyclic group of order pm . For w ∈ T p , we denote by
φw : Z p → C p the locally constant function x −→ wx . If we take f (x) = φw (x)etx , then we
C. S. Ryoo et al. 3
easily see that
t
.
wet − 1
φw (x)etx dμ1 (x) =
Zp
(1.9)
Kim [8] treated the analog of Bernoulli numbers, which is called twisted Bernoulli numbers. We define the twisted Bernoulli polynomials Bn,w (x)
ext
t
=
wet − 1
∞
Bn,w (x)
n =0
tn
.
n!
(1.10)
Using Taylor series of etx in the above equation, we obtain
Zp
xn φw (x)dμ1 (x) = Bn,w ,
(1.11)
where Bn,w = Bn,w (0).
The Euler numbers En are usually defined by means of the following generating function:
∞
eEt =
tn
2
=
En
et + 1 n=0 n!
cf. [1–18] ,
(1.12)
where the symbol En is interpreted to mean that En must be replaced by En when we
expand the one on the left. These numbers are classical and important in mathematics
and in various places like analysis, number theory. Frobenius extended such numbers as
En to the so-called Frobenius-Euler numbers Hn (u) belonging to an algebraic number u
with |u| > 1. Let u be an algebraic number. For u ∈ C with |u| > 1, the Frobenius-Euler
numbers Hn (u) belonging to u are defined by the generating function
∞
eH(u)t =
tn
1−u =
H
(u)
n
e t − u n =0
n!
cf. [6–10] ,
(1.13)
with the usual convention of symbolically replacing H n by Hn . The Euler polynomials
En (x) are defined by
∞
eE(x)t =
tn
2 xt En (x)
e =
t
e +1
n!
n =0
cf. [6–16] .
(1.14)
For u ∈ C with |u| > 1, the Frobenius-Euler polynomials Hn (u,x) belonging to u are defined by
∞
eH(u,x)t =
tn
1 − u xt Hn (u,x)
e =
t
e −u
n!
n =0
cf. [6–18] .
(1.15)
Kim gave a relation between Bn,w and Hn (u), with nth Euler numbers as follows:
Bn,w =
n
Hn−1 (w−1 ),
w−1
if w = 1.
(1.16)
4
Journal of Inequalities and Applications
Now, we consider the case q ∈ (−1,0) corresponding to q-deformed fermionic certain
and annihilation operators and the literature given therein [6–18]. The expression for the
Iq (g) remains the same, so it is tempting to consider the limit q → −1. That is,
I−1 (g) = lim Iq (g) =
q→−1
Zp
g(x)dμ−1 (x) = lim
N →∞
g(x)(−1)x .
(1.17)
0≤x<pN
Let g1 (x) be translation with g1 (x) = g(x + 1). Then we see that
p N −1
I−1 g1 = − lim
N →∞
g(x)(−1)x + 2g(0) = −I−1 ( f ) + 2g(0).
(1.18)
x =0
Therefore, we obtain the following lemma.
Lemma 1.1. For g ∈ UD(Z p ), one has
I−1 g1 + I−1 (g) = 2g(0).
(1.19)
From (1.19), we can easily derive the following theorem.
Theorem 1.2. For g ∈ UD(Z p ), n ∈ N, one has
I−1 gn = (−1)n I−1 (g) + 2
n
−1
(−1)n−1−l g(l),
(1.20)
l=0
where gn (x) = g(x + n).
Corollary 1.3. For g ∈ UD(Z p ), n(= odd) ∈ N, one has
I−1 gn + I−1 (g) = 2
n
−1
(−1)l g(l).
(1.21)
l=0
By Lemma 1.1, we can consider twisted Euler numbers. If we take g(z) = φw (z)etz ,
(w ∈ T p ), then we have
∞
I−1 φw (z)etz =
2
tn
=
E
.
n,w
wet + 1 n=0
n!
(1.22)
Now we define twisted Euler numbers En,w as follows:
∞
Fw (t) =
2
tn
=
E
.
n,w
wet + 1 n=0
n!
(1.23)
Using Taylor series of ezt above, we obtain
En,w =
Zp
wz zn dμ−1 (z).
(1.24)
C. S. Ryoo et al. 5
For w ∈ T p , we introduce the twisted Euler polynomials En,w (z). Twisted Euler polynomials En,w (z) are defined by means of the generating function
∞
Fw (t,z) =
tn
2
En,w (z) ,
ezt = I−1 φw (x)et(z+x) =
t
we + 1
n!
n =0
(1.25)
where En,w (0) = En,w . Using Taylor series of etx in the above equation, we have
En,w (x) =
n
Zp
φw (x)(x + z) dμ−1 (x) =
n n
k =0
k
z n −k
Zp
φw (x)xk dμ−1 (x).
(1.26)
Thus we easily see that
En,w (z) =
n n
k =0
k
zn−k Ek,w .
(1.27)
Let χ be the Dirichlet character with conductor f (= odd) ∈ N. Ryoo et al. [16] studied
the generalized Euler numbers and polynomials. The generalized Euler numbers associated with χ, En,χ , were defined by means of the generating function
Fχ (t) =
2
f −1
a=0
∞
χ(a)(−1)a eat tn
=
E
.
n,χ
eft +1
n!
n =0
(1.28)
Generalized Euler polynomials, En,χ (x), were also defined by means of the generating
function
Fχ (t,z) =
2
f −1
a=0
∞
χ(a)(−1)a eat zt tn
En,χ (z) .
e =
f
t
e +1
n!
n =0
(1.29)
Substituting g(x) = χ(x)φw (x)etx into (1.21), then the generalized twisted Euler numbers
En,χ,w are defined by means of the generating functions
Fχ,w (t) =
X
2
=
φw (x)etx χ(x)dμ−1 (x)
f −1
a=0
∞
eta (−1)a χ(a)φw (a) tn
=
En,χ,w .
f
t
φw ( f )e + 1
n!
n =0
(1.30)
Using the above equation, En,χ,w are defined by
En,χ,w =
X
φw (x)xn χ(x)dμ−1 (x).
(1.31)
Generalized twisted Euler polynomials, En,χ,w (z), are defined by
Fχ,w (t,z) = Fχ,w (t)ezt =
f −1
=
2
a=0
X
φw (x)etx χ(x)dμ−1 (x)etz
∞
eta (−1)a χ(a)φw (a) zt tn
e =
En,χ,w (z) .
f
t
φw ( f )e + 1
n!
n =0
(1.32)
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Journal of Inequalities and Applications
We set
Fχ,w (t,z) =
2
f −1
a=0 (−1)
φw
a χ(a)φ (a)e(a+z)t
w
.
( f )e f t + 1
(1.33)
Using the above equation, we have
∞
En,χ,w (z)
n =0
tn
=
n!
X
φw (x)etx χ(x)dμ−1 (x)etz
=
=
X
φw (x)et(x+z) χ(x)dμ−1 (x)
∞ n =0
=
∞
X
n =0
φw (x)(x + z)n χ(x)dμ−1 (x)
n n
k =0
k
z n −k
X
(1.34)
tn
n!
φw (x)xk χ(x)dμ−1 (x)
tn
.
n!
Using the comparing coefficients t n /n!, we easily see that
En,χ,w (z) =
n n
k
k =0
zn−k Ek,χ,w .
(1.35)
We have the following remark.
Remark 1.4. Note that
(1) if w → 1, then Fχ,w (t,z) → Fχ (t,z) and En,χ,w (z) → En,χ (z);
(2)
f −1
∞
2
tn
En,χ,w (z) =
n!
n =0
a=0
eta (−1)a χ(a)φw (a) zt
e
φw ( f )e f t + 1
∞
∞
tn n tn
=
En,χ,w (z)
z .
n! n=0 n!
n =0
(1.36)
Using the Cauchy product in the right-hand side of the above equation in (2), we obtain
∞
n =0
∞
En,χ,w (z)
tn z n −k t n
=
Ek,χ,w
.
n! n=0 k=0
k!(n − k)!
n
(1.37)
Comparing the coefficients t n on both sides of the above equation, we arrive at (1.35).
2. Twisted zeta function
In this section, we introduce the twisted Euler zeta function and twisted Hurwitz-Euler
zeta function. We derive a new twisted Hurwitz-type l-function which interpolates the
C. S. Ryoo et al. 7
generalized Euler polynomials En,χ,w (x). We give the relation between twisted Euler numbers and twisted l-functions at nonpositive integers. Let χ be the Dirichlet character with
conductor f (= odd) ∈ N. We set
Fχ,w (t) =
2
f −1
a=0
eta (−1)a χ(a)φw (a)
,
φw ( f )e f t + 1
−
π
π
− logw < t < − log .
f
f
(2.1)
By (2.1), we see that
Fχ,w (t) = 2
∞
χ(m)wm (−1)m etm .
(2.2)
m=1
From (1.30) and (2.2), we note that
∞
dk
Fχ,w (t)|t=0 = 2
χ(m)wm (−1)m mk
k
dt
m=1
(k ∈ N).
(2.3)
Therefore, we obtain the following theorem.
Theorem 2.1. For k ∈ N, one has
Ek,χ,w = 2
∞
χ(m)wm (−1)m mk
(k ∈ N).
(2.4)
m=1
Thus we define the twisted Dirichlet-type l-series as follows.
Definition 2.2. For s ∈ C, define the Dirichlet-type l-series related to twisted Euler numbers,
lw (s,χ) = 2
∞
χ(n)(−1)n wn
n =1
ns
.
(2.5)
Theorem 2.3. For k ∈ N, one has
lw (−k,χ) = Ek,χ,w .
(2.6)
Next, we introduce the Hurwitz-type twisted Euler zeta function. Since
∞
Fw (t,z) =
tn
2
En,w (z) ,
ezt =
t
we + 1
n!
n =0
(2.7)
we obtain
Fw (t,z) = 2
∞
(−1)n wn e(n+z)t .
(2.8)
n =0
From (2.8), we note that
∞
dk
Fw (t,z)|t=0 = 2 (−1)n wn (n + z)k
k
dt
n =0
(k ∈ N).
(2.9)
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Journal of Inequalities and Applications
Therefore, we have the following theorem.
Theorem 2.4. For k ∈ N, one has
Ek,w (z) = 2
∞
(−1)n wn (n + z)k
(k ∈ N).
(2.10)
n =0
Thus the twisted Hurwitz-Euler zeta function is defined as follows.
Definition 2.5. Let s ∈ C. Then
ζE,w (s,z) = 2
∞
(−1)n wn
n =0
(n + z)s
.
(2.11)
By Theorem 2.4 and Definition 2.5, we have the following theorem.
Theorem 2.6. For k ∈ N, one obtains
ζE,w (−k,z) = Ek,w (z).
(2.12)
Let us define two-variable twisted Euler numbers attached to χ as follows. By (1.33),
we see that
Fχ,w (t,z) = 2
∞
(−1)n χ(n)wn e(n+z)t .
(2.13)
n =0
From (2.13), we note that
∞
dk
Fχ,w (t,z)|t=0 = 2 (−1)n χ(n)wn (n + z)k
k
dt
n =0
(k ∈ N).
(2.14)
Therefore, we obtain the following theorem.
Theorem 2.7. For k ∈ N, one has
Ek,χ,w (z) = 2
∞
(−1)n χ(n)wn (n + z)k
(k ∈ N).
(2.15)
.
(2.16)
n =0
Hence we define two-variable twisted l-series as follows.
Definition 2.8. For s ∈ C. Then
lw (s,χ | z) = 2
∞
(−1)n χ(n)wn
n =0
(n + z)s
The relation between lw (−k,χ | z) and Ek,χ,w (z) is given by the following theorem.
Theorem 2.9. For k ∈ N, one obtains
lw (−k,χ | z) = Ek,χ,w (z).
(2.17)
C. S. Ryoo et al. 9
Note that
∞
(−1)n χ(n)wn
lw (s,χ | z) = 2
(n + z)s
n =0
f −1
=f
−s
a
(2.18)
a+z
s,
.
f
a
(−1) w χ(a)ζE,w f
a=0
The relation between lw (s,χ | z) and ζE,w (s,z) is given by the following theorem.
Theorem 2.10. For s ∈ C,
lw (s,χ | z) =
f −1
a+z
1 a a
f
(
−
1)
w
χ(a)ζ
s,
.
E,w
f s a=0
f
(2.19)
Observe that, substituting z = 0 into (2.19),
f −1
lw (s,χ) = f −s
(−1)a wa χ(a)ζE,w f s,
a=0
a
.
f
(2.20)
Substituting s = −n with n ∈ N,
f −1
lw (−n,χ | z) = f n
(−1)a wa χ(a)ζE,w f − n,
a=0
a+z
.
f
(2.21)
By Theorem 2.6 and (2.21), we have
f −1
lw (−n,χ | z) = f
n
(−1)a wa χ(a)En,w f
a=0
a+z
.
f
(2.22)
Using (1.27), we get
f −1
lw (−n,χ | z) = f
n
a
a
(−1) w χ(a)
a=0
=f
a
a
(−1) w χ(a)
a=0
=
k
k =0
f −1
n
n n
n n
n n
−k
n n−k
k=0 j =0
k
k =0
k
j
f k−n (a + z)n−k Ek,w f
f
k −n
n
−k
j =0
n − k n −k − j j
z
a Ek,w f
j
(2.23)
f k zn−k− j Ek,w f Sw,χ ( j),
where
f −1
Sw,χ ( j) =
(−1)a wa χ(a)a j .
a=0
By Theorem 2.9 and (2.23), we arrive at the following theorem.
(2.24)
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Journal of Inequalities and Applications
Theorem 2.11. For n ∈ Z+ , one obtains
n n
−k
n n−k
En,χ,w (z) =
k=0 j =0
k
j
f k zn−k− j Ek,w f Sw,χ ( j).
(2.25)
3. Twisted q-Euler zeta function and twisted q-analog Dirichlet l-function
Our primary goal of this section is to define generating functions of the twisted q-Euler
numbers and polynomials. Using these functions, twisted q-zeta function and twisted ql-functions are defined. These functions interpolate twisted q-Euler numbers and generalized twisted q-Euler numbers, respectively. Now, we introduce the generating functions
Fq (t) and Fq (x,t). Ryoo et al. [15] treated the analog of Euler numbers, which is called
q-Euler numbers in this paper. Using p-adic q-integral, we defined the q-Euler numbers
as follows:
En,q =
Zp
[t]nq dμ−q (t),
Thus we obtain
En,q = [2]q
where
n
i
for n ∈ N.
(3.1)
n
1
(−1)l
,
l
1 + ql+1
(3.2)
n n 1
1−q
l =0
is the binomial coefficient (see [1–18]). Using the above equation, we have
Fq (t) = [2]q
∞
(−1)m qm e[m]q t .
(3.3)
m=0
Thus q-Euler numbers, En,q , are defined by means of the generating function
Fq (t) = [2]q
∞
(−1)n qn e[n]q t .
(3.4)
n =0
Note that
∞
n =0
En,q
tn
=
n!
∞
[x]nq
Z p n =0
n!
t n dμ−q (x) =
Zp
e[x]q t dμ−q (x).
(3.5)
Thus we have
Zp
e[x]q t dμ−q (x) = [2]q
∞
(−1)n qn e[n]q t .
(3.6)
n =0
Similarly, the generating function Fq (t,z) of the q-Euler polynomials En,q (z) is defined
analogously as follows:
Fq (t,z) =
∞
n =0
∞
En,q (z)
tn
= [2]q
(−1)n qn e[n+z]q t .
n!
n =0
(3.7)
C. S. Ryoo et al.
11
Now, we introduce twisted q-Euler numbers En,q,w . For w ∈ T p , we define twisted q-Euler
numbers as follows:
En,q,w =
Zp
[x]nq wx dμ−q (x),
for n ∈ N.
(3.8)
Using (3.8), we obtain
En,q,w = [2]q
1
n n 1−q
n
1
(−1)l
.
l
1 + wql+1
l=0
(3.9)
Note that
Fq,w (t) =
∞
En,q,w
n =0
tn
n!
n ∞
1 n n
1
tn
l
= [2]q
(−1)
1 − q l=0 l
1 + wql+1 n!
n =0
= [2]q
∞ ∞
m m
(−1) q w
n=0 m=0
= [2]q
∞
m
t
[m]nq
(3.10)
n
n!
(−1)m qm wm e[m]q t .
m=0
Thus we obtain the generating function of twisted q-Euler numbers En,q,w as follows:
Fq,w (t) = [2]q
∞
(−1)n qn wn e[n]q t =
n =0
∞
En,q,w
n =0
tn
.
n!
(3.11)
Observe that limq→1 En,q,w = En,w . Using (3.11), we easily see that
Fq,w (t) =
∞
∞
En,q,w
n =0
=
∞
Z p n =0
tn =
n! n=0
Zp
wx [x]nq dμ−q (x)
[x]nq n
n!
tn
n!
t dμ−q (x) =
x [x]q t
Zp
w e
(3.12)
dμ−q (x).
Hence we obtain
Zp
wx e[x]q t dμ−q (x) = [2]q
∞
(−1)n qn wn e[n]q t .
(3.13)
n =0
From (3.11), we note that
∞
dk
Fq,w (t,z)|t=0 = [2]q (−1)n qn wn [n]kq
k
dt
n =0
(k ∈ N).
(3.14)
Using the above equation, we are now ready to define twisted q-Euler zeta functions.
12
Journal of Inequalities and Applications
Definition 3.1. Let s ∈ C. Then
ζq,w (s) = [2]q
∞
(−1)n qn wn
[n]sq
n =1
.
(3.15)
Note that ζE,w (s) is a meromorphic function on C. The relation between ζq,w (s) and
Ek,q,w is given by the following theorem.
Theorem 3.2. For k ∈ N, one has
ζq,w (−k) = Ek,q,w .
(3.16)
Observe that ζq,w (s) function interpolates Ek,q,w numbers at nonnegative integers. Using p-adic q-integral, we defined the twisted q-Euler polynomials as follows:
En,q,w (z) =
Zp
wx [x + z]nq dμ−q (x),
for n ∈ N.
(3.17)
Since [x + z]q = [z]q + qz [x]q , we have
En,q,w (z) =
Zp
wx [x + z]nq dμ−q (x)
=
=
Zp
n n
l =0
n l =0
=
wx
qlz [x]lq [z]nq −l dμ−q (x)
n lz n−l
q [z]q
l
n n
l =0
l
l
Zp
wx [x]lq dμ−q (x)
(3.18)
qlz El,q,w [z]nq −l
= qz Eq,w + [z]q
n
,
k must be replaced by E
where the symbol Ek,q,w is interpreted to mean that Eq,w
k,q,w . Using
(3.17), we have
En,q,w (z) = [2]q
1
1−q
n n l=0
n
1
(−1)l qlz
.
l
1 + wql+1
(3.19)
C. S. Ryoo et al.
13
Note that
Fq,w (t,z) =
∞
En,q,w (z)
n =0
tn
n!
n ∞
1 n n
1
tn
l lz
= [2]q
(−1) q
l+1
1 − q l =0 l
1 + wq
n!
n =0
= [2]q
∞ ∞
m m
(−1) q w
n=0 m=0
= [2]q
∞
m
t
[m + z]nq
(3.20)
n
n!
(−1)m qm wm e[m+z]q t .
m=0
Hence we have the generating function of twisted q-Euler polynomials En,q,w (z) as follows:
Fq,w (t,z) = [2]q
∞
(−1)m qm wm e[m+z]q T .
(3.21)
m=0
Observe that limq→1 En,q,w (z) = En,w (z). From (3.21), we note that
∞
dk
Fq,w (t,z)|t=0 = [2]q (−1)n qn wn [n + z]kq
k
dt
n =0
(k ∈ N).
(3.22)
Using the above equation, we are now ready to define the twisted Hurwitz q-Euler zeta
functions.
Definition 3.3. Let s ∈ C. Then
ζq,w (s,z) = [2]q
∞
(−1)n qn wn
n =1
[n + z]sq
.
(3.23)
Note that ζE,w (s,z) is a meromorphic function on C. The relation between ζq,w (s,z)
and Ek,q,w (z) is given by the following theorem.
Theorem 3.4. For k ∈ N, one has
ζq,w (−k,z) = Ek,q,w (z).
(3.24)
Observe that ζq,w (−k,z) function interpolates Ek,q,w (z) numbers at nonnegative integers.
4. Distribution and structure of the zeros
In this section, we investigate the zeros of the twisted q-Euler polynomials En,q,w (z) by
using computer. Let w = e2πi/N in C. We plot the zeros of En,q,w (x), x ∈ C, for N = 1,
q = 1/2 (see Figures 4.1, 4.2, 4.3, and 4.4).
14
Journal of Inequalities and Applications
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
3
4
3
4
Figure 4.1. Zeros of E10,q,w (x).
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
Figure 4.2. Zeros of E20,q,w (x).
Next, we plot the zeros of En,q,w (x), x ∈ C, for N = 2, q = 1/2 (see Figures 4.5, 4.6, 4.7,
and 4.8).
C. S. Ryoo et al.
15
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
3
4
3
4
Figure 4.3. Zeros of E30,q,w (x).
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
Figure 4.4. Zeros of E40,q,w (x).
Finally, we plot the zeros of En,q,w (x), x ∈ C, for N = 4, q = 1/3 (see Figures 4.9, 4.10,
4.11, and 4.12).
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Journal of Inequalities and Applications
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
3
4
3
4
Figure 4.5. Zeros of E10,q,w (x).
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
Figure 4.6. Zeros of E20,q,w (x).
Our numerical results for numbers of real and complex zeros of En,q,w (x), q = 1/2 are
displayed in Table 4.1.
C. S. Ryoo et al.
17
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
3
4
3
4
Figure 4.7. Zeros of E30,q,w (x).
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
Figure 4.8. Zeros of E40,q,w (x).
We will consider the more general open problem. In general, how many roots does
En,q,w (x) have? Prove or disprove: En,q (x) has n distinct solutions. Find the numbers of
complex zeros CEn,q,w (x) of En,q,w (x), Im(x) = 0. Prove or give a counterexample.
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Journal of Inequalities and Applications
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
3
4
3
4
Figure 4.9. Zeros of E10,q,w (x).
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
Figure 4.10. Zeros of E20,q,w (x).
Conjecture. Since n is the degree of the polynomial En,q,w (x), the number of real zeros
REn,q,w (x) lying on the real plane Im(x) = 0 is then REn,q,w (x) = n − CEn,q,w (x) , where CEn,q,w (x)
C. S. Ryoo et al.
19
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
3
4
3
4
Figure 4.11. Zeros of E30,q,w (x).
3
2
Im(z)
1
0
1
2
1
0
1
Re(z)
2
Figure 4.12. Zeros of E40,q,w (x).
denotes complex zeros. See Table 4.1 for tabulated values of REn,q,w (x) and CEn,q,w (x) . The
authors have no doubt that investigation along this line will lead to a new approach
20
Journal of Inequalities and Applications
Table 4.1. Numbers of real and complex zeros of En,q,w (x).
Degree n
1
2
3
4
5
6
7
8
9
10
Real zeros
1
0
1
0
1
0
1
0
1
0
w = e2πi/2
Complex zeros
0
2
2
4
4
6
6
8
8
10
w = e2πi/4
Real zeros
0
0
0
0
0
0
0
0
0
0
Complex zeros
1
2
3
4
5
6
7
8
9
10
Table 4.2. Approximate solutions of En,q,w (x) = 0, w = e2πi/2 .
Degree n
1
2
3
4
5
6
x
−0.584963
−0.403677 − 0.708194i, −0.403677 + 0.708194i
−0.683972, −0.111459 − 1.05549i, −0.111459 + 1.05549i
−0.635616 − 0.40476i, −0.635616 + 0.40476i, 0.158518 − 1.26338i,
0.158518 + 1.26338i
−0.746907, −0.510667 − 0.673i, −0.510667 + 0.673i,
0.39548 − 1.40175i, 0.39548 + 1.40175i
−0.728973 − 0.291464i, −0.728973 + 0.291464i, −0.368723 − 0.86325i,
−0.368723 + 0.86325i, 0.603353 − 1.50045i, 0.603353 + 1.50045i
employing numerical method in the field of research of the En,q,w (x) to appear in mathematics and physics. The reader may refer to [11, 14–16] for the details. We calculated an
approximate solution satisfying En,q,w (x), N = 2,4, q = 1/2, x ∈ C. The results are given
in Tables 4.2 and 4.3.
5. Further remarks and observations
Using p-adic q-fermionic integral, Rim and Kim [13] studied explicit p-adic expansion for alternating sums of powers. In the recent paper [10], Kim and Rim constructed
(h, q)-extensions of the twisted Euler numbers and polynomials. They also defined (h, q)generalizations of the twisted zeta function and L-series. These numbers and polynomials
are considered as the (h, q)-extensions of their previous results. However, these (h, q)Euler numbers and generating functions do not seem to be natural extension of Euler
numbers and polynomials. By this reason, we consider the natural q-extension of Euler
numbers and polynomials. In this paper, we include the numerical computations for our
C. S. Ryoo et al.
21
Table 4.3. Approximate solutions of En,q,w (x) = 0, w = e2πi/4 .
Degree n
1
2
3
4
5
6
x
0.117233 + 0.315473i
−0.349486 − 0.000392286i, 0.499266 + 0.489888i
−0.434656 − 0.317986i, −0.214323 + 0.325767i, 0.807131 + 0.571069i
−0.491443 + 0.120881i, −0.384977 − 0.598516i, −0.0235844 + 0.482636i
1.06026 + 0.618831i
−0.573494 − 0.129254i, −0.422702 + 0.36838i, −0.282405 − 0.820568i,
0.164523 + 0.577386i, 1.27487 + 0.650419i
−0.578686 − 0.346326i, −0.569935 + 0.151674i, −0.330723 + 0.532678i,
−0.160255 − 0.9957i, 0.339376 + 0.642424i, 1.46114 + 0.672881i
twisted q-Euler numbers and polynomials and the Euler numbers and polynomials which
are treated in this paper. In [9], many interesting integral equations related to fermionic
p-adic integrals on Z p are known. We proceed by first constructing generating functions
of the twisted q-Euler polynomials and numbers. Then, by applying Mellin transformation to these generating functions, integral representations of the twisted q-Euler zeta
function (and l-functions) are obtained, which interpolate the (generalized) twisted qEuler numbers at nonpositive integers.
References
[1] M. Cenkci and M. Can, “Some results on q-analogue of the Lerch zeta function,” Advanced
Studies in Contemporary Mathematics, vol. 12, no. 2, pp. 213–223, 2006.
[2] M. Cenkci, M. Can, and V. Kurt, “p-adic interpolation functions and Kummer-type congruences for q-twisted and q-generalized twisted Euler numbers,” Advanced Studies in Contemporary Mathematics, vol. 9, no. 2, pp. 203–216, 2004.
[3] A. Kudo, “A congruence of generalized Bernoulli number for the character of the first kind,”
Advanced Studies in Contemporary Mathematics, vol. 2, pp. 1–8, 2000.
[4] Q.-M. Luo, “Some recursion formulae and relations for Bernoulli numbers and Euler numbers
of higher order,” Advanced Studies in Contemporary Mathematics, vol. 10, no. 1, pp. 63–70, 2005.
[5] Q.-M. Luo and F. Qi, “Relationships between generalized Bernoulli numbers and polynomials
and generalized Euler numbers and polynomials,” Advanced Studies in Contemporary Mathematics, vol. 7, no. 1, pp. 11–18, 2003.
[6] T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp.
288–299, 2002.
[7] T. Kim, “A note on q-Volkenborn integration,” Proceedings of the Jangjeon Mathematical Society,
vol. 8, no. 1, pp. 13–17, 2005.
[8] T. Kim, “q-Euler numbers and polynomials associated with p-adic q-integrals,” Journal of Nonlinear Mathematical Physics, vol. 14, no. 1, pp. 15–27, 2007.
[9] T. Kim, “A note on some formulas for the q-Euler numbers and polynomials,” Proceedings of the
Jangjeon Mathematical Society, vol. 9, pp. 227–232, 2006.
[10] T. Kim and S.-H. Rim, “On the twisted q-Euler numbers and polynomials associated with basic
q-l-functions,” Journal of Mathematical Analysis and Applications, vol. 336, no. 1, pp. 738–744,
2007.
22
Journal of Inequalities and Applications
[11] T. Kim, C. S. Ryoo, L. C. Jang, and S.-H. Rim, “Exploring the q-Riemann zeta function and qBernoulli polynomials,” Discrete Dynamics in Nature and Society, vol. 2005, no. 2, pp. 171–181,
2005.
[12] H. Ozden, Y. Simsek, S.-H. Rim, and I. N. Cangul, “A note on p-adic q-Euler measure,” Advanced
Studies in Contemporary Mathematics, vol. 14, no. 2, pp. 233–239, 2007.
[13] S.-H. Rim and T. Kim, “Explicit p-adic expansion for alternating sums of powers,” Advanced
Studies in Contemporary Mathematics, vol. 14, no. 2, pp. 241–250, 2007.
[14] C. S. Ryoo and T. Kim, “A numerical computation of the structure of the roots of q-Bernoulli
polynomials,” to appear in Journal of Computational and Applied Mathematics.
[15] C. S. Ryoo, T. Kim, and R. P. Agarwal, “A numerical investigation of the roots of q-polynomials,”
International Journal of Computer Mathematics, vol. 83, no. 2, pp. 223–234, 2006.
[16] C. S. Ryoo, T. Kim, and L. C. Jang, “A note on generalized Euler numbers and polynomials,”
International Journal of Computer Mathematics, vol. 84, no. 7, pp. 1099–1111, 2007.
[17] Y. Simsek, “Theorems on twisted L-function and twisted Bernoulli numbers,” Advanced Studies
in Contemporary Mathematics, vol. 11, no. 2, pp. 205–218, 2005.
[18] Y. Simsek and S. Yang, “Transformation of four Titchmarsh-type infinite integrals and generalized Dedekind sums associated with Lambert series,” Advanced Studies in Contemporary Mathematics, vol. 9, no. 2, pp. 195–202, 2004.
C. S. Ryoo: Department of Mathematics, Hannam University, Daejeon 306-791, South Korea
Email address: ryoocs@hannam.ac.kr
T. Kim: School of Electronic Engineering and Computer Science, Kyungpook National University,
Taegu 702-701, South Korea
Email address: tkim@knu.ac.kr
Lee-Chae Jang: Department of Mathematics and Computer Sciences, KonKuk University,
Chungju 308-701, South Korea
Email addresses: leechae.jang@kku.ac.kr; leechae-jang@hanmail.net