Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2008, Article ID 390857, 16 pages
doi:10.1155/2008/390857
Research Article
Multivariate Interpolation Functions
of Higher-Order q-Euler Numbers and
Their Applications
Hacer Ozden,1 Ismail Naci Cangul,1 and Yilmaz Simsek2
1
2
Department of Mathematics, Faculty of Arts and Science, University of Uludag, Bursa 16059, Turkey
Department of Mathematics, Faculty of Arts and Science, University of Akdeniz, Antalya 07058, Turkey
Correspondence should be addressed to Hacer Ozden, hozden@uludag.edu.tr
Received 7 December 2007; Accepted 22 January 2008
Recommended by Paul Eloe
The aim of this paper, firstly, is to construct generating functions of q-Euler numbers and polynomials of higher order by applying the fermionic p-adic q-Volkenborn integral, secondly, to define multivariate q-Euler zeta function Barnes-type Hurwitz q-Euler zeta function and l-function which
interpolate these numbers and polynomials at negative integers, respectively. We give relation between Barnes-type Hurwitz q-Euler zeta function and multivariate q-Euler l-function. Moreover,
complete sums of products of these numbers and polynomials are found. We give some applications related to these numbers and functions as well.
Copyright q 2008 Hacer Ozden 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, definitions, and notations
Let p be a fixed odd prime. Throughout this paper, Zp , Qp , C, and Cp will, respectively, denote
the ring of p-adic rational integers, the field of p-adic rational numbers, the complex number
field, and the completion of the algebraic closure of Qp . Z Z ∪ {0}. Let vp be the normalized exponential valuation of Cp with |p|p p−vp p 1/p cf. 1–28. When we talk about
q-extensions, q is variously considered as an indeterminate, either a complex q ∈ C, or a p-adic
number q ∈ Cp . If q ∈ C, we assume that |q| < 1. If q ∈ Cp , then we assume |q − 1|p < p−1/p−1 so
that qx expx log q for |x|p ≤ 1.
For a fixed positive integer d with p, d 1, set
Xd lim
Z/dpN Z,
←
N
X1 Zp ,
2
Abstract and Applied Analysis
cX∗ a dpZp ,
0<a<dp
a,p1
a dpN Zp {x ∈ X : x ≡ amod dpN },
1.1
where a ∈ Z satisfies the condition 0 ≤ a < dpN cf. 1–28.
The distribution μq a dpN Zp is given as
qa
μq a dpN Zp N dp q
1.2
cf. 4, 10.
We say that f is a uniformly differentiable function at a point a ∈ Zp ; we write f ∈
UDZp if the difference quotient
Ff x, y fx − fy
x−y
1.3
has a limit f a as x, y → a, a. Let f ∈ UDZp . An invariant p-adic q-integral is defined by
Iq f Zp
fxdμq x lim
1
N
p
−1
N→∞ pN q x0
fxqx
1.4
cf. 4, 5, 10, 29, 30.
The q-extension of n ∈ N is defined by
nq 1 − qn
.
1−q
1.5
We note that limq→1 nq n.
Classical Euler numbers are defined by means of the following generating function:
∞
2
tn
En ,
t
e 1 n0 n!
1.6
cf. 1–3, 5, 8, 9, 15, 16, 18–20, 23, 28, 30, where En denotes classical Euler numbers. These
numbers are interpolated by the Euler zeta function which is defined as follows:
ζE s ∞
−1n
n1
ns
, s ∈ C,
1.7
cf. 8, 9, 24, 25, 28.
q-Euler numbers and polynomials have been studied by many mathematicians. These
numbers and polynomials are very important in number theory, mathematical analysis and
statistics, and the other areas.
Hacer Ozden et al.
3
In 16, Ozden and Simsek constructed extensions of q-Euler numbers and polynomials.
In 8, Kim et al. constructed new q-Euler numbers and polynomials which are different from
Ozden and Simsek 16.
In 31, Kim gave a detailed proof of fermionic p-adic q-measures on Zp . He treated
some interesting formulae related q-extension of Euler numbers and polynomials. He defined
fermionic p-adic q-measures on Zp as follows:
−qa
μ−q a dpN Zp N ,
dp −q
1.8
where
n−q 1 − −qn
1q
1.9
cf. 1, 31.
By using the fermionic p-adic q-measures, he defined the fermionic p-adic q-integral on
Zp as follows:
I−q f Zp
fxdμ−q x lim N→∞
p
1
N
N
p
−1
fx−qx
1.10
−q x0
cf. 31.
Observe that I−q f can be written symbolically as
lim Iq f I−q f
q→−q
1.11
cf. 31.
By using fermionic p-adic q-integral on Zp , Kim et al. 8 defined the generating function
of the q-Euler numbers as follows:
Fq t ∞
q1
tn
En,q ,
t
n!
qe 1 n0
1.12
where En,q denotes q-Euler numbers.
Witt’s formula of En x, q was given by Kim et al. 8:
En x, q Zp
x yn dμ−q y,
1.13
where q ∈ Cp and |1 − q|p < 1.
In 16, Ozden and Simsek defined generating function of q-Euler numbers by
Ft, q 2
2
Fq t.
qet 1 q 1
1.14
4
Abstract and Applied Analysis
In 7, 9, Kim defined q-l-functions and q-multiple l-functions. He also gave many applications of these functions.
We summarize our paper as follows. In Section 2, we give some fundamental properties
of the q-Euler numbers and polynomials. We also give some relations related to these numbers
and polynomials. By using generating functions of q-Euler numbers and polynomials of higher
order, we define multivariate q-Euler zeta function Barnes-type Hurwitz q-Euler zeta function and l-function which interpolate these numbers and polynomials at negative integers.
We also give contour integral representation of these functions. In Section 3, we find relation
r
r
between lE,q s, χ and ζq,E s, x. By using these relations, we obtain distribution relations of the
generalized q-Euler numbers and polynomials of higher order. In Section 4, we find complete
sums of products of these numbers and polynomials. We also give some applications related
to these numbers and functions.
2. Some properties of q-Euler numbers and polynomials
For q ∈ C with |q| < 1,
Fq t ∞
q1
tn
En,q
t
n!
qe 1 n0
2.1
cf. 8, where En,q denotes the q-Euler number and |t log q| < π.
Observe that by 2.1 we have
∞
n
q1
1 q−1 −1 t
H
−
q
.
n
n!
qet 1 et q−1 n0
2.2
From 2.1 and 2.2, we note that Hn −q−1 En,q , where Hn −q−1 are called Frobenius Euler
numbers cf. 27, 28.
The q-Euler polynomials are also defined by means of the following generating function
8:
Fq t, x ∞
q 1 xt tn
En,q x ,
e t
n!
qe 1
n0
2.3
where |t log q| < π;
q 1 xt
e eEq xt ,
qet 1
q 1ext qe1Eq xt eEq xt ,
2.4
∞
∞
n
tn
tn q 1 xn q 1 Eq x En,q x
.
n! n0
n!
n0
By comparing the coefficients of tn on both sides of the above equation, we have the following
theorem.
Hacer Ozden et al.
5
Theorem 2.1. Let n be nonnegative integer. Then
n
q 1 Eq x En,q x q 1xn ,
2.5
with the usual convention about replacing Eqn x by En,q x.
By using 2.5, we have
n
En,q x q
Ek,q x q 1xn .
k
k0
n
2.6
From 2.3, by applying Cauchy product and using 2.1, we also obtain
∞
tn
En,q
n!
n0
∞
tn
xn
n!
n0
∞
∞
n
n0
k
k0
x
n−k
Ek,q
∞
tn
tn
En,q x .
n!
n!
2.7
n0
By comparing the coefficients of tn on both sides of the above equation, we have
n
xn−k Ek,q
En,q x k
k0
n
2.8
cf. 8, 14.
By using Theorem 2.1 and 8, equation 3, we obtain
En x, q Zp
x yn dμ−q y
1
lim N N→∞ jp
−q
N
−1
jp
x yn −qx
x0
j−1 jp
−1
1
1
lim N a jx yn −qajx
j−q N→∞ p −qj x0 x0
N
p
j−1
−1
n x
jn
ay
1
lim N −qa
−qj
x
j−q N→∞ p −qj a0
j
x0
N
p
−1
j−1
n x
ay
jn 1
x
−qa lim N −qj
N→∞ p
j−q a0
j
−qj x0
N
j−1
a y jn −qa En
, qj .
j−q a0
j
2.9
6
Abstract and Applied Analysis
By using the above equation, we arrive at the following theorem.
Theorem 2.2. Let j be odd. Then
En,q x j−1
q 1j n ax
a a
j
−1
q
E
.
n,q
j
qj 1 a0
2.10
By simple calculation in 2.3, Ryoo et al. 14 give another proof of Theorem 2.2,which
is given as follows: let j be odd;
∞
En x
n0
q 1 xt
tn
e
n! qet 1
j−1
q1 −1a qa eat ext
1 qj ejt a0
eaxt qj 1
q 1 −1a qa
1 qj ejt qj 1
a0
d−1
∞ a x tn
q 1 n
a a
j
.
−1
q
E
j
n,q
j
j
n!
n0 q 1 a0
j−1
2.11
By comparing the coefficients of tn on both sides of the above equation, we have Theorem 2.2.
By substituting x n, with n ∈ Z into 2.3, then we have
Fq t, n ∞
q 1 nt tk
.
e
E
n
k,q
k!
qet 1
k0
2.12
Thus,
Fq t − qn −1n Fq t, n
∞
∞
q 1 −1l ql elt − q 1 −1ln qln etln
q 1
l0
l0
n−1
∞
−1l ql elt q 1
l0
2.13
−1ln qln etln − q 1
l0
∞
−1ln qln etln .
l0
Hence, by 2.13, we have
Fq t − qn −1n Fq t, n q 1
n−1
−1l ql elt .
2.14
l0
By the generating function of q-Euler numbers and polynomials and by 2.14, we see that
∞
m0
m
∞ n−1
tm t
q 1 ql −1l lm
.
m! m0
m!
l0
Em,q − qn −1n Em,q n
2.15
Hacer Ozden et al.
7
By comparing the coefficients of tn on both sides of 2.15, we obtain the following alternating
sums of powers of consecutive q-integers as follows.
Theorem 2.3 see 14. Let n ∈ Z . Then
n−1
Em,q − qn −1n Em,q n ql −1l lm .
q1
l0
2.16
Remark 2.4. Proof of Theorem 2.3 is similar to that of 14. If we take q → 1 in 2.16, we have
n−1
Em − −1n Em n −1l lm .
2
l0
2.17
The above formula is well known in the number theory and its applications.
Remark 2.5. Generating function of the q-Euler numbers in this paper is different than that
in 29, 31. It is same as in 8. Consequently, all these generating functions in 8, 16, 29, 31
produce different-type q-Euler numbers. But we observe that all these generating functions
were obtained by the same fermionic p-adic q-measures on Zp and the fermionic p-adic qintegral on Zp ; for applications of this integral and measure see for detail 2, 4, 8, 14–19, 23, 25,
29–31.
Now, we consider q-Euler numbers and polynomials of higher order as follows:
∞
n
q1
q1
q1
q1 r r t
,
·
·
·
E
n,q
n!
qet 1
qet 1
qet 1
qet 1
n0
2.18
r times
r
where En,q
are called q-Euler numbers of order r. We also consider q-Euler polynomials of order
r as follows:
∞
q1
q1
q1
q 1r etx
tn
tx
r
·
·
·
e
r ,
E
x
2.19
t
t
t
n,q
qe 1
qe 1
qe 1
n!
qet 1
n0
where |t log q| < π. From these generating functions of q-Euler numbers and polynomials of
higher order, we construct multiple q-Euler zeta functions. First, we investigate the properties
of generating function of q-Euler polynomials of higher order as follows:
r
Fq t, x q1 q1
q 1 tx
e
... t
t
t
qe 1 qe 1
qe 1
r times
∞
∞
r
qj etx
−1n1 qn1 en1 t . . .
−1nr qnr enr t
j
n 0
n 0
j0
r
1
∞
∞
n0
r
r
−1n1 ···nr qjn1 ···nr en1 ···nr xt
j
j0
r
n1 ,n2 ,...,nr 0
En,q x
r
tn
.
n!
2.20
8
Abstract and Applied Analysis
By applying Mellin transformation to 2.20, we have
r
∞
jn1 ···nr n1 ···nr xt
1 ∞ s−1 r
1 ∞s
r
e
t Fq t, xdt t −1n1 ···nr q
dt.
j
Γs 0
Γs
0
n1 ,n2 ,...,nr 0 j0
After some elementary calculations, we obtain
∞
r
∞
−1n1 ···nr qn1 ···nr
r
1 ∞ s−1 r
t Fq −t, xdt qj
s
Γs 0
j
n1 ,n2 ,...,nr 0 n1 ,n2 ,...,nr 0 n1 · · · nr x
j0
r
∞
r −1n1 ···nr qjn1 ···nr
.
j
n1 · · · nr xs
n1 ,n2 ,...,nr 0 j0
2.21
2.22
From 2.22, we define the analytic function which interpolates higher-order q-Euler
numbers at negative integers as follows.
Definition 2.6. For s ∈ C, x ∈ R 0 < x ≤ 1, one defines
n ···nr jn1 ···nr
∞
r
q
r −1 1
r
ζq,E s, x .
j
n
·
·
·
nr xs
1
n1 ,n2 ,...,nr 0 j0
2.23
r
ζq,E s, x is called Barnes-type Hurwitz q-Euler zeta function.
Remark 2.7. By applying the kth derivative operator dk /dtk |t0 on both sides of 2.20, we have
∞
r
k
dk r
r
r
2.24
En,q x k Fq t, x|t0 −1n1 ···nr qjn1 ···nr n1 · · · nr x .
j
dt
n ,n ,...,n 0 j0
1
2
r
By using the above equation, Ryoo et al. 14 and Simsek 23 also define 2.23.
By substituting s −k, k ∈ Z into 2.23 and using 2.24, after some calculations, we
arrive at the following theorem.
Theorem 2.8. Let k ∈ Z . Then
r
r
ζq,E −k, x En,q x.
r
2.25
r
Observe that the function ζq,E s, x interpolates En,q x polynomial at negative integers.
r
By using the complex integral representation of generating function of the polynomials En,q x,
we have
∞ −1n Er x
1
1
n,q
r
ts−1 Fq −t, xdt tns−1 dt,
2.26
Γs C
n!
Γs
C
n0
where C is Hankel’s contour along the cut joining the points z 0 and z ∞ on the real axis,
which starts from the point at ∞, encircles the origin z 0 once in the positive counterclockwise direction, and returns to the point at ∞ see for detail 13, 17, 25, 28. By using
2.26 and Cauchy-Residue theorem, then we arrive at 2.25.
Hacer Ozden et al.
9
r
r
Remark 2.9. ζq,E s, 1 ζq,E s is called Barnes-type q-Euler zeta function; see for detail 14.
r
ζq,E s, x is an analytic function in whole complex s-plane. For s ∈ C,
r
ζEr s, x limζq,E
s, x 2r
q→1
∞
n1 ,n2 ,...,nr 0
−1n1 ···nr
n1 · · · nr x
s .
2.27
If r 1 in the above equation, we have
ζE s, x 2
∞
−1n
s.
n0 n x
2.28
The function ζE s, x is known as classical Hurwitz-type zeta function which interpolates classical Euler numbers at negative integers, cf. 28.
Let χ be Dirichlet’s character with conductor d ∈ Z . The generalized q-Euler numbers
attached to χ of higher order are defined by
∞
q 1 da1 −1a qa χaeta tn
Fq,χ t 2.29
En,χ
d
dt
n!
q e 1
n0
cf. 8, where |t log d| < π. The q-Euler numbers attached to χ of higher order are defined
by
q 1 da1 −1a qa χaeta r
r
Fq,χ t qd edt 1
2.30
∞
n
r t
En,q,χ .
n!
n0
From 2.30, we obtain
r
Fq,χ t
∞
r
n1 ,n2 ,...,nr 1 j0
∞
r
r
χnk −1n1 ···nr qjn1 ···nr en1 ···nr t
j
k1
n
r t
En,q,χ .
2.31
n!
n0
By applying the kth derivative operator dk /dtk |t0 in 2.31, we have
r
r
∞
n1 ,n2 ,...,nr 1 j0
Ek,q,χ dk r
Fq,χ t|t0
dtk
r
r
−1n1 ···nr qjn1 ···nr n1 · · · nr k χnk .
j
k1
2.32
By using 2.32, we define Dirichlet-type multiple Euler q- l-function as follows.
Definition 2.10. Let s ∈ C;
r
lE,q s, χ
∞
r
n1 ,n2 ,...,nr 1 j0
n ···nr jn1 ···nr r
q
r −1 1
k1 χnk .
s
j
n1 · · · nr 2.33
10
Abstract and Applied Analysis
r
Remark 2.11. lE,q s, χ is an analytic function in the whole complex s-plane. From the above
definition,
∞
2−1n1 ···nr rk1 χnk r
r
.
2.34
lE s, χ lim lE,q s, χ q→1
n1 · · · nr s
n1 ,n2 ,...,nr 1
For r 1 in the above equation, we have
lE s, χ ∞
2−1n χn
ns
n1
2.35
.
This function is called Euler l-function.
Here, we observe that by applying Mellin transformation to 2.31, we obtain
1 ∞ s−1 r
r
t Fq,χ −tdt lE,q s, χ.
Γs 0
2.36
This gives us another definition of 2.32.
By substituting s −k, k ∈ Z into 2.33 and using 2.32, we arrive at the following
theorem.
Theorem 2.12. Let k ∈ Z . Then
r
r
lE,q −k, χ Ek,q,χ .
2.37
We note that
r
r
r
limlE,q −k, χ lE −k, χ En,χ ,
q→1
2.38
where En,χ are called classical Euler numbers attached to χ of higher order, cf. 28. By using
2.26, 2.36, we obtain another proof of 2.37.
r
r
3. Relation between lE,q s, χ and ζq,E s, x
Substituting nj aj mj f, where mj 0, 1, 2, 3, . . . , ∞ and aj 1, 2, . . . , f, where χaj mj f χaj and f is odd conductor of χ, 1 ≤ j ≤ r , into 2.33, we have
r
lE,q s, χ 1 qr
−1a1 m1 f···ar mr f qa1 m1 f···ar mr f rk1 χak mk f
a1 m1 f · · · ar mr fs
a1 ,a2 ,...,ar 1 m1 ,m2 ,...,mr 0
f
1 qr f −s
1 r
qf r
× 1 qf
∞
f
−1a1 ···ar qa1 ···ar
a1 ,a2 ,...,ar 1
r
χak k1
∞
−1m1 ···mr qfvm1 f···mr f
a · · · a
s .
1
r
m1 ,m2 ,...,mr 0
m1 · · · mr
f
By substituting 2.23 into the above equation, we arrive at the following theorem.
3.1
Hacer Ozden et al.
11
Theorem 3.1. Let χ be a Dirichlet character with conductor f odd. Then
r
lE,q s, χ f
1 qr f −s
1 r
qf −1a1 ···ar qa1 ···ar
a ··· a r
1
r
χak ζqf ,E s,
.
f
k1
r
a1 ,a2 ,...,ar 1
3.2
By substituting s −k, k ∈ Z, into 3.2, we obtain
r
lE,q −k, χ f
1 qr f k
1 r
qf −1a1 ···ar qa1 ···ar
a1 ,a2 ,...,ar 1
a1 · · · ar r
χak ζqf ,E − k,
.
f
k1
r
3.3
By using 2.25 and 2.37 in the above equation, we obtain distribution relation of the q-Euler
r
numbers attached to χ of higher order, Ek,q,χ , which is given as follows.
Theorem 3.2. The following holds:
r
Ek,q,χ f
1 qr f k
1 r
qf −1a1 ···ar qa1 ···ar
a1 ,a2 ,...,ar 1
r
k1
r
χak En,qf
a · · · a 1
r
.
f
3.4
4. Multivariate p-adic fermionic q-integral on Zp associated with
higher-order q-Euler numbers
In 14, Ryoo et al. defined q-extension of Euler numbers and polynomials of higher order.
They studied Barnes-type q-Euler zeta functions. They also derived sums of products of q-Euler
numbers and polynomials by using fermionic p-adic q-integral. In this section, we assume that
q ∈ Cp with |1 − q|p < 1. By using 1.4, the p-adic fermionic q-integral on Zp is defined by
I−q f Zp
fxdμ−q x
p
−1
1
lim N fx−qx .
N→∞ p
x0
−q
N
4.1
From this integral equation, we have see 1, 2, 4
qI−q f1 I−q f q 1f0,
4.2
where f1 x fx 1. If we take fx etx in 4.2, we have
I−q e tx
cf. 8.
Zp
etx dμ−q x ∞
q1
tn
E
n,q
n!
qet 1 n0
4.3
12
Abstract and Applied Analysis
Now we are ready to give multivariate p-adic fermionic q-integral on Zp as follows see
for detail 14. Let
···
Zrp
Zp
Zp
Zp
r times
Zrp
etx1 ···xr dμ−q x1 · · · dμ−q xr q1
qet 1
···
q1
qet 1
4.4
∞
n
r t
En,q .
n!
n0
r times
From 4.4, we obtain Witt’s formula for q-Euler numbers of higher order as follows.
Theorem 4.1 see 14. Let k ∈ Z . Then
r
x1 · · · xr n dμ−q x1 · · · dμ−q xr En,q .
4.5
Zrp
By 4.4, we obtain
Zrp
etx1 ···xr x dμ−q x1 · · · dμ−q xr ext q 1r
qe 1 · · · qe 1
t
t
∞
r
En,q x
n0
tn
.
n!
4.6
r times
Theorem 4.2 multinomial theorem. The following holds:
n
v
xj
j1
l1 ,l2 ,...,lv ≥0
l1 l2 ···lv n
n
l1 , l2 , . . . , l v
v
xala ,
4.7
a1
where l1 ,l2n,...,lv are the multinomial coefficients, which are defined by
n!
n
l 1 , l2 , . . . , l v
l1 !l2 ! . . . lv !
4.8
(cf. [32, 33]).
Now we give a main theorem of this section, which is called complete sums of products
of q-Euler polynomials of higher order.
Theorem 4.3. For positive integers n, r, one has
r En,q y1
y2 · · · yr l1 ,l2 ,...,lr ≥0
l1 l2 ···lr n
where l1 ,l2n,...,lr is the multinomial coefficient.
n
l 1 , l2 , . . . , l r
r
Elj ,q yj ,
j1
4.9
Hacer Ozden et al.
13
Proof. The proof of this theorem is similar to that of 23. By using Taylor series of etx into 4.6,
and x by y1 y2 · · · yr , then we have
n r
r
r En,q y1 y2 · · · yr ···
yj xj dμ−q xj .
4.10
Zp
Zp
j1
j1
By using 4.7 in the above equation, and after some elementary calculations, we get
r l
n
r En,q y1 y2 · · · yr yj xj j dμ−q xj
.
l 1 , l2 , . . . , l v
Z
l1 ,l2 ,...,lr ≥0
l1 l2 ···lr n
j1
p
4.11
By substituting 2.25 into the above equation, we arrive at the desired result.
By substituting 2.8 into 4.9, then Theorem 4.3 reduces to the following theorem.
Theorem 4.4. For positive integers n, r, one has
r En,q y1
y2 · · · yr l1 ,l2 ,...,lr ≥0
l1 l2 ···lr n
n
l1 , l2 , . . . , l r
lj
lj −k
yj Elj ,q .
k
j1 k0
lj
r 4.12
In 4.10, if we replace y1 y2 · · · yr by x, then we obtain the following corollary.
Corollary 4.5. For n ≥ 0, one has
m
r
x1 · · · xr x dμ−q x1 · · · dμ−q xr En,q x Zrp
l1 ···lr lr 1m
l1 ···lr lr 1m
m
l1 · · · lr1
m
l1 · · · lr1
Zp
x1l1 dμ−q x1 · · ·
Zp
xrlr dμ−q xr xlr1
4.13
El1 ,q El2 ,q · · · Elr ,q xlr1 .
Remark 4.6. By using 4.5–4.7, complete sums of products of q-Euler polynomials of higher
order are also obtained. Proof of Corollary 4.5 was also given by Ryoo et al. 14, which is given
by
r
En,q x
n
k0 l1 ···lr lr1
n
k
xn−k El1 ,q El2 ,q · · · Elr ,q .
k
l
·
·
·
l
1
r1
m
4.14
In 4.13, if we take q → 1, we have
r
En x
l1 ···lr lr1 m
m
l1 · · · lr1
El1 El2 · · · Elr xlr1 .
4.15
For more detailed information about complete sums of products of Euler polynomials and
Bernoulli polynomials, see also 11, 14, 20–24, 34, 35.
14
Abstract and Applied Analysis
Let χ be a Dirichlet character with conductor d ∈ Z . Then
d−1
l0
χxe dμ−q x q 1
tx
X
∞
−1d−1−l ql etl χx
edt qd 1
4.16
tn
En,χ,q .
n!
n0
By using Taylor expansion of etx and then comparing coefficients of tn on both sides, we
arrive at
χxxn dμ−q x En,χ,q
4.17
X
cf. 8.
By 4.16, we have
r
Xr
χxi etx1 ···xr dμ−q x1 · · · dμ−q xr i1
r
Xr
i1
∞
r
χxi etx1 ···xr dμ−q x1 · · · dμ−q xr d−1
r
d−1−a a ta
q e χa r a0 −1
edt qd 1
j0
En,χ,q
n0
r
qj
j
tn
.
n!
4.18
r
Thus we give Witt-type formula of En,χ,q as follows.
Theorem 4.7. Let χ be a Dirichlet character with conductor d ∈ N and let m ≥ 0. Then
r
En,χ,q Xr
x1 · · · x r
r
m χxi dμ−q x1 · · · dμ−q xr .
4.19
i1
By using 3.2, 2.8, we obtain
r
En,q,χ
f
n
f−1
1 qr
r
1 qf a1 ···ar a1 ···ar
−1
q
a1 ,a2 ,...,ar 0
r
n a1 · · · ar n−k r
Ek,qf .
k
f
k0
n
χak k1
4.20
By using 4.7 in the above equation, we have
r
En,q,χ 1 qr
1 ×
r
qf n
k0
f−1
−1a1 ···ar qa1 ···ar
a1 ,a2 ,...,ar 0
l1 ,l2 ,...,lv ≥0
l1 l2 ···lv n−k
n−k
l 1 , l2 , . . . , l v
r
k1
χak v
n ly
r
a f k Ek,qf .
k y1 y
4.21
Hacer Ozden et al.
15
Acknowledgments
The first and the second authors are supported by the research fund of Uludag University
Projects no. F-2006/40 and F-2008/31. The third author is supported by the research fund of
Akdeniz University. The authors would like to thank the referee for their comments.
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