AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing CorporationS108533750839085710.1155/2008/390857390857Research ArticleMultivariate Interpolation Functions of Higher-Order q-Euler Numbers and Their ApplicationsOzdenHacer1CangulIsmail Naci1SimsekYilmaz2EloePaul1Department of MathematicsFaculty of Arts and ScienceUniversity of UludagBursa 16059Turkey2Department of MathematicsFaculty of Arts and ScienceUniversity of AkdenizAntalya 07058Turkey200826022008200807122007220120082008Copyright © 2008This 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.

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.

1. Introduction, Definitions, and Notations

Let p be a fixed odd prime. Throughout this paper, p, p, , and p 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 p. +=+{0}. Let vp be the normalized exponential valuation of p with |p|p=pvp(p)=1/p (cf. ). When we talk about q-extensions, q is variously considered as an indeterminate, either a complex q, or a p-adic number qp. If q, we assume that |q|<1. If qp, then we assume |q1|p<p1/(p1) so that qx=exp(xlogq) for |x|p1.

For a fixed positive integer d with (p,d)=1, set 𝕏d=limN/dpN,𝕏1=p,𝕏=0<a<dp(a,p)=1(a+dpp),a+dpNp={x𝕏:xa(moddpN)}, where a satisfies the condition 0a<dpN (cf. ).

The distribution μq(a+dpNp) is given as μq(a+dpNp)=qa[dpN]q (cf. [4, 10]).

We say that f is a uniformly differentiable function at a point ap; we write fUD(p) if the difference quotient Ff(x,y)=f(x)f(y)xy has a limit f(a) as (x,y)(a,a). Let fUD(p). An invariant p-adic q-integral is defined by Iq(f)=pf(x)dμq(x)=limN1[pN]qx=0pN1f(x)qx (cf. [4, 5, 10, 29, 30]).

The q-extension of n is defined by [n]q=1qn1q. We note that limq1[n]q=n.

Classical Euler numbers are defined by means of the following generating function: 2et+1=n=0Entnn!, (cf. [13, 5, 8, 9, 15, 16, 18, 19, 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)=n=1(1)nns,s, (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.

In , Ozden and Simsek constructed extensions of q-Euler numbers and polynomials. In , Kim et al. constructed new q-Euler numbers and polynomials which are different from Ozden and Simsek .

In , Kim gave a detailed proof of fermionic p-adic q-measures on p. He treated some interesting formulae related q-extension of Euler numbers and polynomials. He defined fermionic p-adic q-measures on p as follows: μq(a+dpNp)=(q)a[dpN]q, where [n]q=1(q)n1+q (cf. [1, 31]).

By using the fermionic p-adic q-measures, he defined the fermionic p-adic q-integral on p as follows: Iq(f)=pf(x)dμq(x)=limN1[pN]qx=0pN1f(x)(q)x (cf. ).

Observe that Iq(f) can be written symbolically as limqqIq(f)=Iq(f) (cf. ).

By using fermionic p-adic q-integral on p, Kim et al.  defined the generating function of the q-Euler numbers as follows: Fq(t)=q+1qet+1=n=0En,qtnn!, where En,q denotes q-Euler numbers.

Witt's formula of En(x,q) was given by Kim et al. :En(x,q)=p(x+y)ndμq(y), where qp and |1q|p<1.

In , Ozden and Simsek defined generating function of q-Euler numbers by F(t,q)=2qet+1=2q+1Fq(t).

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 between lE,q(r)(s,χ) and ζq,E(r)(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 <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M105"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Euler Numbers and Polynomials

For q with |q|<1,Fq(t)=q+1qet+1=n=0En,qtnn! (cf. ), where En,q denotes the q-Euler number and |t+logq|<π.

Observe that by (2.1) we have q+1qet+1=1+q1et+q1=n=0Hn(q1)tnn!. From (2.1) and (2.2), we note that Hn(q1)=En,q, where Hn(q1) are called Frobenius Euler numbers (cf. [27, 28]).

The q-Euler polynomials are also defined by means of the following generating function : Fq(t,x)=q+1qet+1ext=n=0En,q(x)tnn!, where |t+logq|<π;q+1qet+1ext=eEq(x)t,(q+1)ext=qe(1+Eq(x))t+eEq(x)t,(q+1)n=0xntnn!=n=0(q(1+Eq(x))n+En,q(x))tnn!. By comparing the coefficients of tn on both sides of the above equation, we have the following theorem.

Theorem 2.1.

Let n be nonnegative integer. Then q(1+Eq(x))n+En,q(x)=(q+1)xn, with the usual convention about replacing Eqn(x) by En,q(x).

By using (2.5), we have En,q(x)+qk=0n(nk)Ek,q(x)=(q+1)xn. From (2.3), by applying Cauchy product and using (2.1), we also obtain (n=0En,qtnn!)(n=0xntnn!)=n=0(k=0(nk)xnkEk,q)tnn!=n=0En,q(x)tnn!. By comparing the coefficients of tn on both sides of the above equation, we have En,q(x)=k=0n(nk)xnkEk,q (cf. [8, 14]).

By using Theorem 2.1 and [8, equation (3)], we obtain En(x,q)=p(x+y)ndμq(y)=limN1[jpN]qx=0jpN1(x+y)n(q)x=1[j]qlimN1[pN](q)jx=0j1x=0jpN1(a+jx+y)n(q)a+jx=jn[j]qlimN1[pN](q)ja=0j1(q)ax=0pN1(a+yj+x)n((q)j)x=jn[j]qa=0j1(q)alimN1[pN](q)jx=0pN1(a+yj+x)n((q)j)x=jn[j]qa=0j1(q)aEn(a+yj,qj).

By using the above equation, we arrive at the following theorem.

Theorem 2.2.

Let j be odd. Then En,q(x)=(q+1)jnqj+1a=0j1(1)aqaEn,qj(a+xj).

By simple calculation in (2.3), Ryoo et al.  give another proof of Theorem 2.2, which is given as follows: let j be odd; n=0En(x)tnn!=q+1qet+1ext=q+11+qjejta=0j1(1)aqaeatext=(q+1)a=0j1(1)aqa(e(a+x)t1+qjejt)qj+1qj+1=n=0(q+1qj+1jna=0d1(1)aqaEn,qj(a+xj))tnn!. By comparing the coefficients of tn on both sides of the above equation, we have Theorem 2.2.

By substituting x=n, with n+ into (2.3), then we have Fq(t,n)=q+1qet+1ent=k=0Ek,q(n)tkk!. Thus,Fq(t)qn(1)nFq(t,n)=(q+1)l=0(1)lqlelt(q+1)l=0(1)l+nql+net(l+n)=(q+1)l=0n1(1)lqlelt+(q+1)l=0(1)l+nql+net(l+n)(q+1)l=0(1)l+nql+net(l+n). Hence, by (2.13), we have Fq(t)qn(1)nFq(t,n)=(q+1)l=0n1(1)lqlelt. By the generating function of q-Euler numbers and polynomials and by (2.14), we see that m=0(Em,qqn(1)nEm,q(n))tmm!=m=0((q+1)l=0n1ql(1)llm)tmm!. 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 [<xref ref-type="bibr" rid="B20">14</xref>]).

Let n+. Then Em,qqn(1)nEm,q(n)q+1=l=0n1ql(1)llm.

Remark 2.4.

Proof of Theorem 2.3 is similar to that of . If we take q1 in (2.16), we have Em(1)nEm(n)2=l=0n1(1)llm.

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 . 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 p and the fermionic p-adic q-integral on p; for applications of this integral and measure see for detail [2, 4, 8, 1419, 23, 25, 29, 30, 31].

Now, we consider q-Euler numbers and polynomials of higher order as follows: (q+1qet+1)(q+1qet+1)(q+1qet+1)rtimes=(q+1qet+1)r=n=0En,q(r)tnn!, where En,q(r) are called q-Euler numbers of order r. We also consider q-Euler polynomials of order r as follows: (q+1qet+1)(q+1qet+1)(q+1qet+1)etx=(q+1)retx(qet+1)r=n=0En,q(r)(x)tnn!, where |t+logq|<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: Fq(r)(t,x)=q+1qet+1q+1qet+1q+1qet+1rtimesetx=j=0r(rj)qjetxn1=0(1)n1qn1en1tnr=0(1)nrqnrenrt=n1,n2,,nr=0j=0r(rj)(1)n1++nrqj+n1++nre(n1++nr+x)t=n=0En,q(r)(x)tnn!.

By applying Mellin transformation to (2.20), we have 1Γ(s)0ts1Fq(r)(t,x)dt=n1,n2,,nr=0j=0r(rj)1Γ(s)0ts(1)n1++nrqj+n1++nre(n1++nr+x)tdt. After some elementary calculations, we obtain 1Γ(s)0ts1Fq(r)(t,x)dt=j=0r(rj)qjn1,n2,,nr=0n1,n2,,nr=0(1)n1++nrqn1++nr(n1++nr+x)s=n1,n2,,nr=0j=0r(rj)(1)n1++nrqj+n1++nr(n1++nr+x)s.

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,x(0<x1), one defines ζq,E(r)(s,x)=n1,n2,,nr=0j=0r(rj)(1)n1++nrqj+n1++nr(n1++nr+x)s.ζq,E(r)(s,x) is called Barnes-type Hurwitz q-Euler zeta function.

Remark 2.7.

By applying the kth derivative operator dk/dtk|t=0 on both sides of (2.20), we have En,q(r)(x)=dkdtkFqr(t,x)|t=0=n1,n2,...,nr=0j=0r(rj)(1)n1++nrqj+n1++nr(n1++nr+x)k. By using the above equation, Ryoo et al.  and Simsek  also define (2.23).

By substituting s=k,k+ into (2.23) and using (2.24), after some calculations, we arrive at the following theorem.

Theorem 2.8.

Let k+. Then ζq,E(r)(k,x)=En,q(r)(x).

Observe that the function ζq,E(r)(s,x) interpolates En,q(r)(x) polynomial at negative integers. By using the complex integral representation of generating function of the polynomials En,q(r)(x), we have 1Γ(s)Cts1Fq(r)(t,x)dt=n=0(1)nEn,q(r)(x)n!1Γ(s)Ctn+s1dt, 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 (counter-clockwise) 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).

Remark 2.9.

ζq,E(r)(s,1)=ζq,E(r)(s) is called Barnes-type q-Euler zeta function; see for detail . ζq,E(r)(s,x) is an analytic function in whole complex s-plane. For sC,ζEr(s,x)=limq1ζq,Er(s,x)=2rn1,n2,,nr=0(1)n1++nr(n1++nr+x)s. If r=1 in the above equation, we have ζE(s,x)=2n=0(1)n(n+x)s. The function ζE(s,x) is known as classical Hurwitz-type zeta function which interpolates classical Euler numbers at negative integers, cf. .

Let χ be Dirichlet's character with conductor d+. The generalized q-Euler numbers attached to χ of higher order are defined by Fq,χ(t)=(q+1)a=1d(1)aqaχ(a)etaqdedt+1=n=0En,χtnn! (cf. ), where |t+logd|<π. The q-Euler numbers attached to χ of higher order are defined by Fq,χ(r)(t)=((q+1)a=1d(1)aqaχ(a)etaqdedt+1)r=n=0En,q,χ(r)tnn!. From (2.30), we obtain Fq,χ(r)(t)=n1,n2,,nr=1j=0r(rj)(1)n1++nrqj+n1++nre(n1++nr)tk=1rχ(nk)=n=0En,q,χ(r)tnn!. By applying the kth derivative operator dk/dtk|t=0 in (2.31), we have Ek,q,χ(r)=dkdtkFq,χ(r)(t)|t=0=n1,n2,,nr=1j=0r(rj)(1)n1++nrqj+n1++nr(n1++nr)kk=1rχ(nk). By using (2.32), we define Dirichlet-type multiple Euler q-l-function as follows.

Definition 2.10.

Let s;lE,q(r)(s,χ)=n1,n2,,nr=1j=0r(rj)(1)n1++nrqj+n1++nrk=1rχ(nk)(n1++nr)s.

Remark 2.11.

lE,q(r)(s,χ) is an analytic function in the whole complex s-plane. From the above definition, lE(r)(s,χ)=limq1lE,q(r)(s,χ)=n1,n2,,nr=12(1)n1++nrk=1rχ(nk)(n1++nr)s. For r=1 in the above equation, we have lE(s,χ)=n=12(1)nχ(n)ns. This function is called Euler l-function.

Here, we observe that by applying Mellin transformation to (2.31), we obtain 1Γ(s)0ts1Fq,χ(r)(t)dt=lE,q(r)(s,χ). This gives us another definition of (2.32).

By substituting s=k,k+ into (2.33) and using (2.32), we arrive at the following theorem.

Theorem 2.12.

Let k+. Then lE,q(r)(k,χ)=Ek,q,χ(r).

We note that limq1lE,q(r)(k,χ)=lE(r)(k,χ)=En,χ(r), where En,χ are called classical Euler numbers attached to χ of higher order, cf. . By using (2.26), (2.36), we obtain another proof of (2.37).

3. Relation between <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M232"><mml:mrow><mml:msubsup><mml:mi>l</mml:mi><mml:mrow><mml:mi>E</mml:mi><mml:mn>,</mml:mn><mml:mi>q</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>s</mml:mi><mml:mn>,</mml:mn><mml:mi>χ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M233"><mml:mrow><mml:msubsup><mml:mi>ζ</mml:mi><mml:mrow><mml:mi>q</mml:mi><mml:mn>,</mml:mn><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>r</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>s</mml:mi><mml:mn>,</mml:mn><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>

Substituting nj=aj+mjf, where mj=0,1,2,3,, and aj=1,2,,f, where χ(aj+mjf)=χ(aj) and f is odd conductor of χ,1jr , into (2.33), we have lE,q(r)(s,χ)=(1+q)ra1,a2,,ar=1fm1,m2,,mr=0(1)a1+m1f++ar+mrfqa1+m1f++ar+mrfk=1rχ(ak+mkf)(a1+m1f++ar+mrf)s=(1+q)rfs(1+qf)ra1,a2,,ar=1f(1)a1++arqa1++ark=1rχ(ak)×(1+qf)rm1,m2,,mr=0(1)m1++mrqfv+m1f++mrf(a1++arf+m1++mr)s. By substituting (2.23) into the above equation, we arrive at the following theorem.

Theorem 3.1.

Let χ be a Dirichlet character with conductor f(=odd). Then lE,q(r)(s,χ)=(1+q)rfs(1+qf)ra1,a2,,ar=1f(1)a1++arqa1++ark=1rχ(ak)ζqf,E(r)(s,a1++arf).

By substituting s=k, k, into (3.2), we obtain lE,q(r)(k,χ)=(1+q)rfk(1+qf)ra1,a2,,ar=1f(1)a1++arqa1++ark=1rχ(ak)ζqf,E(r)(k,a1++arf). By using (2.25) and (2.37) in the above equation, we obtain distribution relation of the q-Euler numbers attached to χ of higher order, Ek,q,χ(r), which is given as follows.

Theorem 3.2.

The following holds: Ek,q,χ(r)=(1+q)rfk(1+qf)ra1,a2,,ar=1f(1)a1++arqa1++ark=1rχ(ak)En,qf(r)(a1++arf).

4. Multivariate <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M252"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula>-Adic Fermionic <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M253"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Integral on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M254"><mml:mrow><mml:msub><mml:mi>ℤ</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> Associated with Higher-Order <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M255"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Euler Numbers

In , 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 qp with |1q|p<1. By using (1.4), the p-adic fermionic q-integral on p is defined by Iq(f)=pf(x)dμq(x)=limN1[pN]qx=0pN1f(x)(q)x. From this integral equation, we have (see [1, 2, 4]) qIq(f1)+Iq(f)=(q+1)f(0), where f1(x)=f(x+1). If we take f(x)=etx in (4.2), we have Iq(etx)=petxdμq(x)=q+1qet+1=n=0En,qtnn! (cf. ).

Now we are ready to give multivariate p-adic fermionic q-integral on p as follows (see for detail ). Let pr=ppprtimespret(x1++xr)dμq(x1)dμq(xr)=(q+1qet+1)(q+1qet+1)rtimes=n=0En,q(r)tnn!. From (4.4), we obtain Witt's formula for q-Euler numbers of higher order as follows.

Theorem 4.1 (see [<xref ref-type="bibr" rid="B20">14</xref>]).

Let k+. Then pr(x1++xr)ndμq(x1)dμq(xr)=En,q(r).

By (4.4), we obtain pret(x1++xr+x)dμq(x1)dμq(xr)=ext(q+1)r(qet+1)(qet+1)rtimes=n=0En,q(r)(x)tnn!.

Theorem 4.2 (multinomial theorem).

The following holds: (j=1vxj)n=l1,l2,,lv0l1+l2++lv=n(nl1,l2,,lv)a=1vxala, where (nl1,l2,,lv) are the multinomial coefficients, which are defined by (nl1,l2,,lv)=n!l1!l2!lv! (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 En,q(r)(y1+y2++yr)=l1,l2,,lr0l1+l2++lr=n(nl1,l2,,lr)j=1rElj,q(yj), where (nl1,l2,,lr) is the multinomial coefficient.

Proof.

The proof of this theorem is similar to that of . By using Taylor series of etx into (4.6), and x by y1+y2++yr, then we have En,q(r)(y1+y2++yr)=pp(j=1r(yj+xj))nj=1rdμq(xj). By using (4.7) in the above equation, and after some elementary calculations, we get En,q(r)(y1+y2++yr)=l1,l2,,lr0l1+l2++lr=n(nl1,l2,,lv)j=1rp(yj+xj)ljdμq(xj). 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 En,q(r)(y1+y2++yr)=l1,l2,,lr0l1+l2++lr=n(nl1,l2,,lr)j=1rk=0lj(ljk)yjljkElj,q.

In (4.10), if we replace y1+y2++yr by x, then we obtain the following corollary.

Corollary 4.5.

For n0, one has En,q(r)(x)=pr(x1++xr+x)mdμq(x1)dμq(xr)=l1++lr+lr+1=m(ml1lr+1)px1l1dμq(x1)pxrlrdμq(xr)xlr+1=l1++lr+lr+1=m(ml1lr+1)El1,qEl2,qElr,qxlr+1.

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. , which is given by En,q(r)(x)=k=0nl1++lr+lr+1=m(nk)(kl1lr+1)xnkEl1,qEl2,qElr,q. In (4.13), if we take q1, we have En(r)(x)=l1++lr+lr+1=m(ml1lr+1)El1El2Elrxlr+1. For more detailed information about complete sums of products of Euler polynomials and Bernoulli polynomials, see also [11, 14, 2024, 34, 35].

Let χ be a Dirichlet character with conductor d+. Then 𝕏χ(x)etxdμq(x)=(q+1)l=0d1(1)d1lqletlχ(x)edtqd+1=n=0En,χ,qtnn!.

By using Taylor expansion of etx and then comparing coefficients of tn on both sides, we arrive at 𝕏χ(x)xndμq(x)=En,χ,q (cf. ).

By (4.16), we have 𝕏ri=1rχ(xi)et(x1++xr)dμq(x1)dμq(xr)=𝕏ri=1rχ(xi)et(x1++xr)dμq(x1)dμq(xr)=(a=0d1(1)d1aqaetaχ(a)edtqd+1)rj=0r(rj)qj=n=0En,χ,q(r)tnn!. Thus we give Witt-type formula of En,χ,q(r) as follows.

Theorem 4.7.

Let χ be a Dirichlet character with conductor d and let m0. Then En,χ,q(r)=𝕏r(x1++xr)mi=1rχ(xi)dμq(x1)dμq(xr).

By using (3.2), (2.8), we obtain En,q,χ(r)=fn(1+q)r(1+qf)ra1,a2,,ar=0f1(1)a1++arqa1++ark=1rχ(ak)k=0n(nk)(a1++arf)nkEk,qf(r). By using (4.7) in the above equation, we have En,q,χ(r)=(1+q)r(1+qf)ra1,a2,,ar=0f1(1)a1++arqa1++ark=1rχ(ak)×k=0nl1,l2,,lv0l1+l2++lv=nk(nkl1,l2,,lv)(nk)y=1vaylyfkEk,qf(r).

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.