# Multivariate Interpolation Functions  of Higher-Orderq-Euler Numbers and  Their Applications

Authors: Hacer Ozden; Ismail Naci Cangul; Yilmaz Simsek
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

The aim of this paper, firstly, is to construct generating functions ofq-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.


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## Body

## 1. Introduction, Definitions, and Notations

Letp 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=exp⁡(xlog⁡q) for |x|p≤1.For a fixed positive integerd with (p,d)=1, set
𝕏d=lim⁡N←ℤ/dpNℤ,𝕏1=ℤp,(1.1)𝕏∗=⋃0<a<dp(a,p)=1(a+dpℤp),a+dpNℤp={x∈𝕏:x≡a(moddpN)},
where a∈ℤ satisfies the condition 0≤a<dpN (cf. [1–28]).The distributionμq(a+dpNℤp)
is given as
(1.2)μq(a+dpNℤp)=qa[dpN]q
(cf. [4, 10]).We say thatf is a uniformly
differentiable function at a point a∈ℤp;
we write f∈UD(ℤp) if the
difference quotient
(1.3)Ff(x,y)=f(x)−f(y)x−y
has a limit f′(a) as (x,y)→(a,a).   Let f∈UD(ℤp). An invariant p-adic q-integral is defined by
(1.4)Iq(f)=∫ℤpf(x)dμq(x)=lim⁡N→∞1[pN]q∑x=0pN−1f(x)qx
(cf. [4, 5, 10, 29, 30]).Theq-extension
of n∈ℕ is defined by
(1.5)[n]q=1−qn1−q.
We note that lim⁡q→1[n]q=n.Classical Euler numbers are defined by means of the
following generating function:(1.6)2et+1=∑n=0∞Entnn!,
(cf. [1–3, 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:
(1.7)ζE(s)=∑n=1∞(−1)nns,s∈ℂ,
(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 [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:
(1.8)μ−q(a+dpNℤp)=(−q)a[dpN]−q,
where
(1.9)[n]−q=1−(−q)n1+q
(cf. [1, 31]).By using the fermionicp-adic q-measures,
he defined the fermionic p-adic q-integral
on ℤp as
follows:
(1.10)I−q(f)=∫ℤpf(x)dμ−q(x)=lim⁡N→∞1[pN]−q∑x=0pN−1f(x)(−q)x
(cf. [31]).Observe thatI−q(f) can be written
symbolically as
(1.11)lim⁡q→−qIq(f)=I−q(f)
(cf. [31]).By using fermionicp-adic q-integral
on ℤp, Kim et al. [8] defined the generating function of the q-Euler
numbers as follows:
(1.12)Fq(t)=q+1qet+1=∑n=0∞En,qtnn!,
where En,q denotes q-Euler
numbers.Witt's formula ofEn(x,q) was given by Kim et al. [8]:(1.13)En(x,q)=∫ℤp(x+y)ndμ−q(y),
where q∈ℂp and |1−q|p<1.In [16], Ozden and Simsek defined generating function of q-Euler numbers by
(1.14)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 Section2, 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 ofq-Euler Numbers and Polynomials

Forq∈ℂ with |q|<1,(2.1)Fq(t)=q+1qet+1=∑n=0∞En,qtnn!
(cf. [8]), where En,q denotes the q-Euler number and |t+log⁡q|<π.Observe that by (2.1) we
have
(2.2)q+1qet+1=1+q−1et+q−1=∑n=0∞Hn(−q−1)tnn!.
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]).Theq-Euler
polynomials are also defined by means of the following generating function
[8]:
(2.3)Fq(t,x)=q+1qet+1ext=∑n=0∞En,q(x)tnn!,
where |t+log⁡q|<π;(2.4)q+1qet+1ext=eEq(x)t,(q+1)ext=qe(1+Eq(x))t+eEq(x)t,(q+1)∑n=0∞xntnn!=∑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.
Letn be nonnegative
integer. Then (2.5)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
(2.6)En,q(x)+q∑k=0n(nk)Ek,q(x)=(q+1)xn.
From (2.3), by applying
Cauchy product and using (2.1), we also obtain
(2.7)(∑n=0∞En,qtnn!)(∑n=0∞xntnn!)=∑n=0∞(∑k=0∞(nk)xn−kEk,q)tnn!=∑n=0∞En,q(x)tnn!.
By comparing the coefficients of tn on both sides of the above equation, we have
(2.8)En,q(x)=∑k=0n(nk)xn−kEk,q
(cf. [8, 14]).By using Theorem2.1 and [8, equation (3)], we obtain
(2.9)En(x,q)=∫ℤp(x+y)ndμ−q(y)=lim⁡N→∞1[jpN]−q∑x=0jpN−1(x+y)n(−q)x=1[j]−qlim⁡N→∞1[pN](−q)j∑x=0j−1∑x=0jpN−1(a+jx+y)n(−q)a+jx=jn[j]−qlim⁡N→∞1[pN](−q)j∑a=0j−1(−q)a∑x=0pN−1(a+yj+x)n((−q)j)x=jn[j]−q∑a=0j−1(−q)alim⁡N→∞1[pN](−q)j∑x=0pN−1(a+yj+x)n((−q)j)x=jn[j]−q∑a=0j−1(−q)aEn(a+yj,qj).By using the above equation, we arrive at the following theorem.Theorem 2.2.
Letj be odd. Then (2.10)En,q(x)=(q+1)jnqj+1∑a=0j−1(−1)aqaEn,qj(a+xj).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;
(2.11)∑n=0∞En(x)tnn!=q+1qet+1ext=q+11+qjejt∑a=0j−1(−1)aqaeatext=(q+1)∑a=0j−1(−1)aqa(e(a+x)t1+qjejt)qj+1qj+1=∑n=0∞(q+1qj+1jn∑a=0d−1(−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 substitutingx=n, with n∈ℤ+ into (2.3), then we have
(2.12)Fq(t,n)=q+1qet+1ent=∑k=0∞Ek,q(n)tkk!.
Thus,(2.13)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=0n−1(−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
(2.14)Fq(t)−qn(−1)nFq(t,n)=(q+1)∑l=0n−1(−1)lqlelt.
By the generating function of q-Euler numbers and polynomials and by (2.14), we see that
(2.15)∑m=0∞(Em,q−qn(−1)nEm,q(n))tmm!=∑m=0∞((q+1)∑l=0n−1ql(−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 [14]).
Letn∈ℤ+. Then (2.16)Em,q−qn(−1)nEm,q(n)q+1=∑l=0n−1ql(−1)llm.Remark 2.4.
Proof of
Theorem2.3 is similar to that of [14]. If we take q→1 in (2.16), we have
(2.17)Em−(−1)nEm(n)2=∑l=0n−1(−1)llm.The above formula is well known in the number theory
and its applications.Remark 2.5.
Generating
function of theq-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 q-integral on ℤp; for applications of this integral and measure see
for detail [2, 4, 8, 14–19, 23, 25, 29, 30, 31].Now, we considerq-Euler numbers
and polynomials of higher order as follows:
(2.18)(q+1qet+1)(q+1qet+1)⋯(q+1qet+1)︸rtimes=(q+1qet+1)r=∑n=0∞En,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:
(2.19)(q+1qet+1)(q+1qet+1)⋯(q+1qet+1)etx=(q+1)retx(qet+1)r=∑n=0∞En,q(r)(x)tnn!,
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:
(2.20)Fq(r)(t,x)=q+1qet+1q+1qet+1⋯q+1qet+1︸rtimesetx=∑j=0r(rj)qjetx∑n1=0∞(−1)n1qn1en1t⋯∑nr=0∞(−1)nrqnrenrt=∑n1,n2,…,nr=0∞∑j=0r(rj)(−1)n1+⋯+nrqj+n1+⋯+nre(n1+⋯+nr+x)t=∑n=0∞En,q(r)(x)tnn!.By applying Mellin transformation to (2.20), we have
(2.21)1Γ(s)∫0∞ts−1Fq(r)(t,x)dt=∑n1,n2,…,nr=0∞∑j=0r(rj)1Γ(s)∫0∞ts(−1)n1+⋯+nrqj+n1+⋯+nre(n1+⋯+nr+x)tdt.
After some elementary calculations, we obtain
(2.22)1Γ(s)∫0∞ts−1Fq(r)(−t,x)dt=∑j=0r(rj)qj∑n1,n2,…,nr=0∞∑n1,n2,…,nr=0∞(−1)n1+⋯+nrqn1+⋯+nr(n1+⋯+nr+x)s=∑n1,n2,…,nr=0∞∑j=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.
Fors∈ℂ,x∈ℝ(0<x≤1),
one defines
(2.23)ζq,E(r)(s,x)=∑n1,n2,…,nr=0∞∑j=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
thekth derivative
operator dk/dtk|t=0 on both sides
of (2.20), we have
(2.24)En,q(r)(x)=dkdtkFqr(t,x)|t=0=∑n1,n2,...,nr=0∞∑j=0r(rj)(−1)n1+⋯+nrqj+n1+⋯+nr(n1+⋯+nr+x)k.
By using the above equation, Ryoo et al. [14] and Simsek [23] also define (2.23).By substitutings=−k,k∈ℤ+ into (2.23) and
using (2.24), after some calculations, we arrive at the following theorem.Theorem 2.8.
Letk∈ℤ+. Then (2.25)ζq,E(r)(−k,x)=En,q(r)(x).Observe that the functionζq,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
(2.26)1Γ(s)∮Cts−1Fq(r)(−t,x)dt=∑n=0∞(−1)nEn,q(r)(x)n!1Γ(s)∮Ctn+s−1dt,
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 [14]. ζq,E(r)(s,x) is an analytic
function in whole complex s-plane. For s∈C,(2.27)ζEr(s,x)=lim⁡q→1ζq,Er(s,x)=2r∑n1,n2,…,nr=0∞(−1)n1+⋯+nr(n1+⋯+nr+x)s.
If r=1 in the above
equation, we have
(2.28)ζE(s,x)=2∑n=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. [28].Letχ be Dirichlet's
character with conductor d∈ℤ+. The generalized q-Euler numbers attached to χ of higher order are defined by
(2.29)Fq,χ(t)=(q+1)∑a=1d(−1)aqaχ(a)etaqdedt+1=∑n=0∞En,χtnn!
(cf. [8]), where |t+log⁡d|<π. The q-Euler numbers
attached to χ of higher order
are defined by
(2.30)Fq,χ(r)(t)=((q+1)∑a=1d(−1)aqaχ(a)etaqdedt+1)r=∑n=0∞En,q,χ(r)tnn!.
From (2.30), we obtain
(2.31)Fq,χ(r)(t)=∑n1,n2,…,nr=1∞∑j=0r(rj)(−1)n1+⋯+nrqj+n1+⋯+nre(n1+⋯+nr)t∏k=1rχ(nk)=∑n=0∞En,q,χ(r)tnn!.
By applying the kth derivative
operator dk/dtk|t=0 in (2.31), we
have
(2.32)Ek,q,χ(r)=dkdtkFq,χ(r)(t)|t=0=∑n1,n2,…,nr=1∞∑j=0r(rj)(−1)n1+⋯+nrqj+n1+⋯+nr(n1+⋯+nr)k∏k=1rχ(nk).
By using (2.32), we define Dirichlet-type multiple
Euler q-l-function as
follows.Definition 2.10.
Lets∈ℂ;(2.33)lE,q(r)(s,χ)=∑n1,n2,…,nr=1∞∑j=0r(rj)(−1)n1+⋯+nrqj+n1+⋯+nr∏k=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,
(2.34)lE(r)(s,χ)=lim⁡q→1lE,q(r)(s,χ)=∑n1,n2,…,nr=1∞2(−1)n1+⋯+nr∏k=1rχ(nk)(n1+⋯+nr)s.
For r=1 in the above
equation, we have
(2.35)lE(s,χ)=∑n=1∞2(−1)nχ(n)ns.
This function is called Euler l-function.
Here, we observe that by applying Mellin
transformation to (2.31), we obtain
(2.36)1Γ(s)∫0∞ts−1Fq,χ(r)(−t)dt=lE,q(r)(s,χ).
This gives us another definition of (2.32).By substitutings=−k,k∈ℤ+ into (2.33) and
using (2.32), we arrive at the following theorem.Theorem 2.12.
Letk∈ℤ+. Then (2.37)lE,q(r)(−k,χ)=Ek,q,χ(r).We note that(2.38)lim⁡q→1lE,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. [28].
By using (2.26), (2.36),
we obtain another proof of
(2.37).


## 3. Relation betweenlE,q(r)(s,χ) and ζq,E(r)(s,x)

Substitutingnj=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,1≤j≤r
,
into (2.33), we have
(3.1)lE,q(r)(s,χ)=(1+q)r∑a1,a2,…,ar=1f∑m1,m2,…,mr=0∞(−1)a1+m1f+⋯+ar+mrfqa1+m1f+⋯+ar+mrf∏k=1rχ(ak+mkf)(a1+m1f+⋯+ar+mrf)s=(1+q)rf−s(1+qf)r∑a1,a2,…,ar=1f(−1)a1+⋯+arqa1+⋯+ar∏k=1rχ(ak)×(1+qf)r∑m1,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.
Letχ be a Dirichlet
character with conductor f(=odd). Then (3.2)lE,q(r)(s,χ)=(1+q)rf−s(1+qf)r∑a1,a2,…,ar=1f(−1)a1+⋯+arqa1+⋯+ar∏k=1rχ(ak)ζqf,E(r)(s,a1+⋯+arf).By substitutings=−k,k∈ℤ, into (3.2), we obtain
(3.3)lE,q(r)(−k,χ)=(1+q)rfk(1+qf)r∑a1,a2,…,ar=1f(−1)a1+⋯+arqa1+⋯+ar∏k=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:(3.4)Ek,q,χ(r)=(1+q)rfk(1+qf)r∑a1,a2,…,ar=1f(−1)a1+⋯+arqa1+⋯+ar∏k=1rχ(ak)En,qf(r)(a1+⋯+arf).


## 4. Multivariatep-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
(4.1)I−q(f)=∫ℤpf(x)dμ−q(x)=lim⁡N→∞1[pN]−q∑x=0pN−1f(x)(−q)x.
From this integral equation, we have (see [1, 2, 4])
(4.2)qI−q(f1)+I−q(f)=(q+1)f(0),
where f1(x)=f(x+1).  If we take f(x)=etx in (4.2), we
have
(4.3)I−q(etx)=∫ℤpetxdμ−q(x)=q+1qet+1=∑n=0∞En,qtnn!
(cf. [8]).Now we are ready to give multivariatep-adic fermionic q-integral on ℤp as follows (see
for detail [14]). Let
(4.4)∫ℤpr=∫ℤp∫ℤp⋯∫ℤp︸rtimes∫ℤpret(x1+⋯+xr)dμ−q(x1)⋯dμ−q(xr)=(q+1qet+1)⋯(q+1qet+1)︸rtimes=∑n=0∞En,q(r)tnn!.
From (4.4), we obtain Witt's formula for q-Euler numbers
of higher order as follows.Theorem 4.1 (see [14]).
Letk∈ℤ+. Then (4.5)∫ℤpr(x1+⋯+xr)ndμ−q(x1)⋯dμ−q(xr)=En,q(r).By (4.4), we obtain
(4.6)∫ℤpret(x1+⋯+xr+x)dμ−q(x1)⋯dμ−q(xr)=ext(q+1)r(qet+1)⋯(qet+1)︸rtimes=∑n=0∞En,q(r)(x)tnn!.Theorem 4.2 (multinomial theorem).
The
following holds:(4.7)(∑j=1vxj)n=∑l1,l2,…,lv≥0l1+l2+⋯+lv=n(nl1,l2,…,lv)∏a=1vxala,
where (nl1,l2,…,lv) are the
multinomial coefficients, which are defined by
(4.8)(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 ofq-Euler polynomials of higher order.Theorem 4.3.
For positive integersn, r,  one has
(4.9)En,q(r)(y1+y2+⋯+yr)=∑l1,l2,…,lr≥0l1+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 [23]. By using Taylor series of etx into (4.6), and x by y1+y2+⋯+yr, then we have
(4.10)En,q(r)(y1+y2+⋯+yr)=∫ℤp⋯∫ℤp(∑j=1r(yj+xj))n∏j=1rdμ−q(xj).
By using (4.7) in the above
equation, and after some elementary calculations,
we get
(4.11)En,q(r)(y1+y2+⋯+yr)=∑l1,l2,…,lr≥0l1+l2+⋯+lr=n(nl1,l2,…,lv)∏j=1r∫ℤp(yj+xj)ljdμ−q(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
integersn,r,
one has
(4.12)En,q(r)(y1+y2+⋯+yr)=∑l1,l2,…,lr≥0l1+l2+⋯+lr=n(nl1,l2,…,lr)∏j=1r∑k=0lj(ljk)yjlj−kElj,q.In (4.10), if we replace y1+y2+⋯+yr by x, then we obtain the following corollary.Corollary 4.5.
Forn≥0,
one has
(4.13)En,q(r)(x)=∫ℤpr(x1+⋯+xr+x)mdμ−q(x1)⋯dμ−q(xr)=∑l1+⋯+lr+lr+1=m(ml1⋯lr+1)∫ℤpx1l1dμ−q(x1)⋯∫ℤpxrlrdμ−q(xr)xlr+1=∑l1+⋯+lr+lr+1=m(ml1⋯lr+1)El1,qEl2,q⋯Elr,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. [14], which is given by
(4.14)En,q(r)(x)=∑k=0n∑l1+⋯+lr+lr+1=m(nk)(kl1⋯lr+1)xn−kEl1,qEl2,q⋯Elr,q.
In (4.13), if we take q→1, we have
(4.15)En(r)(x)=∑l1+⋯+lr+lr+1=m(ml1⋯lr+1)El1El2⋯Elrxlr+1.
For more detailed information about
complete sums of products of Euler polynomials and Bernoulli
polynomials, see also [11, 14, 20–24, 34, 35].Letχ be a Dirichlet
character with conductor d∈ℤ+. Then
(4.16)∫𝕏χ(x)etxdμ−q(x)=(q+1)∑l=0d−1(−1)d−1−lqletlχ(x)edtqd+1=∑n=0∞En,χ,qtnn!.By using Taylor expansion ofetx and then
comparing coefficients of tn on both sides,
we arrive at
(4.17)∫𝕏χ(x)xndμ−q(x)=En,χ,q
(cf. [8]).By (4.16), we have
(4.18)∫𝕏r∏i=1rχ(xi)et(x1+⋯+xr)dμ−q(x1)⋯dμ−q(xr)=∫𝕏r∏i=1rχ(xi)et(x1+⋯+xr)dμ−q(x1)⋯dμ−q(xr)=(∑a=0d−1(−1)d−1−aqaetaχ(a)edtqd+1)r∑j=0r(rj)qj=∑n=0∞En,χ,q(r)tnn!.
Thus we give Witt-type formula of En,χ,q(r) as follows.Theorem 4.7.
Letχ be a Dirichlet character with conductor d∈ℕ and let m≥0. Then (4.19)En,χ,q(r)=∫𝕏r(x1+⋯+xr)m∏i=1rχ(xi)dμ−q(x1)⋯dμ−q(xr).By using (3.2), (2.8), we obtain
(4.20)En,q,χ(r)=fn(1+q)r(1+qf)r∑a1,a2,…,ar=0f−1(−1)a1+⋯+arqa1+⋯+ar∏k=1rχ(ak)∑k=0n(nk)(a1+⋯+arf)n−kEk,qf(r).
By using (4.7) in the above
equation, we have
(4.21)En,q,χ(r)=(1+q)r(1+qf)r∑a1,a2,…,ar=0f−1(−1)a1+⋯+arqa1+⋯+ar∏k=1rχ(ak)×∑k=0n∑l1,l2,…,lv≥0l1+l2+⋯+lv=n−k(n−kl1,l2,…,lv)(nk)∏y=1vaylyfkEk,qf(r).



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*Source: 390857-2008-03-18.xml*