# On Multiple Twistedp-adic q-Euler ζ-Functions and l-Functions

Authors: Min-Soo Kim; Taekyun Kim; Jin-Woo Son
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

We give the existence of multiple twistedp-adic q-Euler ζ-functions
and l-functions, which are generalization of the twisted p-adic (h,q)-zeta functions
and twisted p-adic (h,q)-Euler l-functions in the work of Ozden and Simsek (2008).


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## Body

## 1. Introduction, Definitions, and Notations

The constantsEn in the Taylor series expansion(1.1)2et+1=∑n=0∞Entnn!are known as the first kind
Euler numbers (cf. [1]). From the generating function of the first kind
Euler numbers, we note that E0=1 and En=−∑k=0n(nk)Ek for n∈ℕ. The first few are 1,−1/2,0,1/4,−1/2,… and E2k=0 for k∈ℕ. Those numbers play an important role in number
theory. For example, the Euler zeta-function essentially equals an Euler
numbers at nonpositive integer:(1.2)ζE(−m)=Emform≥0,where(1.3)ζE(s)=∑n=1∞(−1)nns,s∈ℂ(see [1–10]).Throughout this paperℤ,ℤp,ℚp,
and ℂp will denote the ring of integers, the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of
the algebraic closure of ℚp, respectively. Let vp be the normalized exponential valuation of ℂp with |p|p=1/p. When one talks of q-extension, q is variously considered as an indeterminate, a
complex number q∈ℂ, or a p-adic number q∈ℂp. If q∈ℂp, then we normally assume |1−q|p<1, so that qx=exp(xlogq) for x∈ℤp. If q∈ℂ, then we assume that |q|<1. Also we use the following notations:(1.4)[x]q=1−qx1−q,[x]−q=1−(−q)x1+q,cf. [2–4]. For(1.5)f∈UD(ℤp,ℂp)={f∣f:ℤp→ℂpis  uniformly  differentiable  function},the p-adic q-integral on ℤp was defined by Kim (cf. [2–4]) as follows:(1.6)Iq(f)=∫ℤpf(a)dμq(a)=limN→∞1[dpN]q∑a=0dpN−1f(a)qafor|1−q|p<1.Furthermore, we can
consider the fermionic integral in contrast to the conventional bosonic
integral. That is, I−1(f)=∫ℤpf(a)dμ−1(a) (cf. [5]). From this, we derive(1.7)I−1(f1)+I−1(f)=2f(0),where f1(a)=f(a+1). Substitute f(a)=ξaqαaeat into (1.7). The twisted (α,q)-extension of Euler numbers is defined by
[8](1.8)I−1(ξaqαaeat)=2ξqαet+1=∑n=0∞En,ξ(α)(q)tnn!.For |1−q|p<1, we consider fermionic p-adic q-integral on ℤp which is the q-extension of I−1(f) as follows:(1.9)I−q(f)=∫ℤpf(a)dμ−q(a)=limN→∞1[dpN]−q∑a=0dpN−1f(a)(−q)a(cf. [5]). From (1.9), we can derive the
following formula [5]:(1.10)qI−q(f1)+I−q(f)=[2]qf(0),where f1(a) is translation with f1(a)=f(a+1). If we take f(a)=ξaeat, then we have f1(a)=ξa+1e(a+1)t=ξaeatξet. From (1.10), we derive (ξqet+1)I−q(ξaeat)=[2]q. Hence, we obtain(1.11)I−q(ξaeat)=∫ℤpξaeatdμ−q(a)=[2]qξqet+1.By (1.11), we define the twisted q-Euler numbers, En,q,ξ by means of the following generating function
(cf. [5]):(1.12)[2]qξqet+1=∑n=0∞En,q,ξtnn!.These numbers are interpolated
by the twisted Euler q-zeta function which is defined as
follows:(1.13)ζq,ξ,E(s)=[2]q∑n=1∞(−1)nξnqnns,s∈ℂ.Note that ζq,ξ,E(s) is analytic function in the whole complex
plane ℂ.In view of the functional equation for the twisted
Eulerq-zeta function at nonpositive integers, we
have(1.14)ζq,ξ,E(−m)=Em,q,ξform≥0(cf. [5]).Twistedq-Bernoulli and Euler numbers and polynomials
are very important not only in practically every field of mathematics, in
particular in combinatorial theory, finite difference calculus, numerical
analysis, numbers theory, but also probability theory. Recently the q-extensions of those Euler numbers
(polynomials) and the multiple of q-extensions of those Euler numbers
(polynomials)   have been studied by many authors, (cf. [1–15]). In [8], Ozden and Simsek have studied (h,q)-extensions of twisted Euler numbers and
polynomials by using p-adic q-integral on the ring of p-adic integers ℤp. From their (h,q)-extensions of twisted Euler numbers and
polynomials, they have derived p-adic (h,q)-extensions of Euler zeta function and p-adic (h,q)-extensions of Euler l-functions. They also gave some interesting
relations between their (h,q)-Euler numbers and (h,q)-Euler zeta functions, and found the p-adic twisted interpolation function of the
generalized twisted (h,q)-Euler numbers. In [11], Jang defined twisted q-Euler numbers and polynomials of higher
order, and studied multiple twisted q-Euler zeta functions. He also derived sums of
products of q-Euler numbers and polynomials by using
fermionic p-adic q-integral. In [7, 9], Ozden et al. defined
multivariate Barnes-type Hurwitz q-Euler zeta functions and l-functions. They also gave relation between
multivariate Barnes-type Hurwitz q-Euler zeta functions and multivariate q-Euler l-functions. In [16], Kim constructed multiple p-adic L-functions, which interpolate the Bernoulli numbers
of higher order. He also derived that the values of the partial derivative of
this multiple p-adic L-function at s=0 are given.In this paper, we consider twistedq-Euler numbers and polynomials of higher
order, and study multiple twisted p-adic, q-Euler, ζ-functions, and l-functions, which are generalization of the
twisted p-adic (h,q)-zeta functions and twisted p-adic (h,q)-Euler l-functions in [8].


## 2. Preliminaries

We assume thatq∈ℂ with |q|<1. Let ξ be a primitive r th root of unity.For an integerh, the twisted q-Euler polynomials of higher order (the index h may be negative), En,q,ξ(h)(x), are defined by means of the following
generating function (cf. [11, 14]):(2.1)Fq,ξ(h)(t,x)=[2]q1+ξqet⋯[2]q1+ξqet︸h−timesext=[2]qhetx∑l1=0∞(−ξ)l1ql1el1t⋯∑lh=0∞(−ξ)lhqlhelht=[2]qh∑l1,…,lh=0∞(−ξq)l1+⋯+lhe(l1+⋯+lh+x)t=∑n=0∞En,q,ξ(h)(x)tnn!,where |t+log(ξq)|<π. Note that [2]q=1+q, so [2]q/(1+ξqet)≡(1+q)/(1+ξq)(modt). Of course the explicit formulas in (2.1)
depend on h which is a positive integer. If h=1,q=qα in the above, we obtain the generating
function of the twisted (α,q)-extension of Euler polynomials in [8, cf. Section 1, (1.3)].
In fact, if h>0 then −h<0. Therefore, the generating function Fq,ξ(−h)(t,x) is the form(2.2)Fq,ξ(−h)(t,x)=([2]q1+ξqet)−hetx=(1+ξqet[2]q)hetx=∑n=0∞En,q,ξ(−h)(x)tnn!.The twisted q-Euler numbers of higher order are En,q,ξ(h)=En,q,ξ(h)(0). Then, it is immediate that(2.3)En,q,ξ(h)(x)=∑k=0n(nk)Ek,q,ξ(h)xn−k.From now on, we assumeh>0 and in general whenever h is actually an index then h>0. Jang [11] defined the two-variable multiple twisted q-Euler zeta functions as follows.
Definition 2.1.
Fors∈ℂ and x∈ℝ+={x∈ℝ∣x>0}, one
defines(2.4)ζq,ξ,E(h)(s,x)=[2]qh∑l1,…,lh=0∞(−ξ)l1+⋯+lhql1+⋯+lh(l1+⋯+lh+x)s.ζq,ξ,E(h)(s,x) is an analytical function in the whole complex
plane.The value ofζq,ξ,E(h)(s,x) at nonpositive integers, ℤ+=ℕ∪{0}, is given explicitly as follows.
Theorem 2.2 (see [11]).
Letm∈ℤ+. Then, ζq,ξ,E(h)(−m,x)=Em,q,ξ(h)(x).Letχ be a Dirichlet character with odd conductor d. We define a twisted Dirichlet's type q-Euler polynomials of higher order by means of
the following generating function (cf. [11, 14]):(2.5)Fq,ξ,χ(h)(t,x)=1[d]−qh∑a1,…,ah=0d−1χ(a1+⋯+ah)(−ξq)a1+⋯+ahe(a1+⋯+ah)t1+qd1+ξdqdedt⋯1+qd1+ξdqdedt︸h−timesext=[2]qh∑a1,…,ah=0d−1χ(a1+⋯+ah)(−ξq)a1+⋯+ahe(a1+⋯+ah)t∑x1=0∞(−ξdqdedt)x1⋯∑xh=0∞(−ξdqdedt)xh︸h−timesext=[2]qh∑x1,…,xh=0∞∑a1,…,ah=0d−1χ(a1+dx1+⋯+ah+dxh)(−ξq)a1+dx1+⋯+ah+dxhe(a1+dx1+⋯+ah+dxh)text=[2]qh∑l1,…,lh=0∞χ(l1+⋯+lh)(−ξq)l1+⋯+lhe(x+l1+⋯+lh)t=∑n=0∞En,q,ξ,χ(h)(x)tnn!.We now see that the twisted Dirichlet's typeq-Euler polynomials of higher order are easily
expressed by the twisted q-Euler polynomials of higher order as
follows.
Proposition 2.3.
LetF be an odd multiple of the conductor d. Then,(2.6)En,q,ξ,χ(h)(x)=Fn1[F]−qh∑a1,…,ah=0F−1χ(a1+⋯+ah)(−ξq)a1+⋯+ahEn,qF,ξF(h)(a1+⋯+ah+xF).Proof.
Letd(=odd)∈ℕ. By (2.1) and (2.5), we note that(2.7)Fq,ξ,χ(h)(t,x)=1[d]−qh∑a1,…,ah=0d−1χ(a1+⋯+ah)(−ξq)a1+⋯+ahFqd,ξd(h)(dt,a1+⋯+ah+xd).Then, we have(2.8)En,q,ξ,χ(h)(x)=dn1[d]−qh∑a1,…,ah=0d−1χ(a1+⋯+ah)(−ξq)a1+⋯+ahEn,qd,ξd(h)(a1+⋯+ah+xd).On the other hand, if F=dp, then we get(2.9)1[F]−qh∑a1,…,ah=0F−1χ(a1+⋯+ah)(−ξq)a1+⋯+ahFqF,ξF(h)(Ft,a1+⋯+ah+xF)=∑a1,…,ah=0F−1χ(a1+⋯+ah)(−ξq)a1+⋯+ah([2]qξFqFeFt+1)he(a1+⋯+ah+x)t=∑b1,…,bh=0d−1∑c1,…,ch=0p−1χ(b1+c1d+⋯+bh+chd)(−ξq)b1+c1d+⋯+bh+chd×([2]qξFqFeFt+1)he(b1+c1d+⋯+bh+chd+x)t=1[d]−qh∑b1,…,bh=0d−1χ(b1+⋯+bh)(−ξq)b1+⋯+bhFqd,ξd(h)(dt,b1+⋯+bh+xd).This completes the proof.The two-variable multiple twistedq-Euler l-functions are defined by the following definition.
Definition 2.4 (see [14]).
Letχ be a Dirichlet character. For s∈ℂ and x∈ℝ+, one has(2.10)lq,ξ,E(h)(s,x,χ)=[2]qh∑l1,…,lh=0∞χ(l1+⋯+lh)∏i=1h(−1)liξliqli(l1+⋯+lh+x)s.The value oflq,ξ,E(h)(s,x,χ) at nonpositive integers is given explicitly by the following theorem.
Theorem 2.5 (see [14]).
Letm∈ℤ+. Then lq,ξ,E(h)(−m,x,χ)=Em,q,ξ,χ(h)(x).Proof.
Letχ be a Dirichlet character with odd conductor d and let F be an odd number of multiple d. Set s∈ℂ and x∈ℝ+. Beside the multiple twisted q-Euler l-function lq,ξ,E(h)(s,x,χ), we consider the multiple twisted q-Euler zeta function ζq,ξ,E(h)(s,x) in Definition 2.1. Then(2.11)lq,ξ,E(h)(s,x,χ)=F−s1[F]−qh∑a1,…,ah=0F−1χ(a1+⋯+ah)(−ξq)a1+⋯+ahζqF,ξF,E(h)(s,a1+⋯+ah+xF)(cf. [14]).
In the integral forΓ(s), we make the change of variable y=(x+l1+⋯+lh)t, where l1,…,lh≥0, to obtain(2.12)Γ(s)=∫0∞e−yysdyy=(x+l1+⋯+lh)s∫0∞e−(x+l1+⋯+lh)ttsdtt,or(2.13)(x+l1+⋯+lh)−sΓ(s)=∫0∞e−(x+l1+⋯+lh)ttsdtt.Summing over all l1,…,lh≥0, we find(2.14)[2]qh∑l1,…,lh=0∞(−ξq)l1+⋯+lh(x+l1+⋯+lh)−sΓ(s)=[2]qh∑l1,…,lh=0∞(−ξq)l1+⋯+lh∫0∞e−(x+l1+⋯+lh)ttsdtt.This gives us(2.15)ζq,ξ,E(h)(s,x)Γ(s)=[2]qh∑l1,…,lh=0∞(−ξq)l1+⋯+lh∫0∞e−(x+l1+⋯+lh)ttsdtt=∫0∞ts[2]q1+ξqe−t⋯[2]q1+ξqe−t︸h−timese−xtdtt=∫0∞tsFq,ξ(h)(−t,x)dtt.If we divide the infinite
integral into two parts:(2.16)∫0∞tsFq,ξ(h)(−t,x)dtt=∫01tsFq,ξ(h)(−t,x)dtt+∫1∞tsFq,ξ(h)(−t,x)dtt,it is easily seen that the
second term is an entire function on t.
Consider∫01tsFq,ξ(h)(−t,x)(dt/t). By the definition of En,q,ξ(h)(x), we have(2.17)Fq,ξ(h)(−t,x)=∑n=0∞En,q,ξ(h)(x)(−1)ntnn!=([2]q1+ξqe−t)he−xt.Therefore,(2.18)∫01tsFq,ξ(h)(−t,x)dtt=∑n=0∞En,q,ξ(h)(x)n!(−1)n∫01ts+n−1dt=∑n=0∞En,q,ξ(h)(x)n!(−1)ns+n.This has an analytic
continuation to a meromorphic function s in the entire complex plane. It is holomorphic
except at s=0,−1,−2,…, where it has a pole of order 1. Note that Γ(s) is holomorphic except at s=0,−1,−2,…, where it has a pole of order 1. Γ(s) does not have a zero. Therefore, ζq,ξ,E(h)(s,x) has an analytic continuation to the whole
complex plane. For an integer m∈ℤ+,
we have(2.19)lims→−m(s+m)(ζq,ξ,E(h)(s,x)Γ(s))=Em,q,ξ(h)(x)m!(−1)m.If m∈ℤ+, we have lims→−m(s+m)Γ(s)=(−1)m(1/m!), and thus we obtain(2.20)ζq,ξ,E(h)(−m,x)=Em,q,ξ(h)(x).Consequently, by using
Propositions 2.3 and (2.6) and the above equation, we have(2.21)lq,ξ,E(h)(−m,x,χ)=Fm1[F]−qh∑a1,…,ah=0F−1χ(a1+⋯+ah)(−ξq)a1+⋯+ahEm,qF,ξF(h)(a1+⋯+ah+xF)=Em,q,ξ,χ(h)(x).Therefore, we obtain another
proof of Theorem 2.5.Remark 2.6 (see [11, 14]).
We put(2.22)D=(ddt).Let m∈ℤ+, and let x∈ℝ+. From (2.1) and (2.4), we obtain the following:(2.23)Em,q,ξ(h)(x)=DmFq,ξ(h)(t,x)|t=0=[2]qh∑l1,…,lk=0∞(−ξq)l1+⋯+lh(x+l1+⋯+lh)m=ζq,ξ,E(h)(−m,x).Similarly, by (2.5) and (2.11),
we have(2.24)Em,q,ξ,χ(h)(x)=DmFq,ξ,χ(h)(t,x)|t=0=[2]qh∑l1,…,lh=0∞χ(l1+⋯+lh)∏i=1h(−1)liξliqli(x+l1+⋯+lh)m=lq,ξ,E(h)(−m,x,χ).


## 3. Partial Multiple Twistedq-Euler ζ-Functions

Lets∈ℂ and ai,F∈ℤ with F as an odd integer and 0<ai<F, where i=1,…,h. Then, partial multiple twisted q-Euler ζ-functions are as follows (cf. [14, 16, 18–20]):(3.1)Hq,ξ,E(h)(s,a1,…,ah,x∣F)=[2]qh∑l1,…,lh=0li≡ai(modF),i=1,…,h∞(−ξq)l1+⋯+lh(l1+⋯+lh+x)s.We give a relationship between Hq,ξ,E(h)(s,a1,…,ah,x∣F) and ζqF,ξF,E(h)(s,x) as follows. For i=1,…,h, substituting li=ai+niF with F as an odd into (3.1), we have(3.2)Hq,ξ,E(h)(s,a1,…,ah,x∣F)=[2]qh∑n1,…,nh=0∞(−ξq)a1+n1F+⋯+ah+nhF(a1+n1F+⋯+ah+nhF+x)s=1[F]−qh(−ξq)a1+⋯+ahFs[2]qFh∑n1,…,nh=0∞(−ξFqF)n1+⋯+nh(n1+⋯+nh+(a1+⋯+ah+x)/F)s=1[F]−qh(−ξq)a1+⋯+ahFsζqF,ξF,E(h)(s,a1+⋯+ah+xF).By using (2.3) and Theorem 2.2 and substituting s=−m,m∈ℤ+ in the above, we arrive at the following
theorem.Theorem 3.1.
LetF be an odd integer, s∈ℂ and let x∈ℝ+. Then(3.3)Hq,ξ,E(h)(s,a1,…,ah,x∣F)=1[F]−qh(−ξq)a1+⋯+ahFsζqF,ξF,E(h)(s,a1+⋯+ah+xF).In particular, if m∈ℤ+, then(3.4)Hq,ξ,E(h)(−m,a1,…,ah,x∣F)=1[F]−qh(−ξq)a1+⋯+ah(a1+⋯+ah+x)m∑k=0m(mk)(Fa1+⋯+ah+x)kEk,qF,ξF(h).By using Theorem3.1 and (2.11), we arrive at the
following theorem.Theorem 3.2.
Letχ be a Dirichlet character with conductor d and F as an odd multiple of d. Then,(3.5)lq,ξ,E(h)(s,x,χ)=∑a1,…,ah=0F−1χ(a1+⋯+ah)Hq,ξ,E(h)(s,x,a1,…,ah,x∣F),where s∈ℂ and x∈ℝ+.


## 4. Multiple Twistedp-Adic q-Euler l-Functions

Letp be an odd prime. ℤp,ℚp and ℂp will always denote, respectively, the ring of p-adic integers, the field of p-adic numbers, and the completion of the
algebraic closure of ℚp. Let vp:ℂp→ℚ∪{∞} (ℚ the field of rational numbers) denote the p-adic valuation of ℂp normalized so that vp(p)=1. The absolute value on ℂp will be denoted as |⋅|p, and |x|p=p−vp(x) for x∈ℂp. We let ℤp×={x∈ℤp∣1/x∈ℤp}. A p-adic integer in ℤp× is sometimes called a p-adic unit. For each integer N≥0,CpN will denote the multiplicative group of the
primitive pN-th roots of unity in ℂp×=ℂp∖{0}. Set(4.1)Tp={ξ∈ℂp∣ξpN=1,forsomeN≥0}=⋃N≥0CpN.The dual of ℤp, in the sense of p-adic Pontrjagin duality, is Tp=Cp∞, the direct limit (under inclusion) of cyclic
groups CpN of order pN(N≥0), with the discrete topology.When one talks ofq-extension, q is variously considered as an indeterminate, a
complex number q∈ℂ, or a p-adic number q∈ℂp. If q∈ℂp, then we normally assume |1−q|p<1.We will consider thep-adic analogue of the lq,ξ,E(h)-functions which are introduced in the
previous section. In order to consider p-adic and complex lq,ξ,E(h)-functions simultaneously, we will use an
isomorphism, σ, between the algebraic closure of the rational
numbers in ℂp and the algebraic closure of the rational
numbers within the complex numbers ℂ. Our purpose is to discuss the values of lq,ξ,E(h)-functions, so we will consider σ as fixed throughout this section and use σ to identify p-adic algebraic numbers with complex algebraic
numbers. We will write x=y, when x∈ℂp,y∈ℂ and y=σ(x).Letω be denoted as the Teichmüller character having
conductor p. For an arbitrary character χ, let χn=χω−n, where n∈ℤ, in sense of the product of characters. We put(4.2)〈a〉=ω−1(a)a=aω(a),whenever (a,p)=1. We then have 〈a〉≡1(modpℤp) for these values of a. Note that we extend this notation by
defining(4.3)〈a+pt〉=ω−1(a)(a+pt)for all a∈ℤ with (a,p)=1, and t∈ℂp such that |t|p≤1. Thus, 〈a+pt〉=〈a〉+pω−1(a)t, so that 〈a+pt〉≡1(modpℤp[t]) (cf. [21, 22]).The significance of Theorem3.1 lies in the fact that
the right-hand side is essentially a liner combination of terms of the form(4.4)∑k=0m(mk)(Fa1+⋯+ah+pt)kEk,qF,ξF(h),which makes sense when m is replaced by a p-adic variable and p|F. Set(4.5)D={s∈ℂp∣|s|p≤p(p−2)/(p−1)},(cf. [8, 16, 18–22]). Let F be an odd integer, and let (a1+⋯+ah,p)=1,ai∈ℤ with 0<ai<F for i=1,…,h. Suppose that s∈D and t∈ℂp with |t|p≤1. We apply [18, Proposition 5.8, page 53]  to the
series(4.6)∑k=0∞(sk)(Fa1+⋯+ah+pt)kEk,qF,ξF(h).Let q∈1+Mp where Mp={z∈ℂp∣|z|<1}. In [13], we see that(4.7)|Ek,qF,ξF(h)|p≤1,since ξ∈Tp. Observe that we have for odd p|F,(4.8)|(Fa1+⋯+ah+pt)kEk,qF,ξF(h)|p≤p−k=(1p)k,so that we can take r=1/p and M=1 in [18, Proposition 5.8]. This prove that (4.6) is analytic
in D. Note that 〈a1+⋯+ah+pt〉−s is analytic in D for (a1+⋯+ah,p)=1 and t∈ℂp such that |t|p≤1.Definition 4.1.
LetF be an odd integer, and let (a1+⋯+ah,p)=1,ai∈ℤ with 0<ai<F for i=1,…,h. Suppose that s∈D and t∈ℂp with |t|p≤1. One defines the partial multiple twisted p-adic q-Euler ζ-functions for p∣F:  (4.9)Hp,q,ξ,E(h)(s,a1,…,ah,pt∣F)=(−ξq)a1+⋯+ah[F]−qh〈a1+⋯+ah+pt〉−s∑k=0∞(−sk)(Fa1+⋯+ah+pt)kEk,qF,ξF(h).Theorem 4.2.
LetF be an odd integer with p∣F, and let (a1+⋯+ah,p)=1,ai∈ℤ with 0<ai<F for i=1,…,h. Suppose that s∈D and t∈ℂp with |t|p≤1. Then Hp,q,ξ,E(h)(s,a1,…,ah,pt∣F) is a p-adic analytic function on D such that(4.10)Hp,q,ξ,E(h)(−m,a1,…,ah,pt∣F)=ω−m(a1+⋯+ah)Hq,ξ,E(h)(−m,a1,…,ah,pt∣F)for m∈ℤ+. In particular, if m≡0(modp−1), then(4.11)Hp,q,ξ,E(h)(−m,a1,…,ah,pt∣F)=Hq,ξ,E(h)(−m,a1,…,ah,pt∣F).Proof.
We have already remarked that〈a1+⋯+ah+pt〉−s is analytic in D for (a1+⋯+ah,p)=1 and t∈ℂp such that |t|p≤1. Also we see that (4.6) is analytic in D. It is clear that Hp,q,ξ,E(h) is a p-adic analytic function on D. For s=−m,m∈ℤ+ one has(4.12)Hp,q,ξ,E(h)(−m,a1,…,ah,pt∣F)=(−ξq)a1+⋯+ah[F]−qh(a1+⋯+ah+ptω(a1+⋯+ah))m∑k=0m(mk)(Fa1+⋯+ah+pt)kEk,qF,ξF(h)(wherewe useTheorem3.1)=ω−m(a1+⋯+ah)Hq,ξ,E(h)(−m,a1,…,ah,pt∣F).This completes the proof.Definition 4.3.
Letχ be a Dirichlet character with odd conductor d, and let F be a positive multiple of p and d. We can define the multiple twisted p-adic q-Euler l-function:(4.13)lp,q,ξ,E(h)(s,t,χ)=∑a1,…,ah=0(a1+⋯+ah,p)=1F−1χ(a1+⋯+ah)Hp,q,ξ,E(h)(s,a1,…,ah,pt∣F).Theorem 4.4.
Letχ be a Dirichlet's character with an odd
conductor d, and let F be a positive multiple of p and d. Then lq,ξ,E(h)(s,t,χ) is a p-adic analytic function on D with(4.14)lp,q,ξ,E(h)(−m,t,χ)=Em,q,ξ,χm(h)(pt)−Fm[F]−qhχm(p)∑β∈I0χm(β)(−ξpqp)βEm,qF,ξF(h)(β+tF/p),where m∈ℤ+ and(4.15)I0={1px|x=a1+⋯+ah≡0(modp)forsomea1,…,ahwith0≤a1,…,ah≤F−1}, and in ∑β∈I0 one sums over β=(1/p)x as many times as x is expressed in the form x=a1+⋯+ah by various aj's, and χm=χω−m with ω the Teichmüller in the sense of the product of
characters.Remark 4.5.
Theorem4.4 can be extended to
obtain similar results for the multiple p-adic L-function in [16]. In the case h=1, we note that(4.16)∑β∈I0=∑a=0F/p−1.Observe that if h=1, then(4.17)limh→1lp,q,ξ,E(h)(−m,t,χ)=Em,q,ξ,χm(pt)−Fm[F]−qχm(p)∑a=0F/p−1χm(a)(−ξpqp)aEm,qF,ξF(a+tF/p)=Em,q,ξ,χm(pt)−pmχm(p)[F/p]−qp[F]−q(Fp)m1[F/p]−qp∑a=0F/p−1χm(a)(−ξpqp)aEm,(qp)F/p,(ξp)F/p(a+tF/p)(where we useProposition2.3)=Em,q,ξ,χm(pt)−pm1[p]−qχm(p)Em,qp,ξp,χm(t).This function interpolates the
twisted generalized q-Euler polynomials at negative integers. For lp,q,ξ,E(1)(−m,t,χ), the twisted p-adic q-Euler l-functions similar results were obtained (cf.
see for detail [8,
Theorem 9]). If q→1 in the above, then(4.18)limq→1lp,q,ξ,E(1)(s,t,χ)=lp,ξ,E(s,t,χ)=2∑l=0(l,p)=1∞χ(l)(−1)lξl(t+l)s,|t|p≤1,which is called the twisted p-adic lE-function of two variables.Proof.
The formula forlq,ξ,E(h)(s,x,χ) is p-adic analytic function in D by the Theorem 4.2. On the other hand, by
substituting s=−m,m∈ℤ+, into Definition 4.3, we have(4.19)lp,q,ξ,E(h)(−m,t,χ)=∑a1,…,ah=0(a1+⋯+ah,p)=1F−1χ(a1+⋯+ah)Hp,q,ξ,E(h)(−m,a1,…,ah,pt∣F)=∑a1,…,ah=0F−1χ(a1+⋯+ah)Hp,q,ξ,E(h)(−m,a1,…,ah,pt∣F)−∑a1,…,ah=0(a1+⋯+ah,p)≠1F−1χ(a1+⋯+ah)Hp,q,ξ,E(h)(−m,a1,…,ah,pt∣F).From Theorems 2.2, 3.1,
and 4.2, we obtain(4.20)∑a1,…,ah=0(a1+⋯+ah,p)≠1F−1χ(a1+⋯+ah)Hp,q,ξ,E(h)(−m,a1,…,ah,pt∣F)=∑a1,…,ah=0(a1+⋯+ah,p)≠1F−1χm(a1+⋯+ah)(−ξq)a1+⋯+ahFm[F]−qhEm,qF,ξF(h)(a1+⋯+ah+ptF)=Fm[F]−qhχm(p)∑β∈I0χm(β)(−ξpqp)βEm,qF,ξF(h)(β+tF/p).Therefore,(4.21)lp,q,ξ,E(h)(−m,t,χ)=Em,q,ξ,χm(h)(pt)−Fm[F]−qhχm(p)∑β∈I0χm(β)(−ξpqp)βEm,qF,ξF(h)(β+tF/p).This completes the proof.



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