# On a Two-Variablep-Adic lq-Function

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

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## Abstract

We prove that a two-variablep-adic lq-function has the series expansion lp,q(s,t,χ)=([2]q/[2]F)∑a=1,(p,a)=1F(−1)a(χ(a)qa/〈a+pt〉s)∑m=0∞(−sm)(F/〈a+pt〉)mEm,qF*
which interpolates the values lp,q(−n,t,χ)=En,χn,q∗(pt)−pnχn(p)([2]q/[2]qp)En,χn,qp∗(t), whenever n is a nonpositive integer. The  proof of this original construction is
due to Kubota and Leopoldt in 1964, although the method given in this note
is due to Washington.


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## Body

## 1. Introduction

The ordinary Euler polynomialsEn(t) are defined by the equation(1.1)2etxex+1=∑n=0∞En(t)xnn!.Setting t=1/2 and normalizing by 2n gives the ordinary Euler numbers(1.2)En=2nEn(12).The ordinary Euler polynomials
appear in many classical results (see [1]). In [2], the values of these polynomials at rational arguments
were expressed in term of the Hurwitz zeta function. Congruences for Euler
numbers have also received much attention from the point of view of p-adic interpolation. In [3], Kim et al. recently defined
the natural q-extension of ordinary Euler numbers and
polynomials by p-adic integral representation and proved
properties generalizing those satisfied by En and En(t). They also constructed the one-variable p-adic q-l-function lp,q(s,χ) for Dirichlet characters χ and s∈ℂp with |s|p<p1−(1/(p−1)), with the property that(1.3)lp,q(−n,χ)=En,χω−n,q*−[2]q[2]qp−1pnχω−n(p)En,χω−n,qp*for n=0,1,…, where En,χω−n,q* is a generalized q-Euler number associated with the Dirichlet
characters χω−n (see Section 2 for definitions).In the present paper, we
will construct a specific two-variablep-adic lq-function lp,q(s,t,χ) by means of a method provided in [4–6]. We also prove that lp,q(s,t,χ) is analytic in s and t for s∈ℂp with |s|p<p1−(1/(p−1)) and t∈ℂp with |t|p≤1, which interpolates the values(1.4)lp,q(−n,t,χ)=En,χn,q*(pt)−pnχn(p)[2]q[2]qpEn,χn,qp*(t),whenever n is a nonpositive integer. This two-variable
function is a generalization of the one-variable p-adic q-l-function, which is the function obtained by
putting t=0 in lp,q(s,t,χ) (cf. [3–11]).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. We will use ℤ+ for the set of nonpositive integers. 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. If q∈ℂ, then we assume that |q|<1. Also, we use the following notations:(1.5)[x]q=1−qx1−q,[x]−q=1−(−q)x1+q,(cf.[7, 8]).
Let d be a fixed integer, and let(1.6)X=Xd=lim←N(ℤ/dpNℤ),X*=⋃0<a<dp(a,p)=1a+dpℤp,a+dpNℤp={x∈X∣x≡a(moddpN)},where a∈ℤ lies in 0≤a<dpN.
Let UD(ℤp) be the space of uniformly differentiable
function on ℤp. For f∈UD(ℤp), the p-adic q-integral is defined by(1.7)Iq(f)=∫ℤpf(a)dμq(a)=∫Xf(a)dμq(a)=limN→∞1[dpN]q∑a=0dpN−1f(a)qafor|1−q|p<1.In [8], the bosonic integral was
considered from a more physical point of view to the limit q→1 as follows:(1.8)I1(f)=limq→1Iq(f)=∫ℤpf(a)dμ1(a)=limN→∞1pN∑a=0pN−1f(a).Furthermore, we can consider the
fermionic integral in contrast to the conventional “bosonic integral.” That
is, I−1(f)=∫ℤpf(a)dμ−1(a) (see [9]). From this, we derive I−1(f1)+I−1(f)=2f(0), where f1(a)=f(a+1). Also, we have(1.9)I−1(fn)+(−1)n−1I−1(f)=2∑a=0n−1(−1)n−1−af(a),where fn(a)=f(a+n) and n∈ℤ+ (see [9]). 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:(1.10)I−q(f)=∫ℤpf(a)dμ−q(a)=limN→∞1[dpN]−q∑a=0dpN−1f(a)(−q)a(cf.[3]).


## 2.q-Euler Numbers and Polynomials

In this section, we review some notations and facts in
[3].From (1.10), we can derive the following
formula:(2.1)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)=eax, then we have f1(a)=e(a+1)x=eaxex. From (2.1), we derive (qex+1)I−q(eax)=[2]q. Hence, we obtain(2.2)I−q(eax)=∫ℤpeaxdμ−q(a)=[2]qqex+1.We now set(2.3)[2]qqex+1=∑n=0∞En,q*xnn!.En,q* is called the nth q-Euler number. By (2.2) and (2.3), we see that(2.4)∫ℤpandμ−q(a)=En,q*.From (2.2), we also note
that(2.5)∫ℤpe(t+a)xdμ−q(a)=[2]qqex+1etx.In view of (2.3) and (2.5), we
can consider q-Euler polynomials associated with t as follows:(2.6)[2]qqex+1etx=∑n=0∞En,q*(t)xnn!,∫ℤp(t+a)ndμ−q(a)=En,q*(t).Put limq→1En,q*=En* and limq→1En,q*(t)=En*(t). Then, we have En(t)=En*(t) and(2.7)En=∑m=0n2m(nm)Em*,where En and En(t) are the ordinary Euler numbers and
polynomials. By (2.3) and (2.6), we easily see that En,q*(t)=∑m=0n(nm)tn−mEm,q*. For d∈ℤ+, let fd(a)=f(a+d). Then, we have(2.8)qdI−q(fd)+(−1)d−1I−q(f)=[2]q∑a=0d−1(−1)d−a−1qaf(a),see [3].If d is an odd positive integer, we
have(2.9)qdI−q(fd)+I−q(f)=[2]q∑a=0d−1(−1)aqaf(a).Let χ be a Dirichlet character with conductor d=dχ (=odd) ∈ℤ+. If we take f(a)=χ(a)e(t+a)x, then we have fd(a)=f(a+d)=χ(a)edxe(t+a)x. From (1.7) and (2.9), we
derive(2.10)∫Xχ(a)e(t+a)xdμ−q(a)=[2]q∑a=1d(−1)aqaχ(a)e(t+a)xqdedx+1.In view of (2.10), we also
consider the generalized q-Euler polynomials associated with χ as follows:(2.11)Fχ,q(x,t)=[2]q∑a=1d(−1)aqaχ(a)e(t+a)xqdedx+1=∑n=0∞En,χ,q*(t)xnn!.From (2.10) and (2.11), we derive
the following equation:(2.12)∫Xχ(a)(t+a)ndμ−q(a)=En,χ,q*(t)for n≥0. Put limq→1En,χ,q*(t)=En,χ*(t). On the other hand, the generalized q-Euler polynomials associated with χ are easily expressed as the q-Euler polynomials:(2.13)En,χ,q*(t)=dn[2]q[2]qd∑a=1d(−1)aqaχ(a)En,qd*(a+td),n≥0.
Let χ be a Dirichlet character with conductor d=dχ∈ℤ+. It is well known (see [11, 12]) that, for positive integers m and n,(2.14)∑a=1dnχ(a)am=1m+1(Bm+1,χ(dn)−Bm+1,χ(0)),where Bm+1,χ(t) are the generalized Bernoulli polynomials.
When d=dχ (=odd) ∈ℤ+, note that(2.15)[2]q∑a=1d(−1)aqaχ(a)e(t+a)x(1−(−qdedx)n)1−(−qdedx)=[2]q∑a=1d∑l=0n−1(−1)a+dlqa+dlχ(a+dl)ex(t+a+dl)=[2]q∑a=1dn(−1)aqaχ(a)ex(t+a)=∑m=0∞([2]q∑a=1dn(−1)aqaχ(a)(t+a)m)xmm!.By (2.11), the relation (2.15)
can be rewritten as(2.16)[2]q∑a=1d(−1)aqaχ(a)e(t+a)x(1−(−qdedx)n)1−(−qdedx)=∑m=0∞(Em,χ,q*(t)+(−1)n+1qdnEm,χ,q*(t+dn))xmm!.Now, we give the q-analog of (2.14) for the generalized Euler
polynomials. From (2.15) and (2.16), it is easy to see that(2.17)∑a=1dn(−1)aqaχ(a)(t+a)m=1[2]q(Em,χ,q*(t)+(−1)n+1qdnEm,χ,q*(t+dn))for positive integers m and n. In particular, replacing q by 1 in (2.17), if χ=χ0, the principal character (dχ=1), and t=0, then(2.18)∑a=1n−1(−1)aam=12(Em(0)+(−1)n+1Em(n)).Definition 2.1.
Lets∈ℂ with  Re(s)>1. Let χ be a primitive Dirichlet character with
conductor d=dχ (=odd) ∈ℤ+. One sets(2.19)lq(s,t,χ)=[2]q∑n=0∞(−1)nqnχ(n)(t+n)s,0<t≤1.Remark 2.2.
We assume thatq∈ℂ with |q|<1. Let χ be a primitive Dirichlet character with
conductor d=dχ (=odd) ∈ℤ+. From (2.11), we consider the below integral
which is known as the Mellin transformation of Fχ,q(x,t) (cf. [13])(2.20)1Γ(s)∫0∞xs−1Fχ,q(−x,t)dx=[2]q∑a=1d(−1)aqaχ(a)1Γ(s)∫0∞xs−1e−(t+a)x1−(−qde−dx)dx=[2]q∑a=1d(−1)aqaχ(a+dl)∑l=0∞(−1)dlqdl(a+dl+t)s.We write n=a+dl, where n=1,2,…, and obtain(2.21)1Γ(s)∫0∞xs−1Fχ,q(−x,t)dx=[2]q∑n=0∞(−1)nqnχ(n)(t+n)s=lq(s,t,χ).Note thatlq(s,t,χ) is an analytic function in the whole complex s-plane. By using a geometric series in (2.11),
we obtain(2.22)[2]qetx∑n=0∞(−1)nqnχ(n)enx=∑n=0∞En,χ,q*(t)xnn!.We also note
that(2.23)En,χ,q*(t)=(ddx)k[2]qetx∑n=0∞(−1)nqnχ(n)enx|x=0.By Definition 2.1 and (2.23), we
obtain the following proposition.Proposition 2.3.
Forn∈ℤ+, one has lq(−n,t,χ)=En,χ,q*(t).The values oflq(s,t,χ) at negative integers are algebraic, hence may
be regarded as being in an extension of ℚp. We therefore look for a p-adic function which agrees with lq(s,t,χ) at the negative integers in Section 3.


## 3. A Two-Variablep-Adic lq-Function

We will consider thep-adic analog of the lq-functions which are introduced in the
previous section (see Definition 2.1). Throughout this section we assume that p is an odd prime. Note that there exist φ(p) distinct solutions, modulo p, to the equation xφ(p)−1=0, and each solution must be congruent to one of
the values a∈ℤ, where 1≤a<p,(a,p)=1. Thus, given a∈ℤ with (a,p)=1, there exists a unique ω(a)∈ℤp, where ω(a)φ(p)=1, such that ω(a)≡a(modpℤp). Letting ω(a)=0 for a∈ℤ,
such that (a,p)≠1, it can be seen that ω is actually a Dirichlet character having
conductor dω=p, called the Teichmüller character. Let 〈a〉=ω−1(a)a. Then, 〈a〉≡1(modpℤp). For the context in the sequel, an extension of
the definition of the Teichmüller character is needed. We denote a particular
subring of ℂp as(3.1)R={a∈ℂp∣|a|p≤1}.If t∈ℂp,
such that |t|p≤1, then for any a∈ℤ,a+pt≡a(modpR).
Thus, for t∈ℂp,|t|p≤1,ω(a+pt)=ω(a). Also, for these values of t, let 〈a+pt〉=ω−1(a)(a+pt). Let χ be the Dirichlet character of conductor d=dχ. For n≥1, we define χn to be the primitive character associated with
the character χn:(ℤ/lcm(d,p)ℤ)×→ℂ× defined by χn(a)=χ(a)ω−n(a).We define an interpolation function for generalizedq-Euler polynomials.Definition 3.1.
Letχ be the Dirichlet character with conductor d=dχ (=odd) and let F be a positive integral multiple of p and d. Now, one defines the two-variable p-adic lq-function as follows:(3.2)lp,q(s,t,χ)=[2]q[2]qF∑a=1(p,a)=1F(−1)aχ(a)qa〈a+pt〉−s∑m=0∞(−sm)(F〈a+pt〉)mEm,qF*.Let D={s∈ℂp∣|s|p<p1−(1/(p−1))} and let a∈ℤ,(a,p)=1. For t∈ℂp,|t|p≤1, the same argument as that given in the proof
of the main theorem of [4, 5]
can be used to show that the functions ∑m=0∞(sm)(F/(a+pt))mEm,qF* and 〈a+pt〉s=∑m=0∞(sm)(〈a+pt〉−1)m are analytic for s∈D. According to this method, we see that the
function ∑m=0∞(sm)(F/(a+pt))mEm,qF* is analytic for t∈ℂp,|t|p≤1, whenever s∈D. It readily follows that 〈a+pt〉s=〈a〉s∑m=0∞(sm)(a−1pt)m is analytic for t∈ℂp,|t|p≤1, when s∈D. Therefore,(3.3)lp,q(s,t,χ)is analytic fort∈ℂp,|t|p≤1,
provided s∈D (see [5]).We set(3.4)hp,q(s,t,a∣F)=(−1)aqa〈a+pt〉−s[2]q[2]qF∑m=0∞(−sm)(Fa+pt)mEm,qF*.Thus, we note
that(3.5)hp,q(−n,t,a∣F)=ω−n(a)(−1)aqaFn[2]q[2]qFEn,qF*(a+ptF)for n∈ℤ+. We also consider the two-variable p-adic lq-functions which interpolate the generalized q-Euler polynomials at negative integers as
follows:(3.6)lp,q(s,t,χ)=∑a=1(p,a)=1Fχ(a)hp,q(s,t,a∣F).We will in the process derive an
explicit formula for this function. Before we begin this derivation, we need
the following result concerning generalized q-Euler polynomials.Lemma 3.2.
LetF be a positive integral multiple of p and d=dχ. Then, for each n∈ℤ,n≥0,(3.7)En,χ,q*(t)=Fn[2]q[2]qF∑a=1F(−1)aqaχ(a)En,qF*(a+tF).Proof.
By (2.6) and (2.11), we note that(3.8)Fχ,q(x,t)=[2]q[2]qd∑a=1d(−1)aqaχ(a)∑n=0∞En,qd*(a+td)(dx)nn!.Then, we have(3.9)En,χ,q*(t)=dn[2]q[2]qd∑a=1d(−1)aqaχ(a)En,qd*(a+td).On the other hand, if F=dp, then we get(3.10)[2]q[2]qF∑a=1F(−1)aqaχ(a)∑n=0∞En,qF*(a+tF)(Fx)nn!=[2]q∑a=1d∑b=0p−1(−1)a+bdqa+bdχ(a+bd)e(a+bd)xetxqFeFx+1=[2]q∑a=1d(−1)aqaχ(a)e(a+t)x∑b=0p−1(−1)bdqbdebdx1qFeFx+1=[2]q[2]qd∑a=1d(−1)aqaχ(a)∑n=0∞En,qd*(a+td)(dx)nn!.This completes the proof.Setχn=χω−n. From (3.5) and (3.6), we
obtain(3.11)lp,q(−n,t,χ)=Fn[2]q[2]qF∑a=1(p,a)=1Fχn(a)(−1)aqaEn,qF*(a+ptF)=Fn[2]q[2]qF∑a=1Fχn(a)(−1)aqaEn,qF*(a+ptF)−Fn[2]q[2]qF∑a=1F/pχn(pa)(−1)paqpaEn,qF*(pa+ptF).for n∈ℤ+. From Lemma 3.2, we see that(3.12)En,χn,q*(pt)=Fn[2]q[2]qF∑a=1F(−1)aqaχn(a)En,qF*(a+ptF),En,χn,qp*(t)=(Fp)n[2]qp[2](qp)F/p∑a=1F/p(−1)a(qp)aχn(a)En,(qp)F/p*(a+tF/p).
From (3.3), (3.11), and (3.12), we obtain the
following theorem.Theorem 3.3.
LetF (=odd) be a positive integral multiple of p and dχ. Then, the two-variable p-adic lq-function(3.13)lp,q(s,t,χ)=[2]q[2]qF∑a=1(p,a)=1F(−1)aχ(a)qa〈a+pt〉−s∑m=0∞(−sm)(F〈a+pt〉)mEm,qF*admits an analytic function for t∈ℂp with |t|p≤1 and s∈D and satisfies the relation(3.14)lp,q(−n,t,χ)=En,χn,q*(pt)−pnχn(p)[2]q[2]qpEn,χn,qp*(t)for n∈ℤ+ and t∈ℂp with |t|p≤1.From (3.5) and Theorem 3.3, it follows that hp,q(s,t,a∣F) is analytic for t∈ℂp with |t|p≤1 and s∈D.Remark 3.4.
Let〈a+pt〉=ω−1(a)(a+pt), and let t∈ℂp with |t|p≤1 and s∈D. Then the two-variable p-adic lq-function defined above is redefined by(3.15)lp,q(s,t,χ)=∫X*χ(a)〈a+pt〉−sdμ−q(a),compared to[3,10].Then, we have(3.16)lp,q(−n,t,χ)=∫Xχn(a)(a+pt)ndμ−q(a)−∫Xχn(pa)(pa+pt)ndμ−q(pa)=(2.10)En,χn,q*(pt)−pnχn(p)[2]q[2]qpEn,χn,qp*(t),since X*=X−pX and [2]qpdμ−q(pa)=[2]qdμ−qp(a).Ifq→1 in Theorem 3.3 and Remark 3.4, we obtain the
following corollary.Corollary 3.5.
LetF (=odd) be a positive integral multiple of p and dχ, and let the two-variable p-adic l-function(3.17)lp(s,t,χ)=∑a=1(p,a)=1F(−1)aχ(a)〈a+pt〉−s∑m=0∞(−sm)(F〈a+pt〉)mEm*.Then,
(1)
lp(s,t,χ) is analytic for t∈ℂp with |t|p≤1 and s∈D.(2)
lp(−n,t,χ)=En,χn*(pt)−pnχn(p)En,χn*(t) for n∈ℤ+.(3)
lp(s,t,χ)=∫X*χ(a)〈a+pt〉−sdμ−1(a) for t∈ℂp with |t|p≤1 and s∈D.



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