# Multivariatep-Adic Fermionic q-Integral on ℤp and Related Multiple Zeta-Type Functions

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

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## Abstract

In 2008, Jang et al. constructed generating functions
of the multiple twisted Carlitz's typeq-Bernoulli polynomials and obtained
the distribution relation for them. They also raised the following problem:“are there  analytic multiple twisted Carlitz's typeq-zeta functions which
interpolate multiple twisted Carlitz's type q-Euler (Bernoulli) polynomials?”
The aim of this paper is to give a partial answer to this problem. Furthermore
we derive some interesting identities related to twisted q-extension of Euler
polynomials and multiple twisted Carlitz's type q-Euler polynomials.


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## Body

## 1. Introduction, Definitions, and Notations

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 is 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 pNth roots of unity in ℂp×=ℂp∖{0}.
Set(1.1)Tp={ω∈ℂp∣ωpN=1forsomeN≥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<p−1/(p−1), so that qx=exp(xlogq) for |x|p≤1. If q∈ℂ, then we assume that |q|<1.Let(1.2)ℤp=lim←N(ℤpNℤ),ℤp×=⋃0<a<pa+pℤp,a+pNℤp={x∈ℤp∣x≡a(modpN)},where a∈ℤ lies in 0≤a<pN.We use the following notation:(1.3)[x]q=1−qx1−q.Hence(1.4)limq→1[x]q=xfor any x with |x|p≤1 in the present p-adic case. The distribution μq(a+pNℤp) is given as(1.5)μq(a+pNℤp)=qa[pN]q(cf. [1–9]). For the ordinary p-adic distribution μ0 defined by(1.6)μ0(a+pNℤp)=1pN,we see(1.7)limq→1μq=μ0.We say that f is a uniformly differentiable
function at a
point a∈ℤp, we write f∈UD(ℤp,ℂp) if the difference quotient(1.8)Ff(x,y)=f(x)−f(y)x−yhas a limit f′(a) as  (x,y)→(a,a). Also we
use the following
notation:(1.9)[x]−q=1−(−q)x1+q,
(cf.[1–5]).In [1–3], 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;
and he defined fermionic p-adic q-measures on ℤp as follows:(1.10)μ−q(a+pNℤp)=(−q)a[pN]−q.By using the fermionic p-adic q-measures, he defined the fermionic p-adic q-integral on ℤp as follows:(1.11)I−q(f)=∫ℤpf(x)dμ−q(x)=limN→∞1[pN]−q∑x=0pN−1f(x)(−q)xfor f∈UD(ℤp,ℂp) (cf. [1–3]). Observe that(1.12)I−1(f)=limq→1I−q(f)=∫ℤpf(x)dμ−1(x)=limN→∞∑x=0pN−1f(x)(−1)x.From (1.12), we
obtain(1.13)I−1(f1)+I−1(f)=2f(0),where f1(x)=f(x+1). By substituting f(x)=etx into (1.13), classical Euler numbers are
defined by means of the following generating function:(1.14)∫ℤpetxdμ−1(x)=2et+1=∑n=0∞Entnn!.These numbers are interpolated
by the Euler zeta function which is defined as follows:(1.15)ζE(s)=∑n=1∞(−1)nns,s∈ℂ,(cf. [1–9]). From
(1.12), we also obtain(1.16)qI−q(f1)+I−q(f)=[2]qf(0),where f1(x)=f(x+1). By substituting f(x)=etx into (1.13), q-Euler numbers are defined by means of the
following generating function:(1.17)∫ℤpetxdμ−q(x)=[2]qqet+1=∑n=0∞En,qtnn!.These numbers are interpolated
by the Euler q-zeta function which is defined as
follows:(1.18)ζq,E(s)=[2]q∑n=1∞(−1)nqnns,s∈ℂ,(cf. [4]).In [6], Ozden and Simsek defined generating function of q-Euler numbers by(1.19)2q+1∫ℤpetxdμ−q(x)=2qet+1,which are different from (1.17).
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-integrals on ℤp.In this paper, we define a multiple twisted Carlitz's
typeq-zeta functions, which interpolated multiple
twisted Carlitz's type q-Euler polynomials at negative integers. This
result gave us a partial answer of the problem proposed by  Jang et al.
[10], which is given by: “Are there analytic multiple
twisted Carlitz's typeq-zeta functions which interpolate multiple
twisted Carlitz's type q-Euler (Bernoulli) polynomials?”


## 2. Preliminaries

In [10],
Jang and  Ryoo defined
q-extension of Euler numbers and polynomials of
higher order and studied multivariate q-Euler zeta functions. They also derived sums
of products of q-Euler numbers and polynomials by using
ferminonic p-adic q-integral.In [5, 7], 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 this section, we consider twistedq-extension of Euler numbers and polynomials of
higher order and study multivariate twisted Barnes-type Hurwitz q-Euler zeta functions and l-functions.LetUD(ℤph,ℂp) denote the space of all uniformly (or
strictly) differentiable ℂp-valued functions on ℤph=ℤp×⋯×ℤp︸h-times. For f∈UD(ℤph,ℂp), the p-adic q-integral on ℤph is defined by(2.1)I−q(h)(f)=∫ℤp⋯∫ℤp︸h-timesf(x1,…,xh)dμ−q(x1)⋯dμ−q(xh)=limN→∞1[pN]−qh∑x1=0pN−1⋯∑xh=0pN−1f(x1,…,xh)(−q)x1+⋯+xh(cf. [3]). If q→1, then(2.2)I−1(h)(f)=limq→1I−q(h)(f)=limN→∞∑x1=0pN−1⋯∑xh=0pN−1f(x1,…,xh)(−1)x1+⋯+xh.For a fixed positive integer d with (d,p)=1, we set(2.3)Xp=lim←N(ℤdpNℤ).For f∈UD(ℤph,ℂp),(2.4)I−1(h)(f)=∫Xp⋯∫Xp︸h-timesf(x1,…,xh)dμ−1(x1)⋯dμ−1(xh),(cf. [2]).We setf(x1,…,xh)=ωx1+⋯+xhe(x+x1+⋯+xh)t in (2.2) and (2.4). Then we have(2.5)∫ℤp⋯∫ℤp︸h-timesωx1+⋯+xhe(x+x1+⋯+xh)tdμ−1(x1)⋯dμ−1(xh)=(2ωet+1)⋯(2ωet+1)︸h-times=∑n=0∞En,ω(h)(x)tnn!,where En,ω(h)(x) are the twisted Euler polynomials of order h. From (2.5), we note that(2.6)∫ℤp⋯∫ℤp︸h-timesωx1+⋯+xh(x+x1+⋯+xh)ndμ−1(x1)⋯dμ−1(xh)=En,ω(h)(x).We give an application of the multivariateq-deformed p-adic integral on ℤph in the fermionic sense related to [3]. Let(2.7)∫ℤph=∫ℤp⋯∫ℤp︸h-times.By substituting(2.8)f(x1,…,xh)=ωx1+⋯+xhe(x+x1+⋯+xh)tinto (2.1), we define twisted q-extension of Euler numbers of higher order by
means of the following generating function:(2.9)∫ℤphωx1+⋯+xhe(x1+⋯+xh)tdμ−q(x1)⋯dμ−q(xh)=([2]qωqet+1)⋯([2]qωqet+1)︸h-times=∑n=0∞En,q,ω(h)tnn!.Then we have(2.10)∫ℤphωx1+⋯+xh(x1+⋯+xh)ndμ−q(x1)⋯dμ−q(xh)=En,q,ω(h).From
(2.9), we obtain(2.11)∫ℤphωx1+⋯+xhe(x+x1+⋯+xh)tdμ−q(x1)⋯dμ−q(xh)=[2]qhext(ωqet+1)⋯(ωqet+1)︸h-times=∑n=0∞En,q,ω(h)(x)tnn!,where En,q,ω(h)(x) is called twisted q-extension of Euler polynomials of higher
order (cf. [11]). We
note that if ω=1, then En,q,ω(h)(x)=En,q(h)(x) and En,q,ω(h)=En,q(h) (cf. [6]). We also note that(2.12)En,q,ω(h)(x)=∑k=0n(nk)Ek,q,ω(h)xn−k.The twistedq-extension of Euler polynomials of higher
order, En,q,ω(h)(x), is defined by
means of the following generating function:(2.13)Gq,ω(h)(x,t)=[2]qωqet+1⋯[2]qωqet+1︸h-timesext=[2]qhetx∑l1=0∞(−ω)l1ql1el1t⋯∑lh=0∞(−ω)lhqlhelht=[2]qh∑l1,…,lh=0∞(−ω)l1+⋯+lhql1+⋯+lhe(l1+⋯+lh+x)t=∑n=0∞En,q,ω(h)(x)tnn!,where |t+log(ωq)|<π. From these generating functions of twisted q-extension of Euler polynomials of higher
order, we construct twisted multiple q-Euler zeta functions as follows.Fors∈ℂ and x∈ℝ with 0<x≤1, we define(2.14)ζq,ω,E(h)(s,x)=[2]qh∑l1,…,lh=0∞(−ω)l1+⋯+lhql1+⋯+lh(l1+⋯+lh+x)s.By the mth differentiation on both
sides of (2.13) at t=0, we obtain the following(2.15)Em,q,ω(h)(x)=(ddt)mGq,ω(h)(x,t)|t=0=[2]qh∑l1,…,lh=0∞(−ω)l1+⋯+lhql1+⋯+lh(x+l1+⋯+lh)mfor m=0,1,….From (2.14) and (2.15), we arrive at the
following(2.16)ζq,ω,E(h)(−m,x)=Em,q,ω(h)(x),m=0,1,….We set(2.17)∫Xph=∫Xp⋯∫Xp︸h-times.Let χ be Dirichlet's character with odd conductor d. We define twisted q-extension of generalized Euler polynomials of
higher order by means of the following generating function (cf. [11]):(2.18)∫Xphχ(x1+⋯+xh)ωx1+⋯+xhe(x+x1+⋯+xh)tdμ−q(x1)⋯dμ−q(xh)=∑n=0∞En,q,ω,χ(h)(x)tnn!.Note that(2.19)∑n=0∞En,q,ω,χ(h)(x)tnn!=ext∫Xp⋯∫Xp︸h-timesχ(x1+⋯+xh)ωx1+⋯+xhe(x1+⋯+xh)tdμ−q(x1)⋯dμ−q(xh)=ext1[d]−qhlimN→∞1[pN](−q)d∑a1=0d−1∑x1=0pN−1⋯∑ah=0d−1∑xh=0pN−1χ(a1+dx1+⋯+ah+dxh)×ωa1+dx1+⋯+ah+dxhe(a1+dx1+⋯+ah+dxh)t(−q)a1+dx1+⋯+ah+dxh=ext1[d]−qh∑a1,…,ah=0d−1χ(a1+⋯+ah)ωa1+⋯+ah(−q)a1+⋯+ahe(a1+⋯+ah)t×limN→∞1+qd1+qdpN1+ωdpNqdpNedpN1+ωdqdedt⋯limN→∞1+qd1+qdpN1+ωdpNqdpNedpN1+ωdqdedt︸h−times=ext1[d]−qh∑a1,…,ah=0d−1χ(a1+⋯+ah)ωa1+⋯+ah(−q)a1+⋯+ahe(a1+⋯+ah)t×1+qd1+ωdqdedt⋯1+qd1+ωdqdedt︸h-timessince(2.20)limN→∞qpN=1for|1−q|p<1. This allows us to
rewrite (2.18) as(2.21)∑n=0∞En,q,ω,χ(h)(x)tnn!=ext1[d]−qh∑a1,…,ah=0d−1χ(a1+⋯+ah)ωa1+⋯+ah(−q)a1+⋯+ahe(a1+⋯+ah)t×1+qd1+ωdqdedt⋯1+qd1+ωdqdedt︸h-times=[2]qhext∑a1,…,ah=0d−1χ(a1+⋯+ah)ωa1+⋯+ah(−q)a1+⋯+ahe(a1+⋯+ah)t×∑x1=0∞(−ωdqdedt)x1⋯∑xh=0∞(−ωdqdedt)xh︸h-times=[2]qhext∑x1,…,xh=0∞∑a1,…,ah=0d−1χ(a1+dx1+⋯+ah+dxh)×ωa1+dx1+⋯+ah+dxh(−q)a1+dx1+⋯+ah+dxhe(a1+dx1+⋯+ah+dxh)t=[2]qh∑l1,…,lh=0∞(−1)l1+⋯+lhχ(l1+⋯+lh)ωl1+⋯+lhql1+⋯+lhe(x+l1+⋯+lh)t.By applying the mth derivative operator (d/dt)m|t=0 in the above equation, we have(2.22)Em,q,ω,χ(h)(x)=[2]qh∑l1,…,lh=0∞χ(l1+⋯+lh)∏i=1h(−1)liωliqli(x+l1+⋯+lh)mfor m=0,1,….From these generating functions of twistedq-extension of generalized Euler polynomials of
higher order, we construct twisted multiple q-Euler l-functions as follows. For s∈ℂ and x∈ℝ with 0<x≤1, we define(2.23)lq,ω,E(h)(s,x,χ)=[2]qh∑l1,…,lh=0∞χ(l1+⋯+lh)∏i=1h(−1)liωliqli(l1+⋯+lh+x)s.From (2.22) and (2.23), we
arrive at the following(2.24)lq,ω,E(h)(−m,x,χ)=Em,q,ω,χ(h)(x),m=0,1,….Lets∈ℂ and ai,F∈ℤ with F is an odd integer and 0<ai<F, where i=1,…,h. Then twisted partial multiple q-Euler ζ-functions are as follows:(2.25)Hq,ω,E(h)(s,a1,…,ah,x∣F)=[2]qh∑l1,…,lh=0li≡ai(modF),i=1,…,h∞(−1)l1+⋯+lhωl1+⋯+lhql1+⋯+lh(l1+⋯+lh+x)s.For i=1,…,h, substituting li=ai+niF with F is odd into (2.25), we have(2.26)Hq,ω,E(h)(s,a1,…,ah,x∣F)=[2]qh∑n1,…,nh=0∞(−1)a1+n1F+⋯+ah+nhFωa1+n1F+⋯+ah+nhFqa1+n1F+⋯+ah+nhF(a1+n1F+⋯+ah+nhF+x)s=[2]qh[2]qFh(−ωq)a1+⋯+ahFs[2]qFh∑n1,…,nh=0∞(−1)n1+⋯+nh(ωF)n1+⋯+nh(qF)n1+⋯+nh(n1+⋯+nh+(a1+⋯+ah+x)/F)s=[2]qh[2]qFh(−ωq)a1+⋯+ahFsζqF,ωF,E(h)(s,a1+⋯+ah+xF).Then we obtain(2.27)Hq,ω,E(h)(s,a1,…,ah,x∣F)=[2]qh[2]qFh(−ωq)a1+⋯+ahFsζqF,ωF,E(h)(s,a1+⋯+ah+xF).By using (2.12) and (2.27) and
by substituting s=−m,m=0,1,…, we get(2.28)Hq,ω,E(h)(−m,a1,…,ah,x∣F)=[2]qh[2]qFh(−ωq)a1+⋯+ah(a1+⋯+ah+x)m×∑k=0m(mk)(Fa1+⋯+ah+x)kEk,qF,ωF(h).Therefore, we modify twisted
partial multiple q-Euler zeta functions as
follows:(2.29)Hq,ω,E(h)(s,a1,…,ah,x∣F)=[2]qh[2]qFh(−ωq)a1+⋯+ah(a1+⋯+ah+x)−s×∑k=0∞(−sk)(Fa1+⋯+ah+x)kEk,qF,ωF(h).Let χ be a Dirichlet character with
conductors d and d∣F. From (2.23) and (2.27), we
have(2.30)lq,ω,E(h)(s,x,χ)=[2]qh[2]qFhF−s∑a1,…,ah=0F−1(−ωq)a1+⋯+ah×χ(a1+⋯+ah)ζqF,ωF,E(h)(s,a1+⋯+ah+xF)=∑a1,…,ah=0F−1χ(a1+⋯+ah)Hq,ω,E(h)(s,x,a1,…,ah,x∣F).


## 3. The Multiple Twisted Carlitz's Typeq-Euler Polynomials and q-Zeta Functions

Let us consider the multiple twisted Carlitz's typeq-Euler polynomials as follows:(3.1)En,q,ω(z,h)(x)=∫ℤp⋯∫ℤp︸h-times[x1+⋯+xh+x]qnωx1+⋯+xhqx1(z−1)+⋯+xh(z−h)dμq(x1)⋯dμq(xh)(cf. [1, 3]). These can be written
as(3.2)En,q,ω(z,h)(x)=[2]qh(1−q)n∑i=0n(ni)qix(−1)i11+ωqz+i⋯11+ωqz+i−h+1.We may now mention the following
formulae which are easy to prove:(3.3)ωqzEn,q,ω(z,h)(x+1)+En,q,ω(z,h)(x)=[2]qEn,q,ω(z−1,h−1)(x).From (3.2), we can derive
generating function for the multiple twisted Carlitz's type q-Euler polynomials as follows:(3.4)∑n=0∞En,q,ω(z,h)(x)tnn!=∑n=0∞[2]qh(1−q)n∑i=0n(ni)qix(−1)i11+ωqz+i⋯11+ωqz+i−h+1tnn!=∑n=0∞[2]qh(1−q)n∑i=0n(ni)qix(−1)i∑l1=0∞(−ωqz+i)l1⋯∑lh=0∞(−ωqz+i−h+1)lhtnn!=∑n=0∞[2]qh(1−q)n∑l1,…,lh=0∞(−1)l1+⋯+lh∑i=0n(ni)q(x+l1+⋯+lh)i(−1)i×ωl1+⋯+lhql1z+l2(z−1)+⋯+lh(z−h+1)tnn!=[2]qh∑l1,…,lh=0∞(−1)l1+⋯+lhωl1+⋯+lhql1z+l2(z−1)+⋯+lh(z−h+1)×∑n=0∞[x+l1+⋯+lh]qntnn!=[2]qh∑l1,…,lh=0∞(−ω)l1+⋯+lhql1z+l2(z−1)+⋯+lh(z−h+1)e[x+l1+⋯+lh]qt.Also, an obvious generating
function for the multiple twisted Carlitz's type q-Euler polynomials is obtained, from (3.2),
by(3.5)[2]qhet/(1−q)∑j=0n(−1)jqjx(11−q)j11+ωqz+j⋯11+ωqz+j−h+1=En,q,ω(z,h)(x).From now on, we assume thatq∈ℂ with |q|<1. From (3.2) and (3.4), we note
that(3.6)Gq,ω(z,h)(x,t)=[2]qh∑l1,…,lh=0∞(−ω)l1+⋯+lhql1z+l2(z−1)+⋯+lh(z−h+1)e[x+l1+⋯+lh]qt=∑n=0∞En,q,ω(z,h)(x)tnn!,(3.7)En,q,ω(z,h)(x)=[2]qh(1−q)n∑i=0n(ni)qix(−1)i11+ωqz+i⋯11+ωqz+i−h+1=[2]qh∑l1,…,lh=0∞(−ω)l1+⋯+lhql1z+l2(z−1)+⋯+lh(z−h+1)[x+l1+⋯+lh]qn.Thus we can define the multiple
twisted Carlitz's type q-zeta functions as
follows:(3.8)ζq,ω(z,h)(s,x)=[2]qh∑l1,…,lh=0∞(−ω)l1+⋯+lhql1z+l2(z−1)+⋯+lh(z−h+1)[x+l1+⋯+lh]qs.In [12, Proposition 3], Yamasaki showed that the series ζq,ω(z,h)(s,x) converges absolutely for Re (z)>h−1, and it can be analytically continued to the
whole complex plane ℂ. Note that if h=1, then(3.9)ζq,ω(z,h)(s,x)→ζq,ω(z)(s,x)=[2]q∑l=0∞(−ω)lqlz[x+l]qs.In [13],
Wakayama and  Yamasaki studied q-analogue of the Hurwitz zeta function(3.10)ζ(s,x)=∑n=0∞1(n+x)sdefined by the q-series with two complex variable s,z∈ℂ:  (3.11)ζq(z)(s,x)=∑n=0∞q(n+x)z[x+n]qs,Re(z)>0,and special values at
nonpositive integers of the q-analogue of the Hurwitz zeta function.Therefore, by themth differentiation on both
sides of (3.6) at t=0, we obtain the following:(3.12)Em,q,ω(z,h)(x)=(ddt)mGq,ω(z,h)(x,t)|t=0=[2]qh∑l1,…,lh=0∞(−ω)l1+⋯+lhql1z+l2(z−1)+⋯+lh(z−h+1)[x+l1+⋯+lh]qmfor m=0,1,….From (3.7), (3.8), and
(3.12), we have
(3.13) which shows that the multiple twisted
Carlitz's type q-zeta functions interpolate the multiple
twisted Carlitz's type q-Euler numbers and polynomials. For m=0,1,…, we have(3.13)ζq,ω(z,h)(−m,x)=Em,q,ω(z,h)(x),where x∈ℝ and 0<x≤1.Thus, we derive the analytic multiple twisted
Carlitz's typeq-zeta functions which interpolate multiple
twisted Carlitz's type q-Euler polynomials. This gives a part of
the answer to the question
proposed in [10].


## 4. Remarks

For nonnegative integersm and n, we define the q-binomial coefficient [mn]q by(4.1)[mn]q=(q;q)m(q;q)n(q;q)m−n,where (a;q)m=∏k=0m−1(1−aqk) for m≥1 and (a;q)0=1. For h∈ℕ, it holds that(4.2)∑l1,…,lh≥0l1+⋯+lh=lq−(l1+2l2+⋯+hlh)=q−lh[l+h−1h−1]q(cf. [12, Lamma 2.3]). From
(3.8),
it is easy to see that(4.3)ζq,ω(z,h)(s,x)=[2]qh∑l=0∞∑l1,…,lh≥0l1+⋯+lh=l(−ω)l1+⋯+lhq(z+1)(l1+⋯+lh)−(l1+2l2+⋯+hlh)[x+l1+⋯+lh]qs=[2]qh∑l=0∞(−ω)lq(z+1)l[l+x]qs∑l1,…,lh≥0l1+⋯+lh=lq−(l1+2l2+⋯+hlh)=[2]qh∑l=0∞[l+h−1h−1]q(−ω)lq(z−h+1)l[l+x]qs.We set [m]q!=[m]q[m−1]q⋯[1]q for m∈ℕ. The following identity has been studied in
[12]:(4.4)[l+h−1h−1]q=1[h−1]q!∏j=1h−1([l+x]q−ql+j[x−j]q)=∑k=0h−1ql(h−1−k)Pq,hk(x)[l+x]qk,where Pq,hk(x),0≤k≤h−1, is a function of x defined by(4.5)Pq,hk(x)=(−1)h−1−k[h−1]q!∑1≤m1<⋯<mh−1−k≤h−1qm1+⋯+mh−1−k[x−m1]q⋯[x−mh−1−k]qfor 0≤k≤h−2 and Pq,hh−1(x)=1/[h−1]q!. By using (3.9), (4.3), and
(4.5), we
have(4.6)ζq,ω(z,h)(s,x)=[2]qh∑k=0h−1Pq,hk(x)∑l=0∞(−ω)lq(z−k)l[l+x]qs−k=[2]qh−1∑k=0h−1Pq,hk(x)ζq,ω(z−k)(s−k,x),and so(4.7)ζq,ω(z,h)(−m,x)=[2]qh−1∑k=0h−1Pq,hk(x)ζq,ω(z−k)(−m−k,x).The values of ζq,ω(z,h)(−m,x) at h=2,3 are given explicitly as
follows:(4.8)ζq,ω(z,2)(−m,x)=(1+q)(ζq,ω(z−1)(−m−1,x)−q[x−1]qζq,ω(z)(−m,x)),ζq,ω(z,3)(−m,x)=(1+q){ζq,ω(z−2)(−m−2,x)−(q[x−1]q+q2[x−2]q)ζq,ω(z−1)(−m−1,x)+q3[x−1]q[x−2]qζq,ω(z)(−m,x)}.



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*Source: 304539-2008-06-12.xml*