# A Note on the Multiple Twisted Carlitz's Typeq-Bernoulli Polynomials

Authors: Lee-Chae Jang; Cheon-Seoung Ryoo
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

We give the twisted Carlitz's typeq-Bernoulli polynomials and numbers associated with
p-adic q-inetgrals and
discuss their properties. Furthermore, we define the multiple twisted Carlitz's type
q-Bernoulli polynomials and numbers and obtain the distribution
relation for them.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this
paper ℤp, ℚp, ℂ,
and ℂp will, respectively,
be the ring of p-adic rational  integers,  the field of p-adic rational numbers, the complex number
field, and the p-adic completion of the algebraic closure of ℚp.
The p-adic absolute value in ℂp is normalized so that |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∈ℂ,
one normally assumes |q|<1.
If q∈ℂp,
one normally assumes that |1−q|p<p−1/(p−1) so that qx=exp⁡(xlog⁡q) for each x∈ℤp.
We use the notation
(1.1)[x]q=1−qx1−q,[x]−q=1−(−q)x1+q (cf. [1–20]) for all x∈ℤp.
For a fixed odd positive integer d with (p,d)=1,
let(1.2)X=Xd=limn←⁡ℤdpnℤ,X1=ℤp,X∗=⋃0<a<dp(a,p)=1(a+dpℤp),a+dpnℤp={x∈X∣x≡a(mod⁡dpn)}, where a∈ℤ lies in 0≤a<dpn.
For any n∈ℕ,
(1.3)μq(a+dpnℤp)=qa[dpn]q is known to be a distribution on X (cf. [1–20]).We say thatf is uniformly differentiable function at a
point a∈ℤp and denote this property by f∈UD(ℤp),
if the difference quotients
(1.4)Ff(x,y)=f(x)−f(y)x−y have a limit l=f'(a) as (x,y)→(a,a) (cf. [10–13]). The p-adic q-integral of a function f∈UD(ℤp) was defined as
(1.5)Iq(f)=∫ℤpf(x)dμq(x)=lim⁡n→∞1[pn]q∑x=0pn−1f(x)qx.By usingp-adic q-integrals on ℤp,
it is well known that
(1.6)tet−1=∑n=0∞∫ℤpxndμ1(x)tnn!, where μ1(x+pnℤp)=1/pn.
Then, we note that the Bernoulli numbers Bn were defined as
(1.7)tet−1=∑n=0∞Bntnn!, and hence, we
have
(1.8)Bn=∫ℤpxndμ1(x) for all n∈ℕ∪{0}.
For k∈ℕ and n∈ℕ∪{0},
the multiple Bernoulli polynomials Bn(k)(x) were defined as
(1.9)(tet−1)kext=∑n=0∞Bn(k)(x)tnn! (cf. [2]). We note that
(1.10)(tet−1)kext=∑n=0∞∫ℤp⋯∫ℤp︸k‐times(x+x1+⋯+xk)ndμ1(x1)⋯dμ1(xk). From (1.9) and (1.10), we
obtain
(1.11)Bn(k)(x)=∫ℤp⋯∫ℤp︸k‐times(x+x1+⋯+xk)ndμ1(x1)⋯dμ1(xk). In view of (1.11), the multiple
Carlitz's type q-Bernoulli polynomials were defined
as
(1.12)βn(k,q)(x)=∫ℤp⋯∫ℤp︸k‐times[x+x1+⋯+xk]qndμq(x1)⋯dμq(xk). In this case, x=0,
we write βn(k,q)(0)=βn(k,q),
which were called the Carlitz's type q-Bernoulli numbers. By (1.11) and (1.12), we note
that
(1.13)lim⁡q→1βn(k,q)=Bn(k,1)=Bnk.In Section2, we give the twisted Carlitz's type q-Bernoulli polynomials and numbers associated
with p-adic q-inetgrals and discuss their properties. In
Section 3, we define the multiple twisted Carlitz's type q-Bernoulli polynomials and numbers. We also
obtain the distribution relation for them.


## 2. Twisted Carlitz's Typeq-Bernoulli Polynomials

In this section, we assume thatq∈ℂp with |1−q|p<p−1/(p−1).
By using p-adic q-integral on ℤp,
we derive
(2.1)Iq(f1)=1qIq(f)+(q−1log⁡qf′(0)+(q−1)f(0)), (cf. [8]), where f1(x)=f(x+1).
From (1.5), we can derive
(2.2)qnIq(fn)=Iq(f)+q(q−1)log⁡q(∑i=0n−1f′(i)qi+log⁡q∑i=0n−1f(i)qi), (cf. [8]), where n∈ℕ and fn(x)=f(x+n).LetTp=⋃n≥1Cpn=lim⁡n→∞Cpn=Cp∞ be the locally constant space, where Cpn={w∣wpn=1} is the cyclic group of order pn.
For w∈Tp,
we denote the locally constant function by ϕw:ℤp→ℂp, x→wx.
If we take f(x)=ϕw(x)=wx,
then we have
(2.3)∫ℤpetxϕw(x)dμq(x)=(log⁡q+tqwet−1)q(q−1)log⁡q≡Fwq(t).Now we define the twistedq-Bernoulli polynomials as
follows:
(2.4)Fwq(x,t)=(log⁡q+tqwet−1)q(q−1)log⁡qext=∑n=0∞Bn,wq(x)tnn!. We note that Bn,wq(0)=Bn,wq are called the twisted q-Bernoulli numbers and by substituting w=1, lim⁡q→1Bn,1q=Bn are the familiar Bernoulli numbers. By (2.3),
we obtain the following Witt's type formula for the twisted q-Bernoulli polynomials and numbers.Theorem 2.1.
Forn∈ℕ and w∈Tp,
one has
(2.5)∫ℤp(t+x)nwtdμq(t)=Bn,wq(x).From (2.5), we consider the twisted Carliz's type q-Bernoulli polynomials by using p-adic q-integrals. For w∈Tp,
we define the twisted Carlitz's type q-Bernoulli polynomials as
follows:
(2.6)βn,wq(x)=11−q∫ℤp[t+x]qnwtdμq(t). When x=0,
we write βn,wq(0)=βn,wq which are called twisted Carlitz's type q-Bernoulli numbers. Note that if w=1,
then lim⁡q→1βn,1q=Bn.
From (2.6), we can see that
(2.7)βn,wq(x)=1(1−q)n∑i=0n(ni)qix(−1)i11−qi+1w. From (2.7), we can derive the
generating function for the twisted Carlitz's type q-Bernoulli polynomials as
follows:
(2.8)Gwq(x,t)=∑n=0∞βn,wq(x)tnn!=∑n=0∞(1(1−q)n∑i=0n(ni)qix(−1)i11−qi+1w)tnn!=∑n=0∞(1(1−q)n∑i=0n(ni)qix(−1)i∑l=0∞q(i+1)lwl)tnn!=∑l=0∞(∑n=0∞qlwl(1−q)n∑i=0n(ni)q(x+l)i(−1)i)tnn!=∑l=0∞qlwl∑n=0∞(1−qx+l)n(1−q)ntnn!=∑l=0∞qlwle[x+l]qt. Then it is easily to see
that
(2.9)Gwq(x,t)=∫ℤpe[t+x]qtwtdμq(t).By thekth differentiation on both sides of (2.8) at t=0,
we also have
(2.10)βn,wq(x)=dndtnGwq(x,t)|t=0=∑l=0∞qlwl[x+l]qn for n∈ℕ∪{0}.
We note that
(2.11)βn,wq=βn,wq(0)=∑l=0∞qlwl[l]qn. In view of (2.10), we define
twisted Carlitz's type q-zeta function as follows:
(2.12)ζwq(s,x)=∑l=0∞qlwl[x+l]qs for all s∈ℂ and Re(x)>0.
We note that ζwq(s,x) is analytic function in the whole complex s-plane. We also have the following theorem
in which twisted Carlitz's type q-zeta functions interpolate twisted Carlitz's
type q-Bernoulli numbers and polynomials.Theorem 2.2.
Fork∈ℕ∪{0} and w∈Tp,
one has
(2.13)ζwq(−k,x)=βk,wq(x),ζwq(−k,0)=βk,wq.From (2.11), we obtain the following distribution
relation for the twisted q-Bernoulli polynomials.Theorem 2.3.
Forr∈ℕ, n∈ℕ∪{0},
and w∈Tp,
one has
(2.14)βn,wq(x)=[r]qn−1∑i=0r−1wiqiβn,wrqr(i+xr).Proof.
If we puti+rl=j and i=1⋯r and l=0,1,…,
then by (2.11), we have
(2.15)βn,wq(x)=∑j=0∞wjqj[x+j]qn=∑l=0∞∑i=0r−1wi+rlqi+rl[x+i+rl]qn=(1−qr1−q)n∑i=0r−1wiqi∑l=0∞wrlqrl(1−qr((i+x)/r+l)1−qr)n=[r]qn−1∑i=0r−1wiqiβn,wrqr(i+xr).


## 3. Multiple Twisted Carlitz's Typeq-Bernoulli Polynomials

In this section, we consider the multiple twisted
Carlitz's typeq-Bernoulli polynomials as
follows:
(3.1)βk,w(h,q)(x)=∫ℤp⋯∫ℤp︸h‐times[x1+⋯+xh+x]qnwx1+⋯+xhdμq(x1)⋯dμq(xh)=lim⁡ϱ→∞1[pϱ]qh∑x1⋯xh=0pϱ−1[x+x1+⋯+xh]qnwx1+⋯+xhqx1+⋯+xh,where h∈ℕ, k∈ℕ∪{0},
and w∈Tp.
We note that βn,w(h,q)(0)=βn,w(h,q) are called the multiple twisted Carlitz's type q-Bernoulli numbers. We also obtain the
generating function of the multiple twisted Carlitz's type q-Bernoulli polynomials as
follows:
(3.2)Gw(h,q)(x,t)=∫ℤp⋯∫ℤp︸h‐timese[x1+⋯+xh+x]qtwx1+⋯+xhdμq(x1)⋯dμq(xh)=∑l=0∞∫ℤp⋯∫ℤp︸h‐times[x1+⋯+xh+x]qlwx1+⋯+xhdμq(x1)⋯dμq(xh)tll!=∑l=0∞βl,w(h,q)(x)tll!. Finally, we have the following
distribution relation for the multiple twisted q-Bernoulli polynomials.Theorem 3.1.
For eachw∈Tp,h,r∈ℕ, n∈ℕ∪{0},
and w∈Tp,
(3.3)βn,w(h,q)(x)=[r]qn−h∑j1,…,jh=0r−1wj1+⋯+jhqj1+⋯+jhβn,wr(h,qr)(x+j1+⋯+jhr).Proof.
If we putjk+rlk=xk,jk=0,1,…r−1,
and k=1⋯h,
then by (3.1), we have
(3.4)βk,w(h,q)(x)=∫ℤp⋯∫ℤp︸h‐times[x1+⋯+xh+x]qnwx1+⋯+xhdμq(x1)⋯dμq(xh)=lim⁡ϱ→∞1[rpϱ]qh∑x1⋯xh=0rpϱ−1[x+x1+⋯+xh]qnwx1+⋯+xhqx1+⋯+xh=lim⁡ϱ→∞[r]qn−h1[pϱ]qrh∑j1,…,jh=0r−1∑l1⋯lh=0pϱ−1[x+j1+rl1+⋯+jh+rlh]qn⋅wj1+rl1+⋯+jh+rlhqj1+rl1+⋯+jh+rlh=[r]qn−h∑j1,…,jh=0r−1wj1+⋯+jhqj1+⋯+jh⋅lim⁡ϱ→∞1[pϱ]qrh∑l1⋯lh=0pϱ−1[x+j1+⋯+jhr+l1+⋯+lh]qrnwr(l1+⋯+lh)qr(l1+⋯+lh)=[r]qn−h∑j1,…,jh=0r−1wj1+⋯+jhqj1+⋯+jhβn,wr(h,qr)(x+j1+⋯+jhr).Question 1.
Are there the analytic multiple twisted
Carlitz's typeq-zeta functions which interpolate multiple
twisted Carlitz's type q-Bernoulli polynomials?



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