# Some Relations of the Twistedq-Genocchi Numbers and Polynomials with Weight α and Weak Weight β

Authors: J. Y. Kang; H. Y. Lee; N. S. Jung
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Recently many mathematicians are working on Genocchi polynomials and Genocchi numbers. We define a new type of twistedq-Genocchi numbers and polynomials with weight α and weak weight β and give some interesting relations of the twisted q-Genocchi numbers and polynomials with weight α and weak weight β. Finally, we find relations between twisted q-Genocchi zeta function and twisted Hurwitz q-Genocchi zeta function.


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## Body

## 1. Introduction

The Genocchi numbers and polynomials possess many interesting properties and arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of theq-Genocchi numbers and polynomials (see [1–16]).Throughout this paper we use the following notations. Byℤp we denote the ring of p-adic rational integers, ℚp denotes the field of p-adic rational numbers, ℂp denotes the completion of algebraic closure of ℚp, ℕ denotes the set of natural numbers, ℤ denotes the ring of rational integers, ℚ denotes the field of rational numbers, ℂ denotes the set of complex numbers, and ℤ+=ℕ∪{0}. Let νp be the normalized exponential valuation of ℂp with |p|p=p-νp(p)=p-1. When one talks of q-extension, q is considered in many ways such as an indeterminate, a complex number q∈ℂ, or p-adic number q∈ℂp. If q∈ℂ, one normally assume that |q|<1. If q∈ℂp, we normally assume that |q-1|p<p-1/(p-1) so that qx=exp(xlogq) for |x|p≤1. Throughout this paper we use the following notation:(1.1)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q
(cf. [1–13]).Hence,limq→1[x]=x for any x with |x|p≤1 in the  present p-adic case.For(1.2)f∈UD(Zp)={f∣f:Zp⟶Cpisuniformlydifferentiablefunction},
the fermionic p-adic q-integral on ℤp is defined by Kim as follows:(1.3)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1f(x)(-q)x
(cf. [11–14]).If we takef1(x)=f(x+1) in (1.1), then we easily see that(1.4)qI-q(f1)+I-q(f)=[2]qf(0).
From (1.4), we obtain(1.5)qnI-q(fn)+(-1)n-1I-q(f)=[2]q∑l=0n-1(-1)n-1-lqlf(l),
where fn(x)=f(x+n) (cf. [5–9]).LetCpn={w∣wpn=1} be the cyclic group of order pn and let(1.6)Tp=lim⁡n→∞Cpn=Cp∞=∪n≥0Cpn
be the locally constant space. For w∈Tp, we denote by ϕw:ℤp→ℂp the locally constant function x↦wx.As well-known definition, the Genocchi polynomials are defined by(1.7)F(t)=2tet+1=eGt=∑n=0∞Gntnn!,F(t,x)=2tet+1ext=eG(x)t=∑n=0∞Gn(x)tnn!,
with the usual convention of replacing Gn(x) by Gn(x). Gn(0)=Gn are called the nth Genocchi numbers (cf. [2–5, 14]).These numbers and polynomials are interpolated by the Genocchi zeta function and Hurwitz-type Genocchi zeta function, respectively:(1.8)ζG(s)=2∑n=1∞(-1)nns,ζG(s,x)=2∑n=0∞(-1)n(n+x)s.Our aim in this paper is to define twistedq-Genocchi numbers Gn,q,w(α,β) and polynomials Gn,q,w(α,β)(x) with weight α and weak weight β. We investigate some properties which are related to Gn,q,w(α,β) and Gn,q,w(α,β)(x). We also derive the existence of a specific interpolation function which interpolate Gn,q,w(α,β) and Gn,q,w(α,β)(x) at negative integers.


## 2. Generating Functions of Twistedq-Genocchi Numbers and  Polynomials with Weight α and Weak Weight β

Our primary goal of this section is to define twistedq-Genocchi numbers Gn,q,w(α,β) and polynomials Gn,q,w(α,β)(x) with weight α and weak weight β. We also find generating functions of Gn,q,w(α,β) and Gn,q,w(α,β)(x).Definition 2.1.
Forα,β∈ℚ and q∈ℂp with |1-q|p≤1,
(2.1)Gn,q,w(α,β)=n∫Zpϕw(x)[x]qαn-1dμ-qβ(x).
We call Gn,q,w(α,β) twisted q-Genocchi numbers with weight α and weak weight β.By usingp-adic q-integral on ℤp, we obtain(2.2)n∫Zpϕw(x)[x]qαn-1dμ-qβ(x)=nlim⁡N→∞1[pN]-qβ∑x=0pN-1wx[x]qαn-1(-qβ)x=n[2]qβ∑m=0∞(-1)mqβmwm[m]qαn-1.
From (2.1) and (2.2), we have(2.3)Gn,q,w(α,β)=n[2]qβ(11-qα)n-1∑l=0n-1(n-1l)(-1)l11+wqβ+αl.We set(2.4)Fq,w(α,β)(t)=∑n=0∞Gn,q,w(α,β)tnn!.By using the previous equation and (2.3), we have(2.5)Fq,w(α,β)(t)=∑n=0∞(n[2]qβ(11-qα)n-1∑l=0n-1(n-1l)(-1)l11+wqβ+αl)tnn!=[2]qβt∑m=0∞(-1)mqβmwme[m]qαt.
Thus twisted q-Genocchi numbers Gn,q,w(α,β) with weight α and weak weight β are defined by means of the generating function:(2.6)Fq,w(α,β)(t)=[2]qβt∑n=0∞(-1)nqβnwne[n]qαt.By using (2.2), we have(2.7)∑n=0∞Gn,q,w(α,β)tnn!=t∫Zpϕw(x)e[x]qαtdμ-qβ(x).From (2.5) and (2.7), we have(2.8)∑n=0∞Gn,q,w(α,β)tnn!=[2]qβt∑m=0∞(-1)mqβmwme[m]qαt.Next, we introduce twistedq-Genocchi polynomials Gn,q,w(α,β)(x) with weight α and weak weight β.Definition 2.2.
Forα,β∈ℚ and q∈ℂp with |1-q|p≤1,
(2.9)Gn,q,w(α,β)(x)=n∫Zpϕw(y)[x+y]qαn-1dμ-qβ(y).
We call Gn,q,w(α,β)(x) twisted q-Genocchi polynomials with weight α and weak weight β.By usingp-adic q-integral, we have(2.10)n∫Zpϕw(y)[x+y]qαn-1dμ-qβ(y)=nlim⁡N→∞1[pN]-qβ∑y=0pN-1wy[x+y]qαn-1(-qβ)y=n[2]qβ∑m=0∞(-1)mqβmwm[x+m]qαn-1.By using (2.9) and (2.10), we obtain(2.11)Gn,q,w(α,β)(x)=n[2]qβ(11-qα)n-1∑l=0n-1(n-1l)(-1)lqαxl11+wqβ+αl.We set(2.12)Fq,w(α,β)(t,x)=∑n=0∞Gn,q,w(α,β)(x)tnn!.By using the previous equation and (2.11), we have(2.13)Fq,w(α,β)(t,x)=∑n=0∞(n[2]qβ(11-qα)n-1∑l=0n-1(n-1l)(-1)lqαxl11+wqβ+αl)tnn!=[2]qβt∑m=0∞(-1)mqβmwme[x+m]qαt.Thus twistedq-Genocchi polynomials Gn,q,w(α,β)(x) with weight α and weak weight β are defined by means of the generating function:(2.14)Fq,w(α,β)(t,x)=[2]qβt∑m=0∞(-1)mqβmwme[x+m]qαt.By using (2.9), we have(2.15)∑n=0∞Gn,q,w(α,β)(x)tnn!=t∫Zpϕw(y)e[x+y]qαtdμ-qβ(y).By (2.13) and (2.15) we have(2.16)∑n=0∞Gn,q,w(α,β)(x)tnn!=[2]qβt∑m=0∞(-1)mqβmwme[x+m]qαt.Remark 2.3.
In (2.14), we simply identify that
(2.17)lim⁡q→1Fq,w(α,β)(t,x)=2t∑n=0∞(-1)nwne(x+n)t=Fw(t,x).
Observe that ifq→1, then Fq,w(α,β)(t)→Fw(t) and Fq,w(α,β)(t,x)→Fw(t,x). Note that if q→1 and w=1, then Gn,q,w(α,β)→Gn and Gn,q,w(α,β)(x)→Gn(x).


## 3. Some Relations between Twistedq-Genocchi Numbers and  Polynomials with Weight α and Weak Weight β

By (2.11), we have the following complement relation.Theorem 3.1.
One has the property of complement(3.1)Gn,q-1,w-1(α,β)(1-x)=(-1)nwqα(n-1)Gn,q,w(α,β)(x).
Also, by (2.11), we have the following distribution relation.Theorem 3.2.
For any positive integerm(=odd), one has
(3.2)Gn,q,w(α,β)(x)=[2]qβ[2]qβm[m]qαn-1∑i=0m-1(-1)iwiqβiGn,qm,wm(α,β)(i+xm),n∈Z+.
Let f(x)=twxe[x]qαt. Then by (1.5), left-hand side is in the following form:
(3.3)qβnI-qβ(fn)+(-1)n-1I-qβ(f)=∑m=0∞(qβnwnGm,q,w(α,β)(n)+(-1)n-1Gm,q,w(α,β))tmm!.
And right-hand side in (1.5) is in the following form:
(3.4)[2]qβ∑l=0n-1(-1)n-1-lqβlf(l)=∑m=0∞[2]qβm∑l=0n-1(-1)n-1-lqβlwl[l]qαm-1tmm!.
By (3.3) and (3.4), one easily sees that
(3.5)qβnwnGm,q,w(α,β)(n)+(-1)n-1Gm,q,w(α,β)=[2]qβm∑l=0n-1(-1)n-1-lqβlwl[l]qαm-1.
Hence, we have the following theorem.Theorem 3.3 (Letm∈ℤ+).
Ifn≡0(mod2), then
(3.6)qβnwnGm,q,w(α,β)(n)-Gm,q,w(α,β)=[2]qβm∑l=0n-1(-1)l+1qβlwl[l]qαm-1.
Ifn≡1(mod2), then
(3.7)qβnwnGm,q,w(α,β)(n)+Gm,q,w(α,β)=[2]qβm∑l=0n-1(-1)lqβlwl[l]qαm-1.
Since[x+y]qα=[x]qα+qαx[y]qα, one easily obtains that
(3.8)Gn+1,q,w(α,β)(x)=(n+1)∫Zpϕw(y)[x+y]qαndμ-qβ(y)=(n+1)[2]qβ∑m=0∞(-1)mqβmwm[x+m]qαn.
From (1.4), one notes that
(3.9)[2]qβt=qβ∫Zptw(x+1)e[x+1]qαtdμ-qβ(x)+∫Zptwxe[x]qαtdμ-qβ(x)=∑n=0∞(qβwGn,q,w(α,β)(1)+Gn,q,w(α,β))tnn!.By using comparing coefficients oftn/n! in the previous equation, we easily obtain the following theorem.Theorem 3.4.
Forn∈ℤ+, one has
(3.10)qβwGn,q,w(α,β)(1)+Gn,q,w(α,β)={[2]qβ,ifn=1,0,ifn≠1.
By (3.8) and (3.10), we have the following corollary.Corollary 3.5.
Forn∈ℤ+, one has
(3.11)qβ-αw(qαGq,w(α,β)+1)n+Gn,q,w(α,β)={[2]qβ,ifn=1,0,ifn≠1,
with the usual convention of replacing (Gq,w(α,β))n by Gn,q,w(α,β).


## 4. The Analogue of the Genocchi Zeta Function

By usingq-Genocchi numbers and polynomials with weight α and weak weight β, q-Genocchi zeta function and Hurwitz q-Genocchi zeta functions are defined. These functions interpolate the q-Genocchi numbers and q-Genocchi polynomials with weight α and weak weight β, respectively. In this section we assume that q∈ℂ with |q|<1.From (2.5), we note that(4.1)dk+1dtk+1Fq,w(α,β)(t)|t=0=(k+1)[2]qβ∑m=0∞(-1)mqβmwm[m]qαk=Gk+1,q,w(α,β)(k∈N).
By using the previous equation, we are now ready to define q-Genocchi zeta functions.Definition 4.1.
Lets∈ℂ. One has
(4.2)ζq,w(α,β)(s)=[2]qβ∑n=1∞(-1)nqβnwn[n]qαs.
Note that ζq,w(α,β)(s) is a meromorphic function on ℂ. Observe that if q→1, then limq→1ζq,w(α,β)(s)=ζw(s).Theorem 4.2.
Relation betweenζq,w(α,β)(s) and Gk,q,w(α,β) is given by
(4.3)ζq,w(α,β)(-k)=Gk+1,q,w(α,β)k+1.
Observe that ζq,w(α,β)(s) interpolates Gk,q,w(α,β) at nonnegative integers.By using (2.14), one notes that(4.4)dk+1dtk+1Fq,w(α,β)(t,x)|t=0=(k+1)[2]qβ∑m=0∞(-1)mqβmwm[x+m]qαk=Gk+1,q,w(α,β)(x),(4.5)(ddt)k+1(∑n=0∞Gn,q,w(α,β)(x)tnn!)|t=0=Gk+1,q,w(α,β)(x),fork∈N.
By (4.5), we are now ready to define the twisted Hurwitz q-Genocchi zeta functions.Definition 4.3.
Lets∈ℂ. One has
(4.6)ζq,w(α,β)(s,x)=[2]qβ∑n=0∞(-1)nqβnwn[x+n]qαs.
Note that ζq,w(α,β)(s,x) is a meromorphic function on ℂ. Observe that if q→1, then limq→1ζq,w(α,β)(s,x)=ζw(s,x).Theorem 4.4.
Relation betweenζq,w(α)(s,x) and Gk,q,w(α)(x) is given by
(4.7)ζq,w(α,β)(-k,x)=Gk+1,q,w(α,β)(x)k+1.
Observe that ζq,w(α,β)(-k,x) interpolates Gk,q,w(α,β)(x) at nonnegative integers.



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