# On theq-Genocchi Numbers and Polynomials with Weight α and Weak Weight β

Authors: J. Y. Kang; H. Y. Lee; N. S. Jung
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We construct a new type ofq-Genocchi numbers and polynomials with weight α and weak weight β:Gn,q(α,β),Gn,q(α,β)(x), respectively. Some interesting results and relationships are obtained.


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## Body

## 1. Introduction

The Genocchi numbers and polynomials possess many interesting properties and are arising in many areas of mathematics and physics. Recently, many mathematicians have studied in the area of theq-Genocchi numbers and polynomials (see [1–13]). In this paper, we construct a new type of q-Genocchi numbers Gn,q(α,β) and polynomials Gn,q(α,β)(x) with weight α and weak weight β.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 assumes that |q-1|p<p-(1/p-1) so that qx=exp(xlogq) for |x|p≤1. Throughout this paper, we use the 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. [3–6].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. [3–6]).As-well-known definition, the Genocchi polynomials are defined by(1.6)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). In the special case, x=0, Gn(0)=Gn are called the n-th Genocchi numbers (cf. [1–11]).These numbers and polynomials are interpolated by the Genocchi zeta function and Hurwitz-type Genocchi zeta function, respectively.(1.7)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 q-Genocchi numbers Gn,q(α,β) and polynomials Gn,q(α,β)(x) with weight α and weak weight β. We investigate some properties which are related to q-Genocchi numbers Gn,q(α,β) and polynomials Gn,q(α,β)(x) with weight α and weak weight β. We also derive the existence of a specific interpolation function which interpolates q-Genocchi numbers Gn,q(α,β) and polynomials Gn,q(α,β)(x) with weight α and weak weight β at negative integers.


## 2.q-Genocchi Numbers and Polynomials with Weight α and Weak Weight  β

Our primary goal of this section is to defineq-Genocchi numbers Gn,q(α,β) and polynomials Gn,q(α,β)(x) with weight α and weak weight β. We also find generating functions of q-Genocchi numbers Gn,q(α,β) and polynomials Gn,q(α,β)(x) with weight α and weak weight β.Forα∈ℤ and q∈ℂp with |1-q|p≤1, q-Genocchi numbers Gn,q(α,β) are defined by (2.1)Gn,q(α,β)=n∫Zp[x]qαn-1dμ-qβ(x).
By using p-adic q-integral on ℤp, we obtain (2.2)n∫Zp[x]qαn-1dμ-qβ(x)=nlim⁡N→∞⁡1[pN]-qβ∑x=0pN-1[x]qαn-1(-qβ)x=n[2]qβ(11-qα)n-1∑l=0n-1(n-1l)(-1)l11+qαl+β=n[2]qβ∑m=0∞(-1)mqβm[m]qαn-1.
By (2.1), we have (2.3)Gn,q(α,β)=n[2]qβ∑m=0∞(-1)mqβm[m]qαn-1.
From the above, we can easily obtain that (2.4)Fq(α,β)(t)=∑n=0∞Gn,q(α,β)tnn!=t[2]qβ∑m=0∞(-1)mqβme[m]qαt.
Thus, q-Genocchi numbers Gn,q(α,β) with weight α and weak weight β are defined by means of the generating function (2.5)Fq(α,β)(t)=t[2]qβ∑m=0∞(-1)mqβme[m]qαt.Using similar method as above, we introduceq-Genocchi polynomials Gn,q(α,β)(x) with weight α and weak weight β.Gn,q(α,β)(x) are defined by (2.6)Gn,q(α,β)(x)=n∫Zp[x+y]qαn-1dμ-qβ(y).
By using p-adic q-integral, we have (2.7)Gn,q(α,β)(x)=n[2]qβ(11-qα)n-1∑l=0n-1(n-1l)(-1)lqαxl11+qαl+β.
By using (2.6) and (2.7), we obtain (2.8)Fq(α,β)(t,x)=∑n=0∞Gn,q(α,β)(x)tnn!=t[2]qβ∑m=0∞(-1)mqβme[m+x]qαt.Remark 2.1.
In (2.8), we simply see that
(2.9)lim⁡q→1Fq(α,β)(t,x)=2t∑m=0∞(-1)me(m+x)t=2t1+etext=F(t,x).
Since [x+y]qα=[x]qα+qαx[y]qα, we easily obtain that
(2.10)Gn+1,q(α,β)(x)=(n+1)∫Zp[x+y]qαndμ-qβ(y)=q-αx∑k=0n+1(n+1k)[x]qαn+1-kqαxkGk,q(α,β)=q-αx([x]qα+qαxGq(α,β))n+1=(n+1)[2]qβ∑m=0∞(-1)mqβm[x+m]qαn.
Observe that, if q→1, then Gn,q(α,β)→Gn and Gn,q(α,β)(x)→Gn(x).
By (2.7), we have the following complement relation.Theorem 2.2.
Property of complement(2.11)Gn,q-1(α,β)(1-x)=(-1)n-1qα(n-1)Gn,q(α,β)(x).By (2.7), we have the following distribution relation.Theorem 2.3.
For any positive integerm (=odd), one has
(2.12)Gn,q(α,β)(x)=[2]qβ[2]qβm[m]qαn-1∑i=0m-1(-1)iqβiGn,qm(α,β)(i+xm),n∈Z+.
By (1.5), (2.1), and (2.6), one easily sees that
(2.13)m[2]qβ∑l=0n-1(-1)n-1-lqβl[l]qαm-1=qβnGm,q(α,β)(n)+(-1)n-1Gm,q(α,β).Hence, we have the following theorem.Theorem 2.4.
Letm∈ℤ+.
Ifn≡0(mod2), then
(2.14)qβnGm,q(α,β)(n)-Gm,q(α,β)=m[2]qβ∑l=0n-1(-1)l+1qβl[l]qαm-1.
If n≡1(mod2), then
(2.15)qβnGm,q(α,β)(n)+Gm,q(α,β)=m[2]qβ∑l=0n-1(-1)lqβl[l]qαm-1.
From (1.4), one notes that
(2.16)[2]qβt=qβ∫Zpte[x+1]qαtdμ-qβ(x)+∫Zpte[x]qαtdμ-qβ(x)=∑n=0∞(qβ∫Zpn[x+1]qαn-1dμ-qβ(x)+∫Zpn[x]qαn-1dμ-qβ(x))tnn!=∑n=0∞(qβGn,q(α,β)(1)+Gn,q(α,β))tnn!.Therefore, we obtain the following theorem.Theorem 2.5.
Forn∈ℤ+, one has
(2.17)qβGn,q(α,β)(1)+Gn,q(α,β)={[2]qβ,ifn=1,0,ifn≠1.By Theorem2.4 and (2.10), we have the following corollary.Corollary 2.6.
Forn∈ℤ+, one has
(2.18)qβ-α(qαGq(α,β)+1)n+Gn,q(α,β)={[2]qβ,ifn=1,0,ifn≠1.
with the usual convention of replacing (Gq(α,β))n by Gn,q(α,β).


## 3. 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 (2.4), we note that (3.1)dk+1dtk+1Fq(α,β)(t)|t=0=(k+1)[2]qβ∑m=0∞(-1)mqβm[m]qαk=Gk+1,q(α,β),(k∈N).
By using the above equation, we are now ready to define q-Genocchi zeta functions.Definition 3.1.
Lets∈ℂ. We define
(3.2)ζq(α,β)(s)=[2]qβ∑n=1∞(-1)nqβn[n]qαs.Note thatζq(α,β)(s) is a meromorphic function on ℂ. Note that, if q→1, then ζq(α,β)(s)=ζ(s) which is the Genocchi zeta functions. Relation between ζq(α,β)(s) and Gk,q(α,β) is given by the following theorem.Theorem 3.2.
Fork∈ℕ, we have
(3.3)ζq(α,β)(-k)=Gk+1,q(α,β)k+1.Observe that ζq(α,β)(s) function interpolates Gk,q(α,β) numbers at nonnegative integers. By using (2.3), one notes that
(3.4)dk+1dtk+1Fq(α,β)(t,x)|t=0=(k+1)[2]qβ∑m=0∞(-1)mqβm[x+m]qαk=Gk+1,q(α,β)(x),(k∈N),(3.5)(ddt)k+1(∑n=0∞Gn,q(α,β)(x)tnn!)|t=0=Gk+1,q(α,β)(x),fork∈N.By (3.2) and (3.5), we are now ready to define the Hurwitz q-Genocchi zeta functions.Definition 3.3.
Lets∈ℂ. We define
(3.6)ζq(α,β)(s,x)=[2]qβ∑n=0∞(-1)nqβn[n+x]qαs.
Note that ζq(α,β)(s,x) is a meromorphic function on ℂ.Remark 3.4.
It holds that(3.7)lim⁡q→1ζq(α,β)(s,x)=2∑n=0∞(-1)n(n+x)s.Relation betweenζq(α)(s,x) and Gk,q(α)(x) is given by the following theorem.Theorem 3.5.
Fork∈ℕ, one has
(3.8)ζq(α,β)(-k,x)=Gk+1,q(α,β)(x)k+1.
Observe that ζq(α,β)(-k,x) function interpolates Gk,q(α,β)(x) numbers at nonnegative integers.



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*Source: 743939-2012-05-08.xml*