# On Genocchi Numbers and Polynomials

Authors: Seog-Hoon Rim; Kyoung Ho Park; Eun Jung Moon
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

The main purpose of this paper is to study the distribution of
Genocchi polynomials. Finally, we construct the Genocchi zeta function which
interpolates Genocchi polynomials at negative integers.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this
paper, ℤp,ℚp,ℂ,
and ℂp will, respectively, denote the ring of p-adic rational integers, the field of p-adic rational numbers, the
complex number field, and the completion of the algebraic closure of ℚp.
Let vp be the normalized exponential valuation of ℂp with |p|p=p−vp(p)=1/p.
When one talks about q-extension, q is variously considered as an indeterminate, a
complex, q∈ℂ,
or a p-adic number, q∈ℂp.
If q∈ℂ, one normally assumes |q|<1.
If q∈ℂp,
then we assume |q−1|p<1.
The ordinary Genocchi polynomials are defined as the generating function:(1.1)F(t,x)=2tet+1ext=∑n=0∞Gn(x)tnn!,|t|<π.
For a fixed positive integer d with (p,d)=1,
set(1.2)X=Xd=limN←ℤdPNℤ,X1=ℤp,X*=⋃0<a<dp,(a,p)=1(a+dpℤp),a+dpNℤp={x∈X∣x≡a(moddpN)}(cf. [1–30]), where a∈ℤ satisfies the condition 0≤a<dpN.
We say that f is uniformly differential function at a∈ℤp and write f∈UD(ℤp) if the difference quotients, Ff(x,y)=(f(x)−f(y))/(x−y),
have a limit f'(a) as (x,y)→(a,a).
Throughout this paper, we use the following notation:(1.3)[x]−q=1−(−q)x1+q,[x]q=1−qx1−q.For f∈UD(ℤp),
the fermionic p-adic invariant q-integral on ℤp is defined as(1.4)I−q(f)=∫ℤpf(x)dμ−q(x)=limN→∞1[pN]−q∑x=0pN−1f(x)(−q)x;see [1–27]. Note that(1.5)I−1(f)=limq→1I−q(f)=∫ℤpf(x)dμ−1(x).In this paper, we investigate
some interesting integral equations related to I−1(f).
From these integral equations related to I−1(f),
we can derive many interesting properties of Genocchi numbers and polynomials.
The main purpose of this paper is to derive the distribution relations of the
Genocchi polynomials, and to construct the Genocchi zeta function which
interpolates the Genocchi polynomials at negative
integers.


## 2. Genocchi Numbers and Polynomials

The Genocchi
numbers are defined as(2.1)G0=0,(G+1)n+Gn={2ifn=1,0ifn>1,where Gn is replaced by Gn,
symbolically. The Genocchi polynomials are also defined as(2.2)Gn(x)=∑k=0n(nk)xn−kGk.From (2.1), we note that G1=1,G2=−1,G3=0,G4=1,…,G2k+1=0,
and G2k∈ℤ(k=1,2,…).
The fermionic p-adic invariant integral on ℤp is defined as(2.3)I−1(f)=∫ℤpf(x)dμ−1(x)=limN→∞1[pN]−1∑x=0pN−1f(x)(−1)x,see[1].Let f1(x) be translation with f1(x)=f(x+1). Then we have the following integral equation.
Note that I−1(f1)+I−1(f)=2f(0). From (2.3), we can derive(2.4)t∫ℤpextdμ−1(x)=∑n=0∞Gntnn!=t∑n=0∞Gn+1n+1tnn!,(2.5)t∫ℤpe(x+y)tdμ−1(y)=∑n=0∞Gn(x)tnn!=t∑n=0∞Gn+1(x)n+1tnn!.Thus, we obtain(2.6)∫ℤpxndμ−1(x)=Gn+1n+1,∫ℤp(x+y)ndμ−1(y)=Gn+1(x)n+1.For n∈ℕ,
we have(2.7)∫ℤpf(x+n)dμ−1(x)=(−1)n∫ℤpf(x)dμ−1(x)+2∑ℓ=0n−1(−1)n−1+ℓf(ℓ),see[1–27].By (2.6) and
(2.7), if we take f(x)=xk(k∈ℤ+),
we easily see that(2.8)∫ℤp(x+n)kdμ−1(x)−∫ℤpxkdμ−1(x)=2∑ℓ=0n−1(−1)ℓ−1ℓkifn≡0(mod2).Thus, we have(2.9)Gk+1(n)k+1−Gk+1k+1=2∑ℓ=0n−1(−1)ℓ−1ℓkifn≡0(mod2).If n≡1(mod2),
then we know that(2.10)∫ℤp(x+n)kdμ−1(x)+∫ℤpxkdμ−1(x)=2∑ℓ=0n−1(−1)ℓℓkifn≡1(mod2).Thus, we get(2.11)Gk+1(n)k+1+Gk+1k+1=2∑ℓ=0n−1(−1)ℓℓk,see[1–30].Let χ be the Dirichlet character with conductor d∈ℕ,
with d≡1(mod2).
Then, we consider the generalized Genocchi numbers attached to χ as follows:(2.12)Gn+1,χn+1=∫Xχ(x)xndμ−1(x),G0,χ=0,where n∈ℤ+.
From (2.7) and (2.12), we note that(2.13)t∫Xextχ(x)dμ−1(x)=2∑ℓ=0d−1(−1)ℓχ(ℓ)eℓtedt+1t=∑n=0∞Gn,χn!tn.By (2.12) and (2.13), it is not difficult
to show that(2.14)Gn+1,χn+1=∫Xχ(x)xndμ−1(x)=dn∑a=0d−1χ(a)(−1)a∫ℤp(ad+x)ndμ−1(x)=dn∑a=0d−1χ(a)(−1)aGn+1(a/d)n+1,(2.15)∫X(x+y)ndμ−1(y)=dn∑n=0d−1(−1)a∫ℤp(x+ad+y)ndμ−1(y).By (2.6) and
(2.15), we obtain the
following theorem.Theorem 2.1.
Letd∈ℕ with d≡1(mod2),
and let χ be the Dirichlet character with
conductor d.
Then, one has(2.16)Gn+1(x)n+1=dn∑a=0d−1(−1)aGn+1((x+a)/d)n+1(distributionrelationforGenocchipolynomials),(2.17)Gn+1,χn+1=dn∑a=0d−1χ(a)(−1)aGn+1(a/d)n+1.


## 3. Genocchi Zeta Function

LetF(t,x) be the generating function of Gk(x) in complex plane as follows:(3.1)F(t,x)=2textet+1=∑n=0∞Gn(x)tnn!=t∑n=0∞Gn+1(x)n+1tnn!,|t|<π.Then, we show
that(3.2)F(t,x)=2t∑n=0∞(−1)ne(n+x)t.By (3.1) and (3.2), we easily see
that(3.3)Gk(x)=dkF(t,x)dtk|t=0=2k∑n=0∞(−1)n(n+x)k−1.Therefore, we obtain the
following proposition.Proposition 3.1.
Fork∈ℕ,
one has(3.4)Gk(x)k=2∑n=0∞(−1)n(n+x)k−1.From Proposition3.1, we can derive the Genocchi zeta function which
interpolates Genocchi polynomials at negative
integers.Fors∈ℂ,
we define the Hurwitz-type Genocchi zeta function as follows.Definition 3.2.
Fors∈ℂ,(3.5)ζG(s,x)=2∑n=0∞(−1)n(n+x)s,ζG(s)=2∑n=1∞(−1)nns.By Proposition3.1 and Definition 3.2, we obtain the following theorem.Theorem 3.3.
Fork∈ℕ,
one has(3.6)ζG(1−k,x)=Gk(x)k,ζG(1−k)=Gkk.Letχ be the Dirichlet character with conductor d∈ℕ,
with d≡1(mod2),
and let Fχ(t) be the generating function in ℂ of Gn,χ.
Then, we have(3.7)Fχ(t)=2∑ℓ=0d−1(−1)ℓχ(ℓ)eℓtedt+1t=∑n=0∞Gn,χtnn!,|t|<πd.From (3.7), we
derive(3.8)Fχ(t)=∑n=0∞Gn,χtnn!=∑n=1∞Gn,χtnn!=t∑n=0∞Gn+1,χn+1tnn!=t∑n=0∞(dn∑a=0d−1χ(a)(−1)aGn+1(a/d)n+1)tnn!=∑a=0d−1χ(a)(−1)a(dt∑n=0∞Gn+1(a/d)n+1dntnn!).By (3.1), (3.2), and (3.8), we
easily see that(3.9)Fχ(t)=∑a=0d−1χ(a)(−1)a(2t∑k=0∞(−1)ke(k+a/d)dt)=2t∑k=0∞∑a=0d−1χ(a+dk)(−1)a+dke(dk+a)t=2t∑n=0∞χ(n)(−1)nent=2t∑n=1∞χ(n)(−1)nent.From (3.9), we can derive(3.10)Gk,χ=dkdtkFχ(t)|t=0=k(2∑n=1∞χ(n)(−1)nnk−1).Thus, we have(3.11)Gk,χk=2∑k=0∞χ(n)(−1)nnk−1.Now, we consider the
Dirichlet-type Genocchi ℓ-function in complex plane as follows. For s∈ℂ,
define(3.12)ℓG,χ(s)=2∑n=1∞(−1)nχ(n)ns.By (3.11) and (3.12), we obtain the
following theorem.Theorem 3.4.
Letχ be the Dirichlet character with conductor d∈ℕ,
with d≡1(mod2),
and let k∈ℤ+.
Then, one has(3.13)ℓG,χ(1−k)=Gk,χk.Remark 3.5.
In [1], we can observe the value
of Genocchi zeta function at positive integers as
follows:(3.14)ζG(2n)=(−1)n−1π2n(2−4n)2(2n)!(1−4n)G2n,cf.[1].



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*Source: 898471-2008-09-24.xml*