# New Construction Weighted(h,q)-Genocchi Numbers and Polynomials Related to Zeta Type Functions

Authors: Serkan Araci; Jong Jin Seo; Dilek Erdal
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

The fundamental aim of this paper is to construct(h,q)-Genocchi numbers and polynomials with weight α. We shall obtain some interesting relations
by using  p-adic q-integral on ℤp in the sense of fermionic. Also, we shall derive the (h,q)-extensions of zeta type functions with weight α from the Mellin transformation
of this generating function which interpolates the (h,q)-Genocchi numbers and polynomials with weight α at negative integers.


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## Body

## 1. Introduction, Definitions, and Notations

Letp be a fixed odd prime number. Throughout this paper we use the following notations. ℤp denotes the ring of p-adic rational integers, ℚ denotes the field of rational numbers, ℚp denotes the field of p-adic rational numbers, and ℂp denotes the completion of algebraic closure of ℚp. Let ℕ be the set of natural numbers and ℕ*=ℕ∪{0}. The p-adic absolute value is defined by |p|p=1/p. In this paper, we assume |q-1|p<1 as an indeterminate. In [1–3],   Kim defined the fermionic p-adic q-integral on ℤp as follows:(1.1)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1f(x)(-q)x.[x]q is a q-extension of x which is defined by (1.2)[x]q=1-qx1-q,
see [1–15].Note thatlimq→1[x]q=x.Letfn(x)=f(x+n). By the definition (1.1) we easily get(1.3)-qI-q(f1)=lim⁡N→∞1[pN]-q∑x=0pN-1f(x+1)(-q)x+1=lim⁡N→∞1[pN]-q∑x=0pN-1f(x)(-q)x-(1+q)lim⁡N→∞f(pN)qpN+f(0)1+qpN=I-q(f)-[2]qf(0).Continuing this process, we obtain easily the relation(1.4)qnI-q(fn)+(-1)n-1I-q(f)=[2]q∑l=0n-1(-1)n-l-1qlf(l),(h,q)-Genocchi numbers are defined as follows: (1.5)G0,q(h)=0,qh-2(qGq(h)+1)n+Gn,q(h)={[2]q,ifn=1,0,ifn>1,
with the usual convention about replacing (Gq(h))n by Gn,q(h) (see [6]).In this paper, we constructed(h,q)-Genocchi numbers and polynomials with weight α. By using fermionic p-adic q-integral equations on ℤp, we investigated some interesting identities and relations on the (h,q)-Genocchi numbers and polynomials with weight α. Furthermore, we derive the q-extensions of zeta type functions with weight α from the Mellin transformation of this generating function which interpolates the (h,q)-Genocchi polynomials with weight α.


## 2. On the Weighted(h,q)-Genocchi Numbers and Polynomials

In this section, by using fermionicp-adic q-integral equations on ℤp, some interesting identities and relation on the (h,q)-Genocchi numbers and polynomials with weight α are shown.Definition 2.1.
Letα,n∈ℕ* and h∈ℕ. Then the (h,q)-Genocchi numbers with weight α defined by as follows:
(2.1)G̃n+1,q(α,h)n+1=[2]q∑m=0∞(-1)mqmh[m]qαn.If we takeh=1 to (2.1), then we have, G̃n+1,q(α,1)=G̃n+1,q(α) (see [5]).From (2.1), we obtain (2.2)G̃n+1,q(α,h)n+1=[2]q(1-qα)n∑m=0∞(-1)mqmh(1-qmα)n=[2]q(1-qα)n[∑m=0∞(-1)mqmh∑l=0n(nl)(-1)l(qmα)l]=[2]q(1-qα)n∑l=0n(nl)(-1)l∑m=0∞(-1)mqmαl+mh=[2]q(1-qα)n∑l=0n(nl)(-1)l11+qαl+hTherefore, we obtain the following theorem.Theorem 2.2.
Forα,n∈ℕ* and h∈ℕ. Then
(2.3)G̃n+1,q(α,h)n+1=[2]q(1-qα)n∑l=0n(nl)(-1)l11+qαl+h.In (1.1), one takes f(x)=q(h-1)x[x]qαn,(2.4)∫Zpq(h-1)x[x]qαndμ-q(x)=1(1-qα)n∑l=0n(nl)(-1)l∫Zpqx(αl+h-1)dμ-q(x)=1(1-qα)n∑l=0n(nl)(-1)llim⁡N→∞1[pN]-q∑x=0pN-1(-qαl+h)x=1(1-qα)n∑l=0n(nl)(-1)l(1+q)1+qαl+hlim⁡N→∞1+(qαl+h)pN1+qpN=[2]q(1-qα)n∑l=0n(nl)(-1)l11+qαl+h=G̃n+1,q(α,h)n+1.From [12], we obtain (h,q)-Genocchi numbers with weight α witt's type formula as follows.Theorem 2.3.
Forα,n∈ℕ* and h∈ℕ. Then
(2.5)G̃n+1,q(α,h)n+1=∫Zpq(h-1)x[x]qαndμ-q(x).From (2.1), one easily gets(2.6)∫Zpq(h-1)xet[x]qαdμ-q(x)=t[2]q∑m=0∞(-1)mqmhet[m]qα.By (2.6), one has (2.7)∑n=0∞G̃n,q(α,h)tnn!=t[2]q∑m=0∞(-1)mqmhet[m]qα.Therefore, we obtain the following corollary.Corollary 2.4.
IfG̃0,q(α,h)=0. Let Dq(α,h)(t)=∑n=0∞G̃n,q(α,h)(tn/n!). Then
(2.8)Dq(α,h)(t)=t[2]q∑m=0∞(-1)mqmhet[m]qα.Now, one considers the(h,q)-Genocchi polynomials with weight α as follows:
(2.9)G̃n+1,q(α,h)(x)n+1=∫Zpq(h-1)y[x+y]qαndμ-q(y),n∈N,α∈N*.From (2.9), one sees that
(2.10)G̃n+1,q(α,h)(x)n+1=[2]q(1-qα)n∑l=0n(nl)(-1)lqαlx11+qαl+h=[2]q∑m=0∞(-1)mqmh[m+x]qαn.
Let Dq(α,h)(t,x)=∑n=0∞G̃n,q(α,h)(x)(tn/n!). Then, one has
(2.11)Dq(α,h)(t,x)=t[2]q∑m=0∞(-1)mqmhet[m+x]qα=∑n=0∞G̃n,q(α,h)(x)tnn!.By (1.4), one sees that (2.12)qhnG̃m+1,q(α,h)(n)m+1+(-1)n-1G̃m+1,q(α,h)m+1=[2]q∑l=0n-1(-1)n-l-1qhl[l]qαm.Therefore, we obtain the following theorem.Theorem 2.5.
Form,h∈ℕ, and α,n∈ℕ*, one has
(2.13)qhnG̃m+1,q(α,h)(n)m+1+(-1)n-1G̃m+1,q(α,h)m+1=[2]q∑l=0n-1(-1)n-l-1qhl[l]qαm.In (1.3), it is known that (2.14)qI-q(f1)+I-q(f)=[2]qf(0).If we takef(x)=q(h-1)xet[x]qα, then one has(2.15)[2]q=q∫Zpq(h-1)(x+1)et[x+1]qαdμ-q(x)+∫Zpq(h-1)xet[x]qαdμ-q(x)=∑m=0∞(qhG̃m+1,q(α,h)(1)m+1+G̃m+1,q(α,h)m+1)tmm!.Therefore, by (2.15), we obtain the following theorem.Theorem 2.6.
Forα∈ℕ* and m,h∈ℕ, one has
(2.16)G̃0,q(α,h)=0,qhG̃m+1,q(α,h)(1)m+1+G̃m+1,q(α,h)m+1={[2]q,ifm=0,0,ifm≠0.From (2.9), one can easily derive the following:(2.17)∫Zpq(h-1)y[x+y]qαndμ-q(y)=[d]qαn[d]-q∑a=0d-1(-1)aqha∫Zpqdy(h-1)[x+ad+y]qdαndμ(-q)d(y)=[d]qαn[d]-q∑a=0d-1(-1)aqhaG̃n+1,qd(α,h)((x+a)/d)n+1.Therefore, by (2.17), we obtain the following theorem.Theorem 2.7.
Ford≡1(mod2), n∈ℕ* and α,h∈ℕ(2.18)G̃n+1,q(α,h)(x)=[d]qαn[d]-q∑a=0d-1(-1)aqhaG̃n+1,qd(α,h)(x+ad).


## 3. Interpolation Function of the PolynomialsG̃n,q(α,h)(x)

In this section, we give interpolation function of the generating functions of(h,q)-Genocchi polynomials with weight α. For s∈ℂ and h∈ℕ, by applying the Mellin transformation to (2.11), we obtain (3.1)Iq(α,h)(s,x)=1Γ(s)∫0∞ts-2(-Dq(α,h)(-t,x))dt=[2]q∑m=0∞(-1)mqmh1Γ(s)∫0∞ts-1e-t[m+x]qαdt,
so we have (3.2)Iq(α,h)(s,x)=[2]q∑m=0∞(-1)mqmh[m+x]qαs.We defineq-extension zeta type function as follows.Theorem 3.1.
Fors∈ℂ, h∈ℕ, and α∈ℕ*. One has
(3.3)Iq(α,h)(s,x)=[2]q∑m=0∞(-1)mqmh[m+x]qαs.ℑq(α,h)(s,x) can be continued analytically to an entire function.By subsitutings=-n into (3.3) one easily gets (3.4)Iq(α,h)(-n,x)=G̃n+1,q(α,h)(x)n+1.
We obtain the following theorem.Theorem 3.2.
Forh∈ℕ and q,s∈ℂ, |q|<1. Then one defines
(3.5)Iq(α,h)(-n,x)=G̃n+1,q(α,h)(x)n+1.



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*Source: 487490-2011-09-26.xml*