# A Note on Some Properties of the Weightedq-Genocchi Numbers and Polynomials

Authors: L. C. Jang
Journal: Journal of Applied Mathematics
 (2011)

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## Abstract

We consider the weightedq-Genocchi numbers and polynomials. From the construction of the weighted q-Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials.


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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 integers, the field, of p-adic rational numbers, the complex number field and the completion of algebraic closure of ℚp. Let νp be the normalized exponential valuation of ℂp such that |p|p=p-νp(p)=1/p(see [1–16]).As well-known definition, the Euler numbers and Genocchi numbers are defined by(1.1)2et+1=eEt=∑n=0∞Entnn!,
with the usual convention of replacing En by En and(1.2)2tet+1=eGt=∑n=0∞Gntnn!,
with the usual convention of replacing Gn by Gn. We assume that q∈ℂp with |1-q|p<1 and that the q-number of x is defined by(1.3)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q
(see [1–19]).In [9], Kim introduced ordinary fermionic p-adic integral on ℤp, and he studied some interesting relations and identities related to q-extension of Euler numbers and polynomials. In [8], he also introduced the q-extension of the ordinary fermionic p-adic integral on ℤp and he investigated many physical properties related to q-Euler numbers and polynomials. Recently, Kim firstly introduced the meaning of the weighted q-Euler numbers and polynomials associated with the weighted q-Bernstein polynomials by using the fermionic invariant p-adic integral on ℤp(see [14, 15]). In [16], Ryoo tried to study the weighted q-Euler number and polynomials by the same method of Kim et al. in [14] and the q-extension of the fermionic p-adic invariant integrals on ℤp. As well-known properties, the Genocchi numbers are integers. The first few Genocchi numbers for n=2,4,… are -1,1,-3,17,-155,2073,…. The first few prime Genocchi numbers are −3 and 17, which occur for n=6 and 8. There are no others with n<105. These properties are very important to study in the area of fermionic distribution and p-adic numbers theory. By this reason, many mathematicians and physicians have studied Genocchi and Euler numbers which are in the different areas. By the same motivation, we consider weighted q-Genocchi polynomials and numbers by using the fermionic p-adic q-integral on ℤp which are constructed by Kim and Ryoo (cf. [8, 16]).In this paper, we consider theq-Genocchi numbers and polynomials with weighted α(α∈ℚ). From the construction of the weighted q-Genocchi numbers and polynomials, we investigate many interesting identities and relations satisfied by these new numbers and polynomials.


## 2. The Weightedq-Genocchi Numbers and Polynomials

LetUD(ℤp) be the space of uniformly differentiable functions and, for f∈UD(ℤp), the fermionic p-adic invariant integral of f on ℤp is defined by Kim as follows:(2.1)I-1(f)=∫Zpf(x)dμ-1(x)=lim⁡N→∞1[pN]-1∑x=0pN-1f(x)(-1)x
(see [1–16]). If we take f(x)=text, then we get(2.2)t∫Zpextdμ-1(x)=2tet+1.
By (1.2) and (2.2), we get(2.3)∑n=0∞Gntnn!=∑n=0∞∫Zpxndμ-1(x)tn+1n!=∑n=0∞(n+1)∫Zpxndμ-1(x)tn+1(n+1)!=∑n=1∞n∫Zpxn-1dμ-1(x)tnn!.
From (2.3),(2.4)G0=0,Gnn=∫Zpxn-1dμ-1(x),n∈N.
For f∈UD(ℤp), the fermionic p-adic q-integral of f on ℤp is defined by Kim as follows:(2.5)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1f(x)(-q)x
(see [1–16]). From (2.5), we note that(2.6)qnI-q(fn)=(-1)nI-q(f)+[2]q∑l=0n-1(-1)n-1-lqlf(l),
where n∈ℕ and fn(x)=f(x+n).Forα∈ℚ, we consider the following fermionic p-adic q-integral on ℤp:(2.7)t∫Zpe[x]qαtdμ-q(x)=∑n=0∞G̃n,q(α)tnn!,
where G̃n,q(α) are called the nth q-Genocchi numbers with weight α. From (2.7), we get(2.8)t∫Zpe[x]qαtdμ-q(x)=t∑n=0∞∫Zp[x]qαndμ-q(x)tnn!=∑n=0∞(n+1)∫Zp[x]qαndμ-q(x)tn+1(n+1)!=∑n=1∞n∫Zp[x]qαn-1dμ-q(x)tnn!.
By comparing the coefficients on the both sides of (2.7) and (2.8), we get(2.9)n∫Zp[x]qαn-1dμ-q(x)=G̃n,q(α),n∈N,G̃0,q(α)=0.
From (2.9), we obtain the following theorem.Theorem 2.1.
Forn∈ℕ and α∈ℚ, one has
(2.10)∫Zp[x]qαn-1dμ-q(x)=G̃n,q(α)n,G̃0,q(α)=0.By the definition of fermionicp-adic q-integrals, we get(2.11)t∫Zp[x]qαn-1dμ-q(x)=1(1-qα)n-1∑l=0n-1(n-1l)(-1)l∫Zpqαlxdμ-q(x)=[2]q(1-qα)n-1∑l=0n-1(n-1l)(-1)l11+qαl+1.
Therefore, we obtain the following theorem.Theorem 2.2.
Forn∈ℕ and α∈ℚ, we have
(2.12)G̃n,q(α)n=[2]q(1-q)n-1[α]qn-1∑l=0n-1(n-1l)(-1)l11+qαl+1.By Theorem2.2, we have the generating function of G̃n,q(α) as follows:(2.13)∑n=0∞G̃n,q(α)tnn!=[2]q∑n=0∞n(1-qα)n-1∑l=0n-1(n-1l)(-1)l∑m=0∞(-1)mqαlm+mtnn!=[2]q∑m=0∞(-1)mqm∑n=0∞n(1-qα)n-1∑l=0n-1(n-1l)(-1)lqαlmtnn!=[2]q∑m=0∞(-1)mqm∑n=0∞n(1-qα)n-1(1-qαm)n-1tnn!=[2]q∑m=0∞(-1)mqm∑n=0∞n[m]qαn-1tnn!=[2]q∑m=0∞(-1)mqm∑n=1∞[m]qαn-1tn(n-1)!=[2]q∑m=0∞(-1)mqm∑n=0∞[m]qαntn+1n!=[2]qt∑m=0∞(-1)mqme[m]qαt.
Let F̃q(α)(t) be the generating function of G̃n,q(α). Then, by (2.9) and (2.13), we get(2.14)F̃q(α)(t)=[2]qt∑m=0∞(-1)mqme[m]qαt=∑n=0∞G̃n,q(α)tnn!.Theq-Genocchi polynomials with weight α are defined by(2.15)F̃q(α)(t,x)=t∫Zpe[x+y]qαtdμ-q(y)=∑n=0∞G̃n,q(α)(x)tnn!.
From (2.15), we get(2.16)t∫Zpe[x+y]qαtdμ-q(y)=∑n=0∞∫Zp[x+y]qαndμ-q(y)tn+1n!=∑n=0∞(n+1)∫Zp[x+y]qαndμ-q(y)tn+1(n+1)!=∑n=1∞n∫Zp[x+y]qαn-1dμ-q(y)tnn!.
By (2.15) and (2.16), we obtain the following theorem.Theorem 2.3.
Forn∈ℕ and α∈ℚ, one has
(2.17)n∫Zp[x+y]qαn-1dμ-q(y)=G̃n,q(α)(x),G̃0,q(α)(x)=0.We note that(2.18)∫Zp[x+y]qαn-1dμ-q(y)=∑l=0n-1(n-1l)[x]qαn-1-lqαlx∫Zp[y]qαldμ-q(y)=∑l=0n-1(n-1l)[x]qαn-1-lqαlxG̃l+1,q(α)l+1.
From (2.17) and (2.18), we obtain the following theorem.Theorem 2.4.
Forn∈ℕ and α∈ℚ, one has
(2.19)G̃n,q(α)(x)n=∑l=0n-1(n-1l)[x]qαn-1-lqαlxG̃l+1,q(α)l+1.From (2.15), we note that(2.20)F̃q(α)(t,x)=∑n=0∞G̃n,q(α)(x)tnn!=∑n=0∞n∫Zp[x+y]qαn-1dμ-q(y)tnn!=∑n=0∞n(1(1-qα)n-1∑l=0n-1(n-1l)qαlx(-1)l∫Zpqαlydμ-q(y))tnn!=[2]q∑n=0∞n(1(1-qα)n-1∑l=0n-1(n-1l)qαlx(-1)l1+qαl+1)tnn!=[2]q∑n=0∞n(1(1-qα)n-1∑l=0n-1(n-1l)qαlx(-1)l∑m=0∞(-1)mqαlm+m)tnn!=[2]q∑m=0∞(-1)mqm∑n=0∞n(1(1-qα)n-1∑l=0n-1(n-1l)(-1)lqα(x+m)l)tnn!=[2]q∑m=0∞(-1)mqm∑n=0∞n(1(1-qα)n-1(1-qα(x+m))n-1)tnn!=[2]q∑m=0∞(-1)mqm∑n=0∞n[x+m]qαn-1tnn!=[2]q∑m=0∞(-1)mqm∑n=1∞[x+m]qαn-1tn(n-1)!=[2]q∑m=0∞(-1)mqm∑n=0∞[x+m]qαntn+1n!=[2]qt∑m=0∞(-1)mqme[x+m]qαt.
Therefore, we obtain the following theorem.Theorem 2.5.
Forα∈ℚ, one has
(2.21)F̃q(α)(t,x)=[2]qt∑m=0∞(-1)mqme[x+m]qαt.From (2.15) and (2.21), we obtain that(2.22)G̃n,q(α)(x)=dndtnF̃q(α)(t,x)|t=0=n[2]q∑m=0∞(-1)mqm[x+m]qαm-1=n[2]q1(1-qα)n-1∑l=0n-1(n-1l)qαlx(-1)l1+qαl+1=n[2]q(1-q)n-1[α]qn-1∑l=0n-1(n-1l)(-1)lqαlx11+qαl+1.
Therefore, we obtain the following theorem.Theorem 2.6.
Forn∈ℕ and α∈ℚ, one has
(2.23)G̃n,q(α)(x)=n[2]q(1-q)n-1[α]qn-1∑l=0n-1(n-1l)(-1)lqαlx1+qαl+1.From (2.6), if we take f(x)=[x]qαm=((1-qαx)/(1-qα))m, then we get(2.24)qn∫Zp[x+n]qαmdμ-q(x)=(-1)n∫Zp[x]qαmdμ-q(x)+[2]q∑l=0n-1(-1)lql[l]qαm.
By (2.17) and (2.24), we obtain the following theorem.Theorem 2.7.
Forn∈ℕ,m∈ℤ+=ℕ∪{0}, and α∈ℚ, one has
(2.25)qnG̃m+1,q(α)(n)m+1=(-1)nG̃m+1,q(α)m+1+[2]q∑l=0n-1(-1)lql[l]qαm.We remark that if we taken=2s(s∈ℤ+) in Theorem 2.7, then we have(2.26)q2sG̃m+1,q(α)(2s)m+1=G̃m+1,q(α)m+1+[2]q∑l=02s-1(-1)lql[l]qαm
and if we take n=2s+1(s∈ℤ+) in Theorem 2.7, then we have(2.27)q2s+1G̃m+1,q(α)(2s+1)m+1+G̃m+1,q(α)m+1=[2]q∑l=02s(-1)lql[l]qαm.
From (2.27) with s=0, we obtain the following corollary.Corollary 2.8.
Forα∈ℚ and m∈ℤ+, one has
(2.28)qG̃m+1,q(α)(1)m+1+G̃m+1,q(α)m+1={[2]qifm=0,0ifm>0.From (2.19), we note that(2.29)G̃m+1,q(α)(n)m+1=∑l=0m(ml)[x]qαm-lG̃l+1,q(α)l+1qαlx=1m+1∑l=0m(m+1l+1)[x]qαm-lG̃l+1,q(α)qαlx=1m+1∑l=1m(m+1l)[x]qαm+1-lG̃l,q(α)qα(l-1)x=1qα1m+1∑l=0m(m+1l)[x]qαm+1-lG̃l,q(α)qαlx.
From (2.29), we get(2.30)qαG̃m+1,q(α)(x)=∑l=0m+1(m+1l)[x]qαm+1-lG̃l,q(α)qαlx=([x]qα+qαxG̃q(α))m+1,
with the usual convention about replacing (G̃q(α))n by G̃n,q(α). By (2.28) and (2.30), we get(2.31)q1-αqαG̃m+1,q(α)(1)m+G̃m+1,q(α)m+1=q1-α(1+qαG̃q(α))m+1m+1+G̃m+1,q(α)m+1.
From (2.28) and (2.31), we obtain the following theorem.Theorem 2.9.
Forα∈ℚ and m∈ℤ+, one has
(2.32)q1-α(1+qαG̃q(α))m+1+G̃m+1,q(α)={[2]qifm=0,0ifm>0.



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