# Some Identities on theq-Genocchi Polynomials of Higher-Order and

Authors: Seog-Hoon Rim; Jeong-Hee Jin; Eun-Jung Moon; Sun-Jung Lee
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

A systemic study of some families ofq-Genocchi numbers and families of polynomials of Nörlund type is presented by using the multivariate fermionic p-adic integral on ℤp. The study of these higher-order q-Genocchi numbers and polynomials yields an interesting q-analog of identities for Stirling numbers.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper, ℤp,ℚp,ℂ, and ℂp 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, respectively. Let ℕ be the set of natural numbers and ℤ+=ℕ∪{0}. Let vp be the normalized exponential valuation of ℂp with |p|p=p-vp(p)=1/p.When one talks ofq-extension, q is variously considered as an indeterminate, a complex q∈ℂ, or a p-adic number q∈ℂp. If q∈ℂ, then one  normally assumes |q|<1. If q∈ℂp, then we assume |q-1|p<1. In this paper, we use the following notation:
(1.1)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q,
see [1–10]. Hence limq→1[x]q=x for all x∈ℤp.Theq-factorial is defined as [n]q!=[n]q[n-1]q⋯[2]q[1]q, and the Gaussian binomial coefficient is defined by the  standard rule
(1.2)(nk)q=[n]q![n-k]q![k]q!=[n]q[n-1]q⋯[n-k+1]q![k]q!,
(see [7, 9]). Note that limq→1(nk)q=(nk)=n!/(n-k)!k!=n(n-1)⋯(n-k+1)/k!. It readily follows from (1.2) that
(1.3)(n+1k)q=(nk-1)q+qk(nk)q=qn-k+1(nk-1)q+(nk)q,
(see [4, 7]).Theq-binomial formulas are known,
(1.4)(b;q)n=(1-b)(1-bq)⋯(1-bqn-1)=∑i=0n(ni)qq(i2)(-1)ibi,1(b;q)n=1(1-b)(1-bq)⋯(1-bqn-1)=∑i=0∞(n+i-1i)qbi.We say thatf:ℤp→ℂp is uniformly differentiable function at a point a∈ℤp, and we write f∈UD(ℤp), if the difference quotients Φf:ℤp×ℤp→ℂp such that Φf(x,y)=(f(x)-f(y))/(x-y) have a limit f′(a) as (x,y)→(a,a). For f∈UD(ℤp), the q-deformed fermionic p-adic integral is defined as
(1.5)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1f(x)(-q)x,
(see [7, 9]). Note that
(1.6)I-1(f)=lim⁡q→1I-q(f)=∫Zpf(x)dμ-1(x).
For n∈ℕ, write fn(x)=f(x+n). Then, we have
(1.7)I-1(fn)=(-1)nI-1(f)+2∑l=0n-1(-1)n-1-lf(l).
Using (1.7), we can readily derive the Genocchi polynomials, Gn(x), namely,
(1.8)t∫Zpe(x+y)tdμ-1(y)=2tet+1ext=∑n=0∞Gn(x)tnn!,
(see [1–27]). Note that Gn(0)=Gn are referred to as the nth Genocchi numbers. Let us now introduce the Genocchi polynomials of Nörlund type as follows:
(1.9)tr∫Zp⋯∫Zp︸rtimese(x+x1+⋯+xr)tdμ-1(x1)⋯dμ-1(xr)=(2tet+1)rext=∑n=0∞Gn(r)(x)tnn!,(1.10)(et+12t)rext=∑n=0∞Gn(-r)(x)tnn!,
(see [7, 9]). In the special case x=0,Gn(-r)(0)=Gn(-r), and Gn(r)(0)=Gn(r) are referred to as the Genocchi numbers of Nörlund type. Let (Eh)(x)=h(x+1) be the shift operator. Then, the q-difference operator Δq is defined as
(1.11)Δqn=∏i=1n(E-qi-1I),where(Ih)(x)=h(x),
(see [4, 7, 9]). It follows from (1.11) that
(1.12)f(x)=∑n≥0(xn)qΔqnf(0),
where Δqnf(0)=∑k=0n(nk)qq(k2)f(n-k) (see [5, 6, 10]). The q-Stirling number of the second kind (as defined by Carlitz) is given by
(1.13)S2(n,k;q)=q-(k2)[k]q!∑j=0k(-1)jq(j2)(kj)q[k-j]qn,
(see [7, 10]). By (1.12) and (1.13), we see that
(1.14)S2(n,k;q)=q-(k2)[k]q!Δqk0n,
(see [6, 10]).In this paper, theq-extensions of (1.9) are considered in several ways. Using these q-extensions, we derive some interesting identities and relations for Genocchi polynomials and numbers of Nörlund type. The purpose of this paper is to present a systemic study of some families q-Genocchi numbers and polynomials of Nörlund type by using the multivariate fermionic p-adic integral on ℤp.


## 2.q-Extensions of Genocchi Numbers and Polynomials of Nörlund Type

In this section, we assume thatq∈ℂp with |1-q|p<1. We first consider the q-extensions of (1.8) given by the rule
(2.1)∑n=0∞Gn,q(x)tnn!=t∫Zpe[x+y]qtdμ-1(y)=∑n=0∞2t(1-q)n∑l=0n((nl)(-1)lqlx1+ql)tnn!=2t∑m=0∞(-1)me[m+x]qt.Thus, we obtain the following lemma.Lemma 2.1.
Ifn≥0, then
(2.2)Gn+1,q(x)n+1=2∑m=0∞(-1)m[m+x]qn=2(1-q)n∑l=0n(nl)(-1)lqlx1+ql.By (1.14),
(2.3)[x]qn=∑k=0n(xk)q[k]q!S2(k,n-k;q)q(k2)=∑k=0n[x]q[x-1]q⋯[x-k+1]qq(k2)-(n-k2)[n-k]q!Δqn-k0k=∑k=0nq(k2)-(n-k2)[n-k]q!Δqn-k0k1(1-q)k∑l=0k(kl)qq(l2)(-1)lql(x-k+1).
Thus, we have
(2.4)Gn+1,qn+1=∑k=0nq(k2)S2(k,n-k;q)(1-q)k∑l=0k(kl)qq(l2)(-1)l∑m=0l(lm)(q-1)mGm+1,q(1-k)m+1,
and we obtain the following theorem.Theorem 2.2.
Ifn≥0, then
(2.5)Gn+1,qn+1=∑k=0nq(k2)S2(k,n-k;q)(1-q)k∑l=0k(kl)qq(l2)(-1)l∑m=0l(lm)(q-1)mGm+1,q(1-k)m+1,
where Gn,q=Gn,q(0) stand for the nth Genocchi numbers.Consider aq-extensoin of (1.9) such that G0,q(r)(x)=G1,q(r)(x)=⋯=Gr-1,q(r)(x)=0 and
(2.6)Gn+r,q(r)(x)r!(n+rr)=∫Zp⋯∫Zp[x+x1+⋯+xr]qndμ-1(x1)⋯dμ-1(xr)=2r(1-q)n∑l=0n(nl)(-1)lqlx(11+ql)r=2r∑m=0∞(m+r-1m)(-1)m[m+x]qn.
Let Fq(r)(t,x)=∑n=0∞Gn,q(r)(x)(tn/n!). Then,
(2.7)Fq(r)(t,x)=2rtr∑m=0∞(m+r-1m)(-1)me[m+x]qt.
In the special case x=0, the numbers Gn,q(r)(0)=Gn,q(r) are referred to as q-extension of the Genocchi numbers of order r. In the sense of the q-extension in (1.10), consider the q-extension of Genocchi polynomials of Nörlund type given by
(2.8)Gq(r)(t,x)=Fq(-r)(t,x)=12rtr∑m=0r(rm)e[m+x]qt=∑n=0∞Gn,q(-r)(x)tnn!.
By (2.8), G0,q(-r)(x)=G1,q(-r)(x)=⋯=Gr-1,q(-r)(x)=0 and r!(nr)Gn-r,q(r)(x)=(1/2r)∑m=0r(rm)[m+x]qn. Therefore, we obtain the following theorem.Theorem 2.3.
Forr∈ℕ, and, n≥0, write
(2.9)2rtr∑m=0∞(m+r-1m)(-1)me[m+x]qt=∑n=0∞Gn,q(r)(x)tnn!.
Then,
(2.10)Gn+r,q(r)(x)r!(n+rr)=2r(1-q)n∑l=0n(nl)(-1)lqlx(11+ql)r=2r∑m=0∞(m+r-1m)(-1)m[m+x]qn,r!(nr)Gn-r,q(-r)(x)=12r(1-q)n∑l=0n(nl)(-1)lqlx(1+ql)r=12r∑m=0r(rm)[m+x]qn.The numbersGn,q(-r)(0)=Gn,q(-r) are referred to as the q-extension of Genocchi numbers of Nörlund type. For h∈ℤ and r∈ℕ, introduce the extended higher-order q-Genocchi polynomials as follows:
(2.11)Gn+r,q(h,r)(x)r!(n+rr)=∫Zp⋯∫Zpq∑j=1r(h-j)xj[x+x1+⋯+xr]qndμ-1(x1)⋯dμ-1(xr).
Then,
(2.12)Gn+r,q(h,r)(x)r!(n+rr)=2r(1-q)n∑l=0n(nl)(-1)lqlx(-qh-1+l;q-1)r=2r(1-q)n∑l=0n(nl)(-1)lqlx(-qh-r+l;q)r=2r∑m=0∞(m+r-1m)q(-1)mq(h-r)m[x+m]qn.
Let Fq(h,r)(t,x)=∑n=0∞Gn,q(h,r)(x)(tn/n!). Then, we can readily see that
(2.13)Fq(h,r)(t,x)=2rtr∑m=0∞(m+r-1m)q(-1)mq(h-r)me[x+m]qt.
Therefore, we obtain the following theorem.Theorem 2.4.
Forh∈ℤ and n≥0, let
(2.14)2rtr∑m=0∞(m+r-1m)q(-1)mq(h-r)me[x+m]qt=∑n=0∞Gn,q(h,r)(x)tnn!.
Then,
(2.15)Gn+r,q(h,r)(x)r!(n+rr)=2r(1-q)n∑l=0n(nl)(-1)lqlx(-qh-r+l;q)r=2r∑m=0∞(m+r-1m)q(-1)mq(h-r)m[x+m]qn.Let us now define the extended higher-order Nörlund typeq-Genocchi polynomials as follows:
(2.16)r!(nr)Gn-r,q(h,-r)(x)=1(1-q)n∑l=0n(nl)(-1)lqlx∫Zp⋯∫Zpql(x1+⋯+xr)q∑j=1r(h-j)xjdμ-1(x1)⋯dμ-1(xr).
By (2.16),
(2.17)r!(nr)Gn-r,q(h,-r)(x)=12r(1-q)n∑l=0n(nl)(-1)lqlx(-qh-r+l;q)r=12r∑m=0r(rm)qq(m2)q(h-r)m[m+x]qn.
Let Fq(h,-r)(t,x)=∑n=0∞Gn,q(h,-r)(x)(tn/n!). Then, we have
(2.18)Fq(h,-r)(t,x)=12rtr∑m=0r(rm)qq(m2)q(h-r)me[m+x]qt,
where, G0,q(h,-r)(x)=G1,q(h,-r)(x)=⋯=Gr-1,q(h,-r)(x)=0. Therefore, we obtain the following theorem.Theorem 2.5.
Forh∈ℤ, n≥0, and r∈ℕ, write
(2.19)12rtr∑m=0r(rm)qq(m2)q(h-r)me[m+x]qt=∑n=0∞Gn,q(h,-r)(x)tnn!.
Then,
(2.20)r!(nr)Gn-r,q(h,-r)(x)=12r(1-q)n∑l=0n(nl)(-1)lqlx(-qh-r+l;q)r=12r∑m=0r(rm)qq(m2)q(h-r)m[m+x]qn,
where, G0,q(h,-r)(x)=G1,q(h,-r)(x)=⋯=Gr-1,q(h,-r)(x)=0.Forh=r,
(2.21)Gn+r,q(r,r)(x)r!(n+rr)=2r(1-q)n∑l=0n(nl)(-1)lqlx(-ql;q)r=2r∑m=0∞(m+r-1m)q(-1)m[x+m]qn,(2.22)r!(nr)Gn-r,q(r,-r)(x)=12r(1-q)n∑l=0n(nl)(-1)lqlx(-ql;q)r=12r∑m=0r(rm)qq(m2)[m+x]qn.
It can readily be seen that
(2.23)qmx2r(-qm-r;q)r=∫Zp⋯∫Zpq∑j=1r(m-j)xj+mxdμ-1(x1)⋯dμ-1(xr)=∫Zp⋯∫Zp([x+x1+⋯+xr]q(q-1)+1)mq-∑j=1rjxjdμ-1(x1)⋯dμ-1(xr)=∑l=0m(ml)(q-1)l∫Zp⋯∫Zp[x+x1+⋯+xr]qlq-∑j=1rjxjdμ-1(x1)⋯dμ-1(xr)=∑l=0m(ml)(q-1)lGl+r,q(0,r)(x)r!(l+rr).
By (2.23), qmx2r/(-qm-r;q)r=∑l=0m(ml)(q-1)l(Gl+r,q(0,r)(x)/r!(l+rr)r!(l+rr)). As is known,
(2.24)I-1(f1)+I-1(f)=2f(0),wheref1(x)=f(x+1).
It follows from (2.24) that
(2.25)qh-1∫Zp⋯∫Zp[x+1+x1+⋯+xr]qnq∑j=1r(h-j)xjdμ-1(x1)⋯dμ-1(xr)=-∫Zp⋯∫Zp[x+x1+⋯+xr]qnq∑j=1r(h-j)xjdμ-1(x1)⋯dμ-1(xr)+2∫Zp⋯∫Zp[x+x2+⋯+xr]qnq∑j=1r-1(h-1-j)xj+1dμ-1(x2)⋯dμ-1(xr).
By (2.25),
(2.26)qh-1Gn+r,q(h,r)(x+1)n+r+Gn+r,q(h,r)(x)n+r=2Gn+r-1,q(h-1,r-1)(x).
A simple manipulation shows that
(2.27)qx∫Zp⋯∫Zp[x+x1+⋯+xr]qnq∑j=1r(h-j+1)xjdμ-1(x1)⋯dμ-1(xr)=(q-1)∫Zp⋯∫Zp[x+x1+⋯+xr]qn+1q∑j=1r(h-j)xjdμ-1(x1)⋯dμ-1(xr)+∫Zp⋯∫Zp[x+x1+⋯+xr]qnq∑j=1r(h-j)xjdμ-1(x1)⋯dμ-1(xr).
By (2.27), qx(Gn+r,q(h+1,r)(x)/(n+1))=(q-1)(Gn+r+1,q(h,r)(x)/(n+r+1))+(Gn+r,q(h,r)(x)/(n+1)).Therefore, we obtain the following proposition.Proposition 2.6.
Forh∈ℤ, r∈ℕ and n≥0, the following equations
(2.28)qh-1Gn+r,q(h,r)(x+1)n+r+Gn+r,q(h,r)(x)n+r=2Gn+r-1,q(h-1,r-1)(x),qxGn+r,q(h+1,r)(x)n+1=(q-1)Gn+r+1,q(h,r)(x)n+r+1+Gn+r,q(h,r)(x)n+1
hold. Moreover, (qmx2r)/((-qm-r;q)r)=∑l=0m(ml)(q-1)l(Gl+r,q(0,r)(x)/r!(l+rr)(l+rr)).By (2.21),
(2.29)Gn+r,q-1(r,r)(r-x)r!(n+rr)=2r(1-q-1)n∑l=0n(nl)(-1)lq-l(r-x)(-q-l;q-1)r=(-1)nqn+(r2)2r(1-q)n∑l=0n(nl)(-1)lqx(-ql;q)r=(-1)nqn+(r2)Gn+r,q(r,r)(x)r!(n+rr).
Hence,
(2.30)∫Zp⋯∫Zp[r-x+x1+⋯+xr]q-1nq-∑j=1r(r-j)xjdμ-1(x1)⋯dμ-1(xr)=(-1)nqn+(r2)∫Zp⋯∫Zp[x+x1+⋯+xr]qnq∑j=1r(r-j)xjdμ-1(x1)⋯dμ-1(xr).
For h=r, Gn+r,q-1(r,r)(0)=(-1)nqn+(r2)Gn+r,q(r,r)(r). It also follows from (2.26) that
(2.31)qr-1Gn+r,q(r,r)(x+1)n+r+Gn+r,q(r,r)(x)n+r=2Gn+r-1,q(r-1,r-1)(x).The Stirling numbers of the first kind are defined as(2.32)∏k=1n(1+[k]qz)=∑k=0nS1(n,k;q)zk,
(see[6, 9]),
(2.33)q(m2)(rm)q=q(m2)[r]q⋯[r-m+1]q[m]q!=1[m]q!∏k=0m-1([r]q-[k]q).
It can readily be seen that
(2.34)∏k=0n-1(z-[k]q)=zn∏k=0n-1(1-[k]qz)=∑k=0nS1(n-1,k;q)(-1)kzn-k.
By (2.33) and (2.34),
(2.35)∏k=0m-1([r]q-[k]q)=∑k=0mS1(m-1,k;q)(-1)k[r]qm-k.
Formulas (2.22) and (2.35) imply the following assertion.Proposition 2.7.
Forr∈ℕ and n∈ℤ+,
(2.36)r!(nr)Gn-r,q(r,-r)(x)=12r[m]q!∑m=0r∑k=0mS1(m-1,k;q)(-1)k[r]qm-k[m+x]qn.The generalized Genocchi numbers and polynomials of Nörlund type are defined by(2.37)2rtr(ew1t+1)(ew2t+1)⋯(ewrt+1)ext=∑n=0∞Gn(r)(x∣w1,…,wr)tnn!,
and Gn(r)(w1,…,wr)=Gn(r)(0∣w1,…,wr). We can now also define a q-extension of (2.37) as follows. For w1,…,wr∈ℤp and δ1,…,δr∈ℤ, write (2.38)Gn+r,q(r)(x∣w1,…,wr;δ1,…,δr)r!(n+rr)=∫Zp⋯∫Zp[x1w1+⋯+xrwr+x]qndμ-qδ1(x1)⋯dμ-qδr(xr),andGn+r,q(r)(w1,…,wr;δ1,…,δr)=Gn+r,q(r)(0∣w1,…,wr;δ1,…,δr). Thus,
(2.39)Gn+r,q(r)(x∣w1,…,wr;δ1,…,δr)r!(n+rr)=[2]qδ1⋯[2]qδr(1-q)n∑l=0n(nl)(-1)lqlx(1+qδ1+lw1)⋯(1+qδr+lwr).
Another q-extension of Nörlund type generalized Genocchi numbers and polynomials is also of interest, namely,
(2.40)Gn+r,q*(r)(x∣w1,…,wr;δ1,…,δr)r!(n+rr)=∫Zp⋯∫Zp[x1w1+⋯+xrwr+x]qnqδ1x1+⋯+δrxrdμ-1(x1)⋯dμ-1(xr),
and Gn+r,q*(r)(w1,…,wr;δ1,…,δr)=Gn+r,q*(r)(0∣w1,…,wr;δ1,…,δr). By (2.40),
(2.41)Gn+r,q*(r)(x∣w1,…,wr;δ1,…,δr)r!(n+rr)=2r(1-q)n∑l=0n(nl)(-1)lqlx(1+qδ1+lw1)⋯(1+qδr+lwr).


## 3. Further Remarks

Forh=0, consider the following polynomials Gn+r,q(0,r)(x)/r!(n+rr) and r!(nr)Gn+r,q(0,-r)(x):
(3.1)Gn+r,q(0,r)(x)r!(n+rr)=∫Zp⋯∫Zp[x+x1+⋯+xr]qnq-∑j=1rjxjdμ-1(x1)⋯dμ-1(xr),r!(nr)Gn-r,q(0,-r)(x)=1(1-q)n∑l=0n(nl)(-1)lqlx∫Zp⋯∫Zpql(x1+⋯+xr)q-∑j=1rjxjdμ-1(x1)⋯dμ-1(xr).
Then,
(3.2)Gn+r,q(0,r)(x)r!(n+rr)=2r(1-q)n∑l=0n(nl)(-1)lqlx(-ql-r;q)r=2r∑m=0∞(m+r-1m)qq-rm(-1)m[x+m]qnr!(nr)Gn-r,q(0,-r)(x)=12r(1-q)n∑l=0n(nl)(-1)lqlx(-ql-r;q)r=12r∑m=0r(rm)qq(m2)q-rm[m+x]qn.
Let Fq(0,r)(t,x)=∑n=0∞Gn,q(0,r)(x)(tn/n!) and let Fq(0,-r)(t,x)=∑n=0∞Gn,q(0,-r)(x)(tn/n!). Then,
(3.3)Fq(0,r)(t,x)=2rtr∑m=0∞(m+r-1m)qq-rm(-1)me[x+m]qt,Fq(0,-r)(t,x)=12rtr∑m=0r(rm)qq(m2)q-rme[m+x]qt.
Consider the following polynomials:
(3.4)Gn+1,q(h,1)(x)n+1=∫Zpqx1(h-1)[x+x1]qndμ-1(x1)=2(1-q)n∑l=0n(nl)(-1)lqlx1+ql+h-1.
A simple calculation of the fermionic p-adic invariant integral on ℤp show that
(3.5)qx∫Zp[x+x1]qnqx1(h-1)dμ-1(x1)=(q-1)∫Zp[x+x1]qn+1qx1(h-2)dμ-1(x1)+∫Zp[x+x1]qnqx1(h-2)dμ-1(x1).
By (3.5), qxGn+1,q(h,1)(x)=(q-1)(Gn+2,q(h-1,1)(x)/2(n+2))+Gn+1,q(h-1,1)(x). It can readily be proved that
(3.6)∫Zp[x+x1]qnqx1(h-1)dμ-1(x1)=∑j=0n(nj)[x]qn-jqjx∫Zp[x1]qjqx1(h-1)dμ-1(x1).
By (3.6), Gn+1,q(h,1)(x)/(n+1)=∑j=0n(nj)[x]qn-jqjx(Gj+1,q(h,1)/(j+1)). Using (2.24), we can also prove that
(3.7)∫Zp[x+x1+1]qnq(x1+1)(h-1)dμ-1(x1)+∫Zp[x+x1]qnqx1(h-1)dμ-1(x1)=2[x]qn.
Thus,  qh-1(Gn+1,q(h,1)(x)/(n+1))+(Gn+1,q(h,1)(x)/(n+1))=2[x]qn. For x=0, we have qh-1(Gn+1,q(h,1)(1)/(n+1))+(Gn+1,q(h,1)/(n+1))=2δn,0, where δn,0 is the Kronecker delta.It is easy to see thatG1,q(h,1)=∫ℤpqx1(h-1)dμ-1(x1)=2/(1+qh-1)=2/([2]qh-1). By (3.4),
(3.8)Gn+1,q-1(h,1)(1-x)n+1=∫Zp[1-x+x1]q-1nq-x1(h-1)dμ-1(x1)=(-1)nqn+h-12(1-q)n∑l=0n(nl)(-1)lqlx1+ql+h-1=(-1)nqn+h-1Gn+1,q(h,1)(x)n+1.
In particular, if x=1, then Gn+1,q-1(h,1)(0)/(n+1)=(-1)nqn+h-1(Gn+1,q(h,1)(1)/(n+1))=(-1)n-1qn(Gn+1,q(h,1)/(n+1)) for n≥1.Recently, Kim has studiedp-adic fermionic integral on ℤp connected with the problems of mathematical physics (see [6, 10, 11]), and our result are closely related to his results. In the future, we will try to study p-adic stochastic problems associated with our theorems. For example, p-adic q-Bernstein polynomials seem to be closely related to our results (see [6, 14, 20]).



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*Source: 860280-2010-12-29.xml*