# Novel Identities forq-Genocchi Numbers and Polynomials

Authors: Serkan Araci
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

The essential aim of this paper is to introduce novel identities forq-Genocchi numbers and polynomials by using  the method by T. Kim et al. (article in press). We show that these polynomials are related to p-adic analogue of Bernstein polynomials. Also, we derive relations between q-Genocchi and q-Bernoulli numbers.


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## Body

## 1. Preliminaries

Imagine thatp be a fixed odd prime number. We now start with definition of the following notations. Let ℚp be the field p-adic rational numbers and let ℂp be the completion of algebraic closure of ℚp.Thus,(1.1)Qp={x=∑n=-k∞anpn:0≤an<p}.Thenℤp is integral domain, which is defined by
(1.2)Zp={x=∑n=0∞anpn:0≤an<p},
or
(1.3)Zp={x∈Qp:|x|p≤1}.We assume thatq∈ℂp with |1-q|p<1 as an indeterminate. The p-adic absolute value |·|p, is normally defined by
(1.4)|x|p=p-r,
where x=pr(s/t) with (p,s)=(p,t)=(s,t)=1, and r∈ℚ.[
x
]
q is a q-extension of x, which is defined by
(1.5)[x]q=1-qx1-q,
we note that limq→1[x]q=x (see [1–17]).Throughout this paper, we use notation ofℕ*:=ℕ∪{0}, where ℕ denotes set of Natural numbers.We say thatf is a uniformly differentiable function at a point a∈ℤp, if the difference quotient,
(1.6)Ff(x,y)=f(x)-f(y)x-y,
has a limit f′(a) as (x,y)→(a,a), and we denote this by f∈UD(ℤp).  Then, for f∈UD(ℤp), we can start with the following expression:
(1.7)1[pN]q∑0≤ξ<pN‍f(ξ)qξ=∑0≤ξ<pN‍f(ξ)μq(ξ+pNZp),
which represents a p-adic q-analogue of Riemann sums for f.  The integral of f on ℤp will be defined as the limit (N→∞) of these sums, when it exists. The p-adic q-integral of function f∈UD(ℤp) is defined by Kim in [7, 12] as
(1.8)Iq(f)=∫Zp‍f(ξ)dμq(ξ)=lim⁡N→∞1[pN]q∑ξ=0pN-1‍f(ξ)qξ.The bosonic integral is considered as a bosonic limitq→1, I1(f)=limq→1Iq(f). Similarly, the fermionic p-adic integral on ℤp is introduced by Kim as follows:
(1.9)I-q(f)=∫Zp‍f(ξ)dμ-q(ξ)
(for more details, see [13–16]).From (1.9), it is well-known equality that
(1.10)qI-q(f1)+I-q(f)=[2]qf(0),
where f1(x)=f(x+1) (for details, see [2, 3, 8, 9, 12, 13, 15–17]).Theq-Genocchi polynomials with wegiht 0 are introduced as
(1.11)G~n+1,q(x)n+1=∫Zp‍(x+ξ)ndμ-q(ξ).From (1.11), we have
(1.12)G~n,q(x)=∑l=0n‍(nl)xlG~n-l,q,
where G~n,q(0):=G~n,q are called q-Genocchi numbers with weight 0. Then, q-Genocchi numbers are defined as
(1.13)G~0,q=0,q(G~q+1)n+G~n,q={[2]q,ifn=1,0,ifn≠1,
with the usual convention about replacing (G~q)n by G~n,q is used (for details, see [3]).LetUD(ℤp) be the space of continuous functions on ℤp. For f∈UD(ℤp), p-adic analogue of Bernstein operator for f is defined by
(1.14)Bn(f,x)=∑k=0nf(kn)Bk,n(x)=∑k=0nf(kn)(nk)xk(1-x)n-k,
where n,k∈ℕ*. Here, Bk,n(x) is called p-adic Bernstein polynomials, which are defined by
(1.15)Bk,n(x)=(nk)xk(1-x)n-k,x∈[0,1]
(for details, see [1, 4, 5, 7]).Theq-Bernoulli polynomials and numbers with weight 0 are defined by Kim et al., respectively,
(1.16)B~n,q(x)=limn→∞1[pn]q∑y=0pn-1(x+y)nqy=∫ℤp(x+ξ)ndμq(ξ),B~n,q=∫Zpξndμq(ξ)
(for more information, see [10]).The author, by using derivative operator, will investigate some interesting identities on theq-Genocchi numbers and polynomials arising from their generating function. Also, the author derives some relations between q-Genocchi numbers and q-Bernoulli numbers by using Kim’s q-Volkenborn integral and fermionic p-adic q-integral on ℤp.


## 2. Novel Properties ofq-Genocchi Numbers and Polynomials with Weight 0

Letf(x)=et(x+y). Then, by using (1.10), we easily procure the following:
(2.1)∫Zp‍et(x+ξ)dμ-q(ξ)=[2]qqet+1ext.From the last equality, by (1.11), we get Araci, Acikgoz, and Qi’s q-Genocchi polynomials with weight 0 in [3] as follows:
(2.2)[2]qtqet+1ext=∑n=0∞G~n,q(x)tnn!,|log⁡⁡q+t|<π.Here, we assume thatx is a fixed parameter. Let
(2.3)F~q(x,t)=[2]qqet+1ext=∑n=0∞G~n,q(x)tn-1n!.
Thus, by expression of (2.3), we can readily see the following:
(2.4)qetF~q(x,t)+F~q(x,t)=[2]qext.Last from equality, taking derivative operatorD as D=d/dt on the both sides of (2.4), then, we easily see that
(2.5)qet(D+I)kF~q(x,t)+DkF~q(x,t)=[2]qxkext,
where k∈ℕ* and I is identity operator. By multiplying e-t on both sides of (2.5), we get
(2.6)q(D+I)kF~q(x,t)+e-tDkF~q(x,t)=[2]qxke(x-1)t.Let us take derivative operatorDm(m∈ℕ) on the both sides of (2.6). Then, we get
(2.7)qetDm(D+I)kF~q(x,t)+Dk(D-I)mF~q(x,t)=[2]qxk(x-1)mext.LetG[0] (not G(0)) be the constant term in a Laurent series of G(t) in (2.3). Then, we get
(2.8)∑j=0k‍(kj)(qetDk+m-jF~q(x,t))[0]+∑j=0m‍(mj)(-1)j(Dk+m-jF~q(x,t))[0]=[2]qxk(x-1)m.By (2.3), we easily see
(2.9)(DNF~q(x,t))[0]=G~N+1,q(x)N+1,(etDNF~q(x,t))[0]=G~N+1,q(x)N+1.We see that the members of (2.11) are proportional to the Bernstein polynomials with the following theorem.Theorem 2.1.
Fork,m∈ℕ, one has
(2.10)[2]q(-1)mBk,m+k(x)(m+kk)=∑j=0max{k,m}‍q(kj)+(-1)j(mj)k+m-j+1G~k+m-j+1,q(x).Proof.
By expressions of (2.8) and (2.9), we see that
(2.11)∑j=0max⁡{k,m}‍q(kj)+(-1)j(mj)k+m-j+1G~k+m-j+1,q(x)=[2]qxk(x-1)m.
By applying basic operations to above equality, we can easily reach to the desired result.As a special case, we derive the following.Corollary 2.2.
Fork∈ℕ, one has
(2.12)[2]q(-1)kBk,2k(x)(2kk)=[2]q∑j=0[k/2]‍(k2j)2k-2j+1G~2k-2j+1,q(x)+(2kk)(q-1)∑j=0[(k-1)/2]‍(k2j+1)2k-2jG~2k-2j,q(x).Proof.
Whenk=m into (2.10), we derive the following identity:
(2.13)(-1)kBk,2k(x)=(2kk)1+q∑j=0k[q(kj)+(-1)j(kj)]G~2k-j+1,q(x)2k-j+1=(2kk)∑j=0[k/2](k2j)2k-2j+1G~2k-2j+1,q(x)+(2kk)q-1q+1∑j=0[(k-1)/2](k2j+1)2k-2jG~2k-2j,q(x).
Here, [x] is greatest integer ≤x. Then, we complete the proof of Theorem.From (2.2), we note that
(2.14)ddx(G~n,q(x))=n∑l=0n-1(n-1l)G~l,qxn-1-l=nG~n-1,q(x).By (2.14) and (1.11), we easily see that
(2.15)∫01‍G~n,q(x)dx=G~n+1,q(1)-G~n+1,qn+1=-[2]q-1G~n+1,qn+1=-[2]q-1∫Zp‍ξndμ-q(ξ).Now, let us consider definition of integral from0 to 1 in (2.11), then we have
(2.16)-[2]q-1∑j=0max⁡{k,m}q(kj)+(-1)j(mj)k+m-j+1G~k+m-j+2,qk+m-j+2=[2]q(-1)mB(k+1,m+1)=[2]q(-1)mΓ(k+1)Γ(m+1)Γ(k+m+2),
where B(k+1,m+1) is beta function which is defined by
(2.17)B(k+1,m+1)=∫01‍xk(1-x)mdx=1(k+m+1)(k+mm),k>0,m>0.As a result, we obtain the following theorem.Theorem 2.3.
Form,k∈ℕ, one has
(2.18)∑j=1max⁡{k,m}q(kj)+(-1)j(mj)k+m-j+1G~k+m-j+2,qk+m-j+2=q(-1)m+1(k+m+1)(k+mk)-[2]qk+m+1G~k+m+2,qk+m+2.Proof.
By taking integral from0 to 1 in (2.11), we easily reach to desired result.Substitutingm=k+1 into Theorem 2.1, we readily get
(2.19)∑j=1k+1q(kj)+(-1)j(k+1j)2k-j+3G~2k-j+3,q=q(-1)k(2k+2)(2k+1k)-[2]q2k+2G~2k+3,q2k+3.By differentiating both sides of (2.11) with respect to t, we have the following:
(2.20)∑j=0max⁡{k,m}{q(kj)+(-1)j(mj)}G~k+m-j,q(x)=[2]qxk-1(x-1)m-1((k+m)x-k).We now give interesting theorem forq-Genocchi numbers with weight 0 as follows.Theorem 2.4.
Fork∈ℕ, one has
(2.21)[2]q∑j=0[k/2](k2j)2k-2j+1G~2k-2j+2,q2k-2j+2+(q-1)∑j=0[(k-1)/2](k2j+1)2k-2jG~2k-2j+1,q2k-2j+1=q(-1)k+1(2k+1)(2kk).Proof.
It is proved by using definition of integral on the both sides in the following equality, that is,(2.22)∑j=0k‍q(kj)+(-1)j(kj)2k-j+1{∫01‍G~2k-j+1,q(x)dx}=[2]q{∫01‍xk(x-1)kdx}.
Last from equality, we discover the following:
(2.23)[2]q∑j=0[k/2]‍(k2j){∫01‍G~2k-2j+1,q(x)dx}2k-2j+1+(q-1)∑j=0[(k-1)/2]‍(k2j+1){∫01‍G~2k-2j,q(x)dx}2k-2j=[2]q(-1)k{∫01‍xk(1-x)kdx}.
Then, taking integral from 0 to 1 both sides of last equality, we get
(2.24)-[2]q-1[2]q∑j=0[k/2]‍(k2j)2k-2j+1G~2k-2j+2,q2k-2j+2+[2]q-1(1-q)∑j=0[(k-1)/2]‍(k2j+1)2k-2jG~2k-2j+1,q2k-2j+1-[2]q-1=[2]q(-1)kB(k+1,k+1)=[2]q(-1)k(2k+1)(2kk).
Thus, we complete the proof of the theorem.Theorem 2.5.
Fork∈ℕ, one has
(2.25)[2]q∑j=0[(k+1)/2]‍(k2j)2k-2j+1G~2k-2j+1,q(x)+∑j=1[k/2]‍(k2j-1)2k-2j+1G~2k-2j+1,q(x)-∑j=0[k/2]‍(k2j)2k-2jG~2k-2j,q(x)+(q-1)[2]q∑j=0[k/2]‍(k2j+1){G~2k-2j,q(x)[2]q(2k-2j)+G~2k-2j+1,q(x)[2]q2(2k-2j+1)}=xk(x-1)k([2]qx-q).Proof.
In view of (2.2) and (2.23), we discover the following applications:
(2.26)∑j=0k+1‍[q(kj)+(-1)j(k+1j)]G~2k-j+1,q(x)2k-j+1=[2]qG~2k+1,q(x)2k+1∑j=0k+1‍+∑j=1[(k+1)/2]‍[q(k2j)+(k2j)+(k2j-1)]G~2k-2j+1,q2k-2j+1(x)∑j=0k+1‍+∑j=0[k/2]‍[q(k2j+1)-(k2j+1)-(k2j)]G~2k-2j,q(x)2k-2j=-{∑j=0[(k+1)/2]‍(k2j)G~2k-2j,q(x)2k-2j+q-11+q∑j=0[k/2]‍(k2j+1)G~2k-2j+1,q(x)2k-2j+1}+[2]q∑j=0[(k+1)/2]‍(k2j)∑j=0k+1‍×G~2k-2j+1,q(x)2k-2j+1+∑j=1[(k+1)/2]‍(k2j-1)G~2k-2j+1,q(x)2k-2j+1-∑j=0[k/2]‍(k2j)G~2k-2j,q(x)2k-2j∑j=0k+1‍+(q-1)∑j=0[k/2]‍(k2j+1)G~2k-2j,q(x)2k-2j+q-11+q∑j=0[k/2]‍(k2j+1)G~2k-2j+1,q(x)2k-2j+1.
Thus, we give evidence of the theorem.Asq→1 into Theorem 2.5, it leads to the following interesting property.Corollary 2.6.
Fork∈ℕ, one has
(2.27)∑j=0[(k+1)/2]‍(k2j)G2k-2j+1(x)2k+1-2j+∑j=1[k/2]‍(k2j-1)G2k-2j+1(x)4k-4j+2-∑j=0[k/2]‍(k2j)G2k-2j,q(x)4k-4j(k2j)=xk(x-1)k(x-12),
where Gn(x) is ordinary Genocchi polynomials, which is defined by the means of the following generating function [9]:
(2.28)∑n=0∞‍Gn(x)tnn!=2tet+1ext,|t|<π.


## 3. Some Identitiesq-Genocchi Numbers and q-Bernoulli Numbers by Using Kim’s p-Adic q-Integrals on ℤp

In this section, we considerq-Genocchi numbers and q-Bernoulli numbers by means of p-adic q-integral on ℤp. Now, we start with the following theorem.Theorem 3.1.
Form,k∈ℕ, one has
(3.1)∑j=0max⁡{k,m}q(kj)+(-1)j(mj)k+m-j+1∑l=0k+m-j+1(k+m-j+1l)l+1G~k+m-j-l+1,qG~l+1,q=[2]q∑l=0m(ml)(-1)m-lG~l+k+1,ql+k+1.Proof.
Form,k∈ℕ, then by (2.11),
(3.2)I1=[2]q∫Zp‍xk(x-1)mdμ-q(x)=[2]q∑l=0m‍(ml)(-1)m-l∫ℤp‍xl+kdμ-q(x)=[2]q∑l=0m‍(ml)(-1)m-lG~l+k+1,ql+k+1.
On the other hand, the right hand side of (2.11),
(3.3)I2=∑j=0max⁡{k,m}‍q(kj)+(-1)j(mj)k+m-j+1∑l=0k+m-j+1‍(k+m-j+1l)G~k+m-j-l+1,q∫ℤp‍xldμ-q(x)=∑j=0max{k,m}‍q(kj)+(-1)j(mj)k+m-j+1∑l=0k+m-j+1‍(k+m-j+1l)l+1G~k+m-j-l+1,qG~l+1,q.
Combining I1 and I2, we arrive to the proof of the theorem.Theorem 3.2.
Fork∈ℕ, one has
(3.4)∑l=0k‍(kl)(-1)k-l{[2]qG~k+l+2,qk+l+2-qG~k+l+1,qk+l+1}=[2]q∑j=0[k/2]‍(k2j)2k-2j+1∑l=02k-2j+1‍(2k-2j+1l)l+1G~2k+1-2j-l,qG~l+1,q+∑j=1[k/2]‍(k2j-1)2k-2j+1∑l=02k-2j+1‍(2k-2j+1l)l+1G~2k+1-2j-l,qG~l+1,q+q-11+q∑j=0[k/2]‍(k2j+1)Tk,jq,
here Tk,jq=[2]q∑l=02k-2j‍((2k-2jl)/(2k-2j))(G~l+1,qG~2k-2j-l,q/(l+1))+∑l=02k-2j+1‍((2k-2j+1l)/(2k-2j+1))(G~l+1,qG~2k-2j-l+1,q/(l+1)).Proof.
Let us take fermionicp-adic q-inetgral on ℤp left-hand side of Theorem 2.5, we get
(3.5)I3=∫Zp‍xk(x-1)k([2]qx-q)dμ-q(x)=[2]q∑l=0k‍(kl)(-1)k-l∫ℤp‍xk+l+1dμ-q(x)-q∑l=0k‍(kl)(-1)k-l∫ℤp‍xk+ldμ-q(x)=[2]q∑l=0k‍(kl)(-1)k-lG~k+l+2,qk+l+2-q∑l=0k‍(kl)(-1)k-lG~k+l+1,qk+l+1.
In other word, we consider the right-hand side of Theorem 2.5 as follows:
(3.6)I4=[2]q∑j=0[k/2]‍(k2j)2k-2j+1∑l=02k-2j+1‍(2k-2j+1l)G~2k+1-2j-l,q∫ℤp‍xldμ-q(x)+∑j=1[k/2]‍(k2j-1)2k-2j+1∑l=02k-2j+1‍(2k-2j+1l)G~2k+1-2j-l,q∫ℤp‍xldμ-q(x)+∑j=0[k/2]‍(k2j+1){(q-1)∑j=02k-2j‍(2k-2j-ll)2k-2jG~2k-2j-l,q∫ℤp‍xldμ-q(x)+q-11+q∑l=02k-2j+1‍(2k-2j+1l)2k-2j+1G~2k-2j-l+1∫ℤp‍xldμ-q(x)}=[2]q∑j=0[k/2]‍(k2j)2k-2j+1∑l=02k-2j+1‍(2k-2j-l+1l)l+1G~2k+1-2j-l,qG~l+1,q+∑j=1[k/2]‍(k2j-1)2k-2j+1∑l=02k-2j+1‍(2k-2j-l+1l)l+1G~2k+1-2j-l,qG~l+1,q+∑j=0[k/2]‍(k2j+1){(q-1)∑j=02k-2j‍(2k-2j-ll)2k-2jG~2k-2j-l,qG~l+1,ql+1+q-11+q∑l=02k-2j+1‍(2k-2j-l+1l)2k-2j+1G~2k-2j-l+1G~l+1,ql+1}.
EquatingI3 and I4, we complete the proof of the theorem.Asq→1 in the above theorem, we reach interesting property in Analytic Numbers Theory concerning ordinary Genocchi polynomials.Corollary 3.3.
Fork∈ℕ, one has
(3.7)∑l=0k‍(kl)(-1)k-l{2Gk+l+2k+l+2-Gk+l+1k+l+1}=2∑j=0[k/2]‍(k2j)2k-2j+1∑l=02k-2j+1‍(2k-2j+1l)l+1G2k+1-2j-lGl+1+∑j=1[k/2]‍(k2j-1)2k-2j+1∑l=02k-2j+1‍(2k-2j+1l)l+1G2k+1-2j-lGl+1.Theorem 3.4.
Form,k∈ℕ, one has
(3.8)[2]q∑l=0m‍(ml)(-1)m-lB~l+k,q=∑j=0max{k,m}‍q(kj)+(-1)j(mj)k+m-j+1∑l=0k+m-j+1‍(k+m-j+1l)G~k+m+1-j-l,qB~l,q.Proof.
We consider (2.11) and (2.2) by means of q-Volkenborn integral. Then, by (2.11), we see
(3.9)[2]q∫Zp‍xk(x-1)mdμq(x)=[2]q∑l=0m‍(ml)(-1)m-l∫ℤp‍xl+kdμq(x)=[2]q∑l=0m‍(ml)(-1)m-lB~l+k,q.
On the other hand,
(3.10)∑j=0max⁡{k,m}‍q(kj)+(-1)j(mj)k+m-j+1∑l=0k+m-j+1‍(k+m-j+1l)G~k+m+1-j-l,q∫ℤp‍xldμq(x)=∑j=0max{k,m}‍q(kj)+(-1)j(mj)k+m-j+1∑l=0k+m-j+1‍(k+m-j+1l)G~k+m+1-j-l,qB~l,q.
Therefore, we get the proof of theorem.Corollary 3.5.
Fork∈ℕ, one gets
(3.11)∑l=0k‍(kl)(-1)k-l{[2]qB~k+l+1,q-qB~k+l,q}=[2]q∑j=0[k/2]‍(k2j)2k-2j+1∑l=02k-2j+1‍(2k-2j+1l)G~2k+1-2j-l,qB~l,q+∑j=1[k/2]‍(k2j-1)2k-2j+1∑l=0k‍×∑l=02k-2j+1‍(2k-2j+1l)G~2k+1-2j-l,qB~l,q+(q-1q+1)∑j=0[k/2]‍(k2j+1)Sk,jq,
where Sk,jq=[2]q∑j=02k-2j(1/(2k-2j))(2k-2jl)G~2k-2j-l,qB~l,q+∑l=02k-2j+1(1/(2k-2j+1))(2k-2j+1l)G~2k-2j-l+1B~l,q.Proof.
By usingp-adic q-integral on ℤp left-hand side of Theorem 2.5, we get
(3.12)I5=[2]q∫Zp‍xk(x-1)k([2]x-q)dμq(x)=[2]q∑l=0k‍(kl)(-1)k-l∫ℤp‍xk+l+1dμq(x)-q∑l=0k‍(kl)(-1)k-l∫ℤp‍xk+ldμq(x)=[2]q∑l=0k‍(kl)(-1)k-lB~k+l+1,q-q∑l=0k‍(kl)(-1)k-lB~k+l,q.
Also, we compute the right-hand side of Theorem 2.5 as follows:
(3.13)I6=[2]q∑j=0[k/2]‍12k-2j+1(k2j)∑l=02k-2j+1‍(2k-2j+1l)G~2k+1-2j-l,q∫ℤp‍xldμq(x)+∑j=1[k/2]‍12k-2j+1(k2j-1)∑l=02k-2j+1‍(2k-2j+1l)G~2k+1-2j-l,q∫ℤp‍xldμq(x)+∑j=0[k/2]‍(k2j+1){(q-1)∑j=02k-2j‍12k-2j(2k-2jl)G~2k-2j-l,q∫ℤp‍xldμq(x)+q-11+q∑l=02k-2j+1‍12k-2j+1(2k-2j+1l)G~2k-2j-l+1∫ℤp‍xldμq(x)}=[2]q∑j=0[k/2]‍12k-2j+1(k2j)∑l=02k-2j+1‍(2k-2j+1l)G~2k+1-2j-l,qB~l,q+∑j=1[k/2]‍12k-2j+1(k2j-1)∑l=02k-2j+1‍(2k-2j+1l)G~2k+1-2j-l,qB~l,q+∑j=0[k/2]‍(k2j+1){(q-1)∑j=02k-2j‍12k-2j(2k-2jl)G~2k-2j-l,qB~l,q+q-11+q∑l=02k-2j+1‍12k-2j+1(2k-2j+1l)G~2k-2j-l+1B~l,q}.
EquatingI5 and I6, we get the proof of Corollary.Asq→1 in the above theorem, we easily derive the following corollary.Corollary 3.6.
Fork∈ℕ, one has
(3.14)∑l=0k‍(kl)(-1)k-l{2Bk+l+1-Bk+l}=2∑j=0[k/2]‍(k2j)2k-2j+1∑l=02k-2j+1‍(2k-2j+1l)G2k+1-2j-lBl+∑j=1[k/2]‍(k2j-1)2k-2j+1∑l=02k-2j+1‍(2k-2j+1l)G2k+1-2j-lBl.



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*Source: 214961-2012-07-24.xml*