JFSA Journal of Function Spaces and Applications 1758-4965 0972-6802 Hindawi Publishing Corporation 214961 10.1155/2012/214961 214961 Research Article Novel Identities for q-Genocchi Numbers and Polynomials Araci Serkan Ólafsson Gestur 1 Department of Mathematics Faculty of Science and Arts University of Gaziantep 27310 Gaziantep Turkey gantep.edu.tr 2012 24 7 2012 2012 26 02 2012 25 04 2012 09 05 2012 2012 Copyright © 2012 Serkan Araci. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The essential aim of this paper is to introduce novel identities for q-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.

1. Preliminaries

Imagine that p be a fixed odd prime number. We now start with definition of the following notations. Let p be the field p-adic rational numbers and let p be the completion of algebraic closure of p.

Thus, (1.1)Qp={x=n=-kanpn:0an<p}.

Then p is integral domain, which is defined by (1.2)Zp={x=n=0anpn:0an<p}, or (1.3)Zp={xQp:|x|p1}.

We assume that qp 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.

[ x ] q is a q-extension of x, which is defined by (1.5)[x]q=1-qx1-q, we note that limq1[x]q=x (see ).

Throughout this paper, we use notation of *:={0}, where denotes set of Natural numbers.

We say that f is a uniformly differentiable function at a point ap, 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 fUD(p). Then, for fUD(p), we can start with the following expression: (1.7)1[pN]q0ξ<pNf(ξ)qξ=0ξ<pNf(ξ)μq(ξ+pNZp), which represents a p-adic q-analogue of Riemann sums for f. The integral of f on p will be defined as the limit (N) of these sums, when it exists. The p-adic q-integral of function fUD(p) is defined by Kim in [7, 12] as (1.8)Iq(f)=Zpf(ξ)dμq(ξ)=limN1[pN]qξ=0pN-1f(ξ)qξ.

The bosonic integral is considered as a bosonic limit q1, I1(f)=limq1Iq(f). Similarly, the fermionic p-adic integral on p is introduced by Kim as follows: (1.9)I-q(f)=Zpf(ξ)dμ-q(ξ) (for more details, see ).

From (1.9), it is well-known equality that (1.10)qI-q(f1)+I-q(f)=qf(0), where f1(x)=f(x+1) (for details, see [2, 3, 8, 9, 12, 13, 1517]).

The q-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={q,ifn=1,0,ifn1, with the usual convention about replacing (G~q)n by G~n,q is used (for details, see ).

Let UD(p) be the space of continuous functions on p. For fUD(p), 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*. 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]).

The q-Bernoulli polynomials and numbers with weight 0 are defined by Kim et al., respectively, (1.16)B~n,q(x)=limn1[pn]qy=0pn-1(x+y)nqy=p(x+ξ)ndμq(ξ),B~n,q=Zpξndμq(ξ) (for more information, see ).

The author, by using derivative operator, will investigate some interesting identities on the q-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 p.

2. Novel Properties of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M82"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Genocchi Numbers and Polynomials with Weight 0

Let f(x)=et(x+y). Then, by using (1.10), we easily procure the following: (2.1)Zpet(x+ξ)dμ-q(ξ)=qqet+1ext.

From the last equality, by (1.11), we get Araci, Acikgoz, and Qi’s q-Genocchi polynomials with weight 0 in  as follows: (2.2)qtqet+1ext=n=0G~n,q(x)tnn!,|logq+t|<π.

Here, we assume that x is a fixed parameter. Let (2.3)F~q(x,t)=qqet+1ext=n=0G~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)=qext.

Last from equality, taking derivative operator D 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)=qxkext, where k* 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)=qxke(x-1)t.

Let us take derivative operator Dm(m) 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)=qxk(x-1)mext.

Let G (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))+j=0m(mj)(-1)j(Dk+m-jF~q(x,t))=qxk(x-1)m.

By (2.3), we easily see (2.9)(DNF~q(x,t))=G~N+1,q(x)N+1,(etDNF~q(x,t))=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.

For k,m, one has (2.10)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)=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.

For k, one has (2.12)q(-1)kBk,2k(x)(2kk)=qj=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.

When k=m into (2.10), we derive the following identity: (2.13)(-1)kBk,2k(x)=(2kk)1+qj=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+1j=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))=nl=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=-q-1G~n+1,qn+1=-q-1Zpξndμ-q(ξ).

Now, let us consider definition of integral from 0 to 1 in (2.11), then we have (2.16)-q-1j=0max{k,m}q(kj)+(-1)j(mj)k+m-j+1G~k+m-j+2,qk+m-j+2=q(-1)mB(k+1,m+1)=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.

For m,k, 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)-qk+m+1G~k+m+2,qk+m+2.

Proof.

By taking integral from 0 to 1 in (2.11), we easily reach to desired result.

Substituting m=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)-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)=qxk-1(x-1)m-1((k+m)x-k).

We now give interesting theorem for q-Genocchi numbers with weight 0 as follows.

Theorem 2.4.

For k, one has (2.21)qj=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}=q{01xk(x-1)kdx}. Last from equality, we discover the following: (2.23)qj=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=q(-1)k{01xk(1-x)kdx}. Then, taking integral from 0 to 1 both sides of last equality, we get (2.24)-q-1qj=0[k/2](k2j)2k-2j+1G~2k-2j+2,q2k-2j+2+q-1(1-q)j=0[(k-1)/2](k2j+1)2k-2jG~2k-2j+1,q2k-2j+1-q-1=q(-1)kB(k+1,k+1)=q(-1)k(2k+1)(2kk).

Thus, we complete the proof of the theorem.

Theorem 2.5.

For k, one has (2.25)qj=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)qj=0[k/2](k2j+1){G~2k-2j,q(x)q(2k-2j)+G~2k-2j+1,q(x)q2(2k-2j+1)}=xk(x-1)k(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=qG~2k+1,q(x)2k+1j=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+qj=0[k/2](k2j+1)G~2k-2j+1,q(x)2k-2j+1}+qj=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-2jj=0k+1+(q-1)j=0[k/2](k2j+1)G~2k-2j,q(x)2k-2j+q-11+qj=0[k/2](k2j+1)G~2k-2j+1,q(x)2k-2j+1.

Thus, we give evidence of the theorem.

As q1 into Theorem 2.5, it leads to the following interesting property.

Corollary 2.6.

For k, 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 : (2.28)n=0Gn(x)tnn!=2tet+1ext,|t|<π.

3. Some Identities <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M146"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Genocchi Numbers and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M147"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Bernoulli Numbers by Using Kim’s <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M148"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula>-Adic <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M149"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Integrals on <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M150"><mml:mrow><mml:msub><mml:mrow><mml:mi>ℤ</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

In this section, we consider q-Genocchi numbers and q-Bernoulli numbers by means of p-adic q-integral on p. Now, we start with the following theorem.

Theorem 3.1.

For m,k, one has (3.1)j=0max{k,m}q(kj)+(-1)j(mj)k+m-j+1l=0k+m-j+1(k+m-j+1l)l+1G~k+m-j-l+1,qG~l+1,q=ql=0m(ml)(-1)m-lG~l+k+1,ql+k+1.

Proof.

For m,k, then by (2.11), (3.2)I1=qZpxk(x-1)mdμ-q(x)=ql=0m(ml)(-1)m-lpxl+kdμ-q(x)=ql=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+1l=0k+m-j+1(k+m-j+1l)G~k+m-j-l+1,qpxldμ-q(x)=j=0max{k,m}q(kj)+(-1)j(mj)k+m-j+1l=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.

For k, one has (3.4)l=0k(kl)(-1)k-l{qG~k+l+2,qk+l+2-qG~k+l+1,qk+l+1}=qj=0[k/2](k2j)2k-2j+1l=02k-2j+1(2k-2j+1l)l+1G~2k+1-2j-l,qG~l+1,q+j=1[k/2](k2j-1)2k-2j+1l=02k-2j+1(2k-2j+1l)l+1G~2k+1-2j-l,qG~l+1,q+q-11+qj=0[k/2](k2j+1)Tk,jq, here Tk,jq=ql=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 fermionic p-adic q-inetgral on p left-hand side of Theorem 2.5, we get (3.5)I3=Zpxk(x-1)k(qx-q)dμ-q(x)=ql=0k(kl)(-1)k-lpxk+l+1dμ-q(x)-ql=0k(kl)(-1)k-lpxk+ldμ-q(x)=ql=0k(kl)(-1)k-lG~k+l+2,qk+l+2-ql=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=qj=0[k/2](k2j)2k-2j+1l=02k-2j+1(2k-2j+1l)G~2k+1-2j-l,qpxldμ-q(x)+j=1[k/2](k2j-1)2k-2j+1l=02k-2j+1(2k-2j+1l)G~2k+1-2j-l,qpxldμ-q(x)+j=0[k/2](k2j+1){(q-1)j=02k-2j(2k-2j-ll)2k-2jG~2k-2j-l,qpxldμ-q(x)+q-11+ql=02k-2j+1(2k-2j+1l)2k-2j+1G~2k-2j-l+1pxldμ-q(x)}=qj=0[k/2](k2j)2k-2j+1l=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+1l=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+ql=02k-2j+1(2k-2j-l+1l)2k-2j+1G~2k-2j-l+1G~l+1,ql+1}.

Equating I3 and I4, we complete the proof of the theorem.

As q1 in the above theorem, we reach interesting property in Analytic Numbers Theory concerning ordinary Genocchi polynomials.

Corollary 3.3.

For k, one has (3.7)l=0k(kl)(-1)k-l{2Gk+l+2k+l+2-Gk+l+1k+l+1}=2j=0[k/2](k2j)2k-2j+1l=02k-2j+1(2k-2j+1l)l+1G2k+1-2j-lGl+1+j=1[k/2](k2j-1)2k-2j+1l=02k-2j+1(2k-2j+1l)l+1G2k+1-2j-lGl+1.

Theorem 3.4.

For m,k, one has (3.8)ql=0m(ml)(-1)m-lB~l+k,q=j=0max{k,m}q(kj)+(-1)j(mj)k+m-j+1l=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)qZpxk(x-1)mdμq(x)=ql=0m(ml)(-1)m-lpxl+kdμq(x)=ql=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+1l=0k+m-j+1(k+m-j+1l)G~k+m+1-j-l,qpxldμq(x)=j=0max{k,m}q(kj)+(-1)j(mj)k+m-j+1l=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.

For k, one gets (3.11)l=0k(kl)(-1)k-l{qB~k+l+1,q-qB~k+l,q}=qj=0[k/2](k2j)2k-2j+1l=02k-2j+1(2k-2j+1l)G~2k+1-2j-l,qB~l,q+j=1[k/2](k2j-1)2k-2j+1l=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=qj=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 using p-adic q-integral on p left-hand side of Theorem 2.5, we get (3.12)I5=qZpxk(x-1)k(x-q)dμq(x)=ql=0k(kl)(-1)k-lpxk+l+1dμq(x)-ql=0k(kl)(-1)k-lpxk+ldμq(x)=ql=0k(kl)(-1)k-lB~k+l+1,q-ql=0k(kl)(-1)k-lB~k+l,q. Also, we compute the right-hand side of Theorem 2.5 as follows: (3.13)I6=qj=0[k/2]12k-2j+1(k2j)l=02k-2j+1(2k-2j+1l)G~2k+1-2j-l,qpxldμq(x)+j=1[k/2]12k-2j+1(k2j-1)l=02k-2j+1(2k-2j+1l)G~2k+1-2j-l,qpxldμq(x)+j=0[k/2](k2j+1){(q-1)j=02k-2j12k-2j(2k-2jl)G~2k-2j-l,qpxldμq(x)+q-11+ql=02k-2j+112k-2j+1(2k-2j+1l)G~2k-2j-l+1pxldμq(x)}=qj=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+ql=02k-2j+112k-2j+1(2k-2j+1l)G~2k-2j-l+1B~l,q}.

Equating I5 and I6, we get the proof of Corollary.

As q1 in the above theorem, we easily derive the following corollary.

Corollary 3.6.

For k, one has (3.14)l=0k(kl)(-1)k-l{2Bk+l+1-Bk+l}=2j=0[k/2](k2j)2k-2j+1l=02k-2j+1(2k-2j+1l)G2k+1-2j-lBl+j=1[k/2](k2j-1)2k-2j+1l=02k-2j+1(2k-2j+1l)G2k+1-2j-lBl.

Acknowledgment

The author would like to express his sincere gratitude to the referee for his/her valuable comments and suggestions which have improved the presentation of the paper.

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