APPMATHISRN Applied Mathematics2090-55722090-5564International Scholarly Research Network27178410.5402/2011/271784271784Research ArticleSome New Identities on the q-Genocchi Numbers and Polynomials with Weight αRimSeog-HoonJeongJooheeBellouquidA.Department of Mathematics EducationKyungpook National UniversityDaegu 702-701Republic of Koreaknu.ac.kr201107122011201120092011151120112011Copyright © 2011 Seog-Hoon Rim and Joohee Jeong.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.

We construct a new type of q-Genocchi numbers and polynomials with weight α. From these q-Genocchi numbers and polynomials with weight α, we establish some interesting identities and relations.

1. Introduction

Let p be a fixed odd prime number. Throughout this paper, p, p, and p will, respectively denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of the algebraic closure of p. Let be the set of natural numbers and +={0}. Let vp be the normalized exponential valuation of p with |p|p=p-vp(p)=1/p. When one talks of q-extension, q is variously considered as an indeterminate, a complex q, or a p-adic number q. In this paper, we assume that qp with |1-q|p<1. As a definition of q-numbers, we use the notation of q-number of [x]q=1-qx1-q,[x]-q=1-(-q)x1+q(cf. ). Note that limq1[x]q=x. Let C(p) be the space of continuous functions on p. For fC(p), the p-adic invariant integral on p is defined by Kim [1, 3], I-q(f)=Zpf(x)dμ-q(x)=limN1[pN]-qx=0pN-1f(x)(-q)x. From (1.2), we have the well-known integral equation qnI-1(fn)+(-1)n-1Iq(f)=ql=0n-1(-1)lqlf(l)(see [1, 3]), where fn(x)=f(x+n), (n).

For α, in , the q-Genocchi polynomials with weight α are introduced by tZpe[x+y]qαtdμ-q(y)=n=0G̃n,q(α)(x)tnn!. By comparing the coefficients of both sides of (1.4), we have G̃0,q(α)(x)=0,G̃n+1,q(α)(x)(n+1)=Zp[x+y]qαndμ-q(y),    for  nN.

In the special case, x=0, G̃n,q(α)(0)=G̃n,q(α)   are called the nth q-Genocchi numbers with weight α.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M49"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Genocchi Numbers and Polynomials with Weight <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M50"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math></inline-formula>

In this section, we show some new identities on the q-Genocchi numbers and polynomials with weight α. And we establish the distribution relation for q-Genocchi polynomials with weight α.

From (1.5), we can easily see that G̃n+1,q(α)(x)n+1=q[α]qn(1-q)nl=0n(nl)(-1)lqαlx1+qαl+1. From (1.5) and (2.1), we note that G̃n+1,q(α)(x)n+1=Zp[x+y]qαndμ-1(y)=l=0n(nl)[x]qαn-lqαlxZp[y]qαldμ-q(y)=l=0n(nl)[x]qαn-lqαlxG̃l+1,q(α)l+1. Note that (1/(l+1))(nl)=(1/(n+1))(n+1l+1) and from (2.2), we have the relation of polynomials and numbers, G̃n+1,q(α)(x)n+1=1(n+1)l=0n(n+1l+1)[x]qαn+1-lqαlxG̃l+1,q(α)=1(n+1)qαxl=0n+1(n+1l+1)[x]qαn+1-lqα(l+1)xG̃l+1,q(α)=1(n+1)qαx([x]qα+qαlxG̃q(α))n+1, with the usual convention of replacing (G̃q(α))n by (G̃n,q(α)).

Thus, by (2.3), we have a theorem.

Theorem 2.1.

For α and n+, one has qαxG̃n+1,q(α)(x)=(n+1)qαx([x]qα+qαxG̃q(α))n+1=l=0n+1(n+1l)[x]qαn+1-lqαxG̃l,q(α).

In (1.3), if we take n=1, qI-1(f1)+I-1(f)=q.

We apply f(x)=e[x]qαt with (1.5), and we have the following: q=n=0(qZp[x+1]qαndμ-q(x)+Zp[x]qαndμ-q(x))tnn!=n=0(qG̃n+1,q(α)(1)n+1+G̃n+1,q(α)n+1)tnn!. By comparing the coefficients on both the sides in (2.6), we get qG̃n+1,q(α)(1)n+1+G̃n+1,q(α)n+1={qif  n=0,0if  n>0. From (2.2) and (2.7), we can derive the following: G̃1,q(α)(1)=1,q1-α(qαG̃q(α)+1)n+G̃n,q(α)=0if  nN, with the usual convention of replacing (G̃q(α))n by G̃n,q(α).

For a fixed odd positive integer d with (p,d)=1, we set X=Xd=limNZdpNZ,X1=Zp,X*=0<a<dp,(a,p)=1(a+dpZp),a+dpNZp={xXxa  (mod  dpN)}, where a satisfies the condition 0a<dpN. For the distribution relation for the q-Genocchi polynomials with weight α, we consider the following: Zp[x+y]qαndμ-q(y)=Zp[n+y]qαxndμ-q(y)=[d]qαn[d]-qa=0d-1(-1)aqaZp[x+ad+y]qαdndμ-q(y). By (1.5) and (2.10), we get a theorem.

Theorem 2.2.

For α and n+, d with d1(mod  2), one has G̃n+1,q(α)(x)n+1=[d]qαn[d]-qa=0d-1(-1)aqaG̃n+1,qd(α)(x+aa).

3. Higher-Order <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M85"><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:math></inline-formula>-Genocchi Numbers and Polynomials with Weight <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M86"><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:math></inline-formula>

In this section, we define higher-order q-Genocchi polynomials G̃n+1,q(α)(h,kx) and numbers G̃n+1,q(α)(h,k) with weight α. We find an integral equation for higher-order q-Genocchi numbers with weight α. And we establish a combination property.

Let α and h,k+, for n+, then we define higher-order q-Genocchi polynomials with weight α as follows: G̃n+1,q(α)(h,kx)(n+1)=ZpZpk-times[x1+x2++xk+x]qαnq(h-1)x1++(h-k)xkdμ-q(x1)dμ-q(xk)=qk[α]qn(1-q)nl=0n(nl)(-1)lqαlx(1+qαl+h)(1+qαl+h-k+1)=qk[α]qn(1-q)ml=0n(nl)(-1)lqαlx(-qαl+h:q-1)k, where (x:q)h=i=0h-1(1-xqi).

In the special case, x=0, G̃n+1,q(α)(h,k|0)=G̃n+1,q(α)(h,k) are called the (n+1)th (h,k)-Genocchi numbers with weight α.

In (3.1), apply the following identity: [x1+x2++xk]qα(1-qα)+qα(x1+x2++xk)=1, and we have a theorem.

Theorem 3.1.

For α and h,k+, one has G̃n+1,q(α)(h,k)n+1=(1-qα)G̃n+2,q(α)(h,k)n+2+G̃n+1,q(α)(h+α,k)n+1.

We consider, for α and h,k+, j=0i(ij)(qα-1)jG̃n+j-i+1(α)(h-α,k)n+j-i+1=j=0i(ij)(qα-1)jZpZpk-times[l=1kxl]qαn-i-jql=1k(h-α-l)xldμ-q(x1)dμ-q(xk)=j=0i(ij)(qα-1)jZpZpk-times[l=1kxl]qαn-i+jql=1k(h-l)xldμ-q(x1)dμ-q(xk)=j=0i(ij)(qα-1)jG̃n+j-i+1(α)(h,k)n+j-i+1. Therefore, we obtain the following combinatorial property.

Theorem 3.2.

For α and h,k+, one has j=0i(ij)(qα-1)jG̃n+j-i+1(α)(h-α,k)n+j-i+1=j=0i-1(i-1j)(qα-1)jG̃n+j-i+1(α)(h,k)n+j-i+1.

BayadA.KimT.Identities involving values of Bernstein, q-Bernoulli, and q-Euler polynomialsRussian Journal of Mathematical Physics2011182133143281098710.1134/S1061920811020014JangL.-C.KimT.LeeD.-H.ParkD.-W.An application of polylogarithms in the analogs of Genocchi numbersNotes on Number Theory and Discrete Mathematics20017365692020979KimT.Some identities on the q-Euler polynomials of higher order and q-Stirling numbers by the fermionic p-adic integral on pRussian Journal of Mathematical Physics2009164484491258780510.1134/S1061920809040037KimT.q-Volkenborn integrationRussian Journal of Mathematical Physics2002932882991965383ZBL1092.11045KimT.A note on p-adic q-integral on p associated with q-Euler numbersAdvanced Studies in Contemporary Mathematics20071521331372356172KimT.A note on q-Volkenborn integrationProceedings of the Jangjeon Mathematical Society20058113172150959KimT.On the q-extension of Euler and Genocchi numbersJournal of Mathematical Analysis and Applications2007326214581465228099610.1016/j.jmaa.2006.03.037KimT.A note on q-Bernstein polynomialsRussian Journal of Mathematical Physics20111817382278390510.1134/S1061920811010080KimT.ChoiJ.KimY. H.RyooC. S.On the fermionic p-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomialsJournal of Inequalities and Applications201020101286424710.1155/2010/8642472749172KimT.ChoiJ.KimY. H.RyooC. S.A note on the weighted p-adic q-Euler measure on pAdvanced Studies in Contemporary Mathematics2011123540RimS.-H.ParkK. H.MoonE. J.On Genocchi numbers and polynomialsAbstract and Applied Analysis20082008710.1155/2008/8984718984712439256ZBL1217.11024