# Some Identities on the Twisted(h,q)-Genocchi Numbers and Polynomials Associated with q-Bernstein Polynomials

Authors: Seog-Hoon Rim; Sun-Jung Lee
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We give some interesting identities on the
twisted (h,q)-Genocchi numbers and polynomials associated with q-Bernstein polynomials.


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## Body

## 1. Introduction

Letp be a fixed odd prime number. Throughout this paper, we always make use of the following notations: ℤ denotes the ring of rational integers, ℤp denotes the ring of p-adic rational integer, ℚp denotes the ring of p-adic rational numbers, and ℂp denotes the completion of algebraic closure of ℚp, respectively. Let ℕ be the set of natural numbers and ℤ+=ℕ⋃{0}. Let Cpn={ζ∣ζpn=1} be the cyclic group of order pn and let(1.1)Tp=⋃n≥1Cpn=lim⁡n→∞Cpn=Cp∞.
The p-adic absolute value is defined by |x|=1/pr, where x=pr(s/t) (r∈ℚ and s,t∈ℤ with (s,t) = (p,s) = (p,t) = 1). In this paper we assume that q∈ℂp with |q-1|p<1 as an indeterminate.Theq-number is defined by(1.2)[x]q=1-qx1-q
(see [1–15]). Note that limq→1[x]q=x. Let UD(ℤp) be the space of uniformly differentiable function on ℤp. For f∈UD(ℤp), Kim defined the fermionic p-adic q-integral on ℤp as follows:(1.3)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1f(x)(-q)x
(see [2–6, 8–15]). From (1.3), we note that(1.4)qnI-q(fn)=(-1)nI-q(f)+[2]q∑l=0n-1(-1)n-1-lqlf(l)
(see [4–6, 8–12]), where fn(x)=f(x+n) for n∈ℕ. For k,n∈ℤ+ and x∈[0,1], Kim defined the q-Bernstein polynomials of the degree n as follows:(1.5)Bk,n(x,q)=(nk)[x]qk[1-x]q-1n-k,
(see [13–15]). For h∈ℤ and ζ∈Tp, let us consider the twisted (h,q)-Genocchi polynomials as follows:(1.6)t∫Zpe[x+y]qtζyq(h-1)ydμ-q(y)=∑n=0∞Gn,q,ζ(h)(x)tnn!.
Then, Gn,q,ζ(h)(x) is called nth twisted (h,q)-Genocchi polynomials.In the special case,x=0 and Gn,q,ζ(h)(0)=Gn,q,ζ(h) are called the nth twisted (h,q)-Genocchi numbers.In this paper, we give the fermionicp-adic integral representation of q-Bernstein polynomial, which are defined by Kim [13], associated with twisted (h,q)-Genocchi numbers and polynomials. And we construct some interesting properties of q-Bernstein polynomials associated with twisted (h,q)-Genocchi numbers and polynomials.


## 2. On the Twisted(h,q)-Genocchi Numbers and Polynomials

From (1.6), we note that(2.1)Gn+1,q,ζ(h)(x)n+1=∫Zp[x+y]qnζyq(h-1)ydμ-q(y)=∫Zp([x]q+qx[y]q)nζyq(h-1)ydμ-q(y)=∑l=0n(nl)[x]qn-lqlx∫Zp[y]qlζyq(h-1)ydμ-q(y)=∑l=0n(nl)[x]qn-lqlxGl+1,q,ζ(h)l+1.
We also have(2.2)Gn,q,ζ(h)(x)=q-x∑l=0n(nl)[x]qn-lqlxGl,q,ζ(h).
Therefore, we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+ and ζ∈Tp, one has
(2.3)Gn,q,ζ(h)(x)=q-x([x]q+qxGq,ζ(h))n
with usual convention about replacing (Gq,ζ(h))n by Gn,q,ζh.
By (1.6) and (2.1) one gets
(2.4)Gn+1,q-1,ζ-1(h)(1-x)n+1=∫Zp[1-x+y]q-1nζ-yq-(h-1)ydμ-q-1(y)=[2]q(1-q-1)n∑l=0n(nl)(-1)nqh-1ζqlx1+qh+lζ=(-1)nqn+h-1ζ([2]q(1-q)n∑l=0n(nl)(-1)lqlx1+qh+lζ)=(-1)nζqn+h-1Gn+1,q,ζ(h)(x)n+1.Therefore, we obtain the following theorem.Theorem 2.2.
Forn∈ℤ+ and ζ∈Tp, one has
(2.5)Gn,q-1,ζ-1(h)(1-x)=(-1)n-1ζqn+h-2Gn,q,ζ(h).
From (1.5), one gets the following recurrence formula:
(2.6)qhζGn,q,ζ(h)(1)+Gn,q,ζ(h)={[2]qifn=1,0ifn>1.Therefore, we obtain the following theorem.Theorem 2.3.
Forn∈ℤ+ and ζ∈Tp, one has
(2.7)G0,q,ζ=0,qh-1ζ(qGq,ζ(h)+1)n+Gn,q,ζ(h)={[2]qifn=1,0ifn>1
with usual convention about replacing (Gq,ζ(h))n by Gn,q,ζh.From Theorem2.3, we note that(2.8)q2hζ2Gn,q,ζ(h)(2)-qhζn[2]q=-qh-1ζ∑l=0n(nl)qlGl,q,ζ(h)=-qh-1ζ(qGq,ζ(h)+1)n=Gn,q,ζ(h)ifn>1.Therefore, we obtain the following theorem.Theorem 2.4.
Forn∈ℤ+ and ζ∈Tp, one has
(2.9)q2hζ2Gn,q,ζ(h)(2)=Gn,q,ζ(h)+nqhζ[2]q.Remark 2.5.
We note that Theorem2.4 also can be proved by using fermionic integral equation (1.4) in case of n=2.By (2.4) and Theorem 2.2, we get(2.10)Gn+1,q-1,ζ-1(h)(2)n+1=(-1)nqn+h-1ζGn+1,q,ζ(h)(-1)n+1=(-1)nqn+h-1ζ∫Zp[x-1]qnζxq(h-1)xdμ-q(x)=qh-1ζ∫Zp[1-x]q-1nζxq(h-1)xdμ-q(x).Therefore, we obtain the following theorem.Theorem 2.6.
Forn∈ℤ+ and ζ∈Tp, one has
(2.11)(n+1)qh-1ζ∫Zp[1-x]q-1nζxq(h-1)xdμ-q(x)=Gn+1,q-1,ζ-1(h)(2).Letn∈ℕ. By Theorems 2.4 and 2.6, we get(2.12)(n+1)qh-1ζ∫Zp[1-x]q-1nζxq(h-1)xdμ-q(x)=q2hζ2Gn+1,q-1,ζ-1(h)+(n+1)qh-1ζ[2]q.
Therefore, we obtain the following corollary.Corollary 2.7.
Forn∈ℤ+ and ζ∈Tp, one has
(2.13)∫Zp[1-x]q-1nζxq(h-1)xdμ-q(x)=qh+1ζGn+1,q-1,ζ-1(h)n+1+[2]q.
By (1.5), we get the symmetry of q-Bernstein polynomials as follows:
(2.14)Bk,n(x,q)=Bn-k,n(1-x,q-1)
(see [11]).Thus, by Corollary2.7 and (2.14), we get(2.15)∫ZpBk,n(x,q)q(h-1)xζxdμ-q(x)=∫ZpBn-k,n(1-x,q-1)q(h-1)xζxdμ-q(x)=(nk)∑l=0k(kl)(-1)k-l∫Zp[1-x]q-1n-lq(h-1)xζxdμ-1(x)=(nk)∑l=0k(kl)(-1)k-l(qh+1ζGn-l+1,q-1,ζ-1(h)n-l+1+[2]q)={qh+1ζGn+1,q-1,ζ-1(h)n+1+[2]qifk=0,qh+1ζ(nk)∑l=0k(kl)(-1)k-lGn-l+1,q-1,ζ-1(h)n-l+1ifk>0.From (2.15), we have the following theorem.Theorem 2.8.
Forn∈ℤ+ and ζ∈Tp, one has
(2.16)∫ZpBk,n(x,q)q(h-1)xζxdμ-q(x)={qh+1ζGn+1,q-1,ζ-1(h)n+1+[2]qifk=0,qh+1ζ(nk)∑l=0k(kl)(-1)k-lGn-l+1,q-1,ζ-1(h)n-l+1ifk>0.
Forn,k∈ℤ+ with n>k, fermionic p-adic invariant integral for multiplication of two q-Bernstein polynomials on ℤp can be given by the following:
(2.17)∫ZpBk,n(x,q)q(h-1)xζxdμ-q(x)=∫Zp(nk)[x]qk[1-x]q-1n-kq(h-1)xζxdμ-q(x)=∫Zp(nk)[x]qk(1-[x]q)n-kq(h-1)xζxdμ-1(x)=(nk)∑l=0n-k(n-kl)(-1)l∫Zp[x]qk+lq(h-1)xζxdμ-1(x).From Theorem2.8 and (2.17), we have the following corollary.Corollary 2.9.
Forn∈ℤ+ and ζ∈Tp, one has
(2.18)∑l=0n-k(n-kl)(-1)lGk+l+1,q,ζ(h)k+l+1={qh+1ζGn+1,q-1,ζ-1(h)n+1+[2]qifk=0,qh+1ζ∑l=0k(kl)(-1)k-lGn-l+1,q-1,ζ-1(h)n-l+1ifk>0.
Letn1,n2,k∈ℤ+ with n1+n2>2k. Then we get
(2.19)∫ZpBk,n1(x,q)Bk,n2(x,q)q(h-1)xζxdμ-q(x)=(n1k)(n2k)∫Zp∑l=02k(2kl)(-1)2k-l[1-x]q-1n1+n2-lq(h-1)xζxdμ-q(x)=(n1k)(n2k)∑l=02k(2kl)(-1)2k-l(Gn1+n2-l+1,q-1,ζ-1(h)n1+n2-l+1qh+1ζ+[2]q).From (2.19), we have the following theorem.Theorem 2.10.
Forn∈ℤ+ and ζ∈Tp, one has
(2.20)∫ZpBk,n1(x,q)Bk,n2(x,q)q(h-1)xζxdμ-q(x)={qh+1ζGn1+n2+1,q-1,ζ-1(h)n1+n2+1+[2]qifk=0,(n1k)(n2k)∑l=02k(2kl)(-1)2k-lGn1+n2-l+1,q-1,ζ-1(h)n1+n2-l+1ifk>0.
Letn1,n2,k∈ℤ+ with n1+n2>2k, fermionic p-adic invariant integral for multiplication of two q-Bernstein polynomials on ℤp can be given by the following:
(2.21)∫ZpBk,n1(x,q)Bk,n2(x,q)q(h-1)xζxdμ-q(x)=(n1k)(n2k)∫Zp∑l=0n1+n2-2k(-1)l(n1+n2-2kl)[x]q2k+lq(h-1)xζxdμ-q(x)=(n1k)(n2k)∑l=0n1+n2-2k(-1)l(n1+n2-2kl)G2k+l+1,q,ζ(h)2k+l+1.From Theorem2.10 and (2.21), we have the following corollary.Corollary 2.11.
Forn1,n2,k∈ℤ+ and n1+n2>2k, one has
(2.22)∑l=0n1+n2-2k(n1+n2-2kl)(-1)lG2k+l+1,q,ζ(h)2k+l+1={qh+1ζGn1+n2+1,q-1,ζ-1(h)n1+n2+1+[2]qifk=0,∑l=0n1+n2-2k(n1+n2-2kl)(-1)2k-lGn1+n2-l+1,q-1,ζ-1(h)n1+n2-l+1ifk>0.



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*Source: 482840-2011-12-22.xml*