# Some Identities of the Twistedq-Genocchi Numbers and Polynomials with Weight α and q-Bernstein Polynomials with Weight α

Authors: H. Y. Lee; N. S. Jung; C. S. Ryoo
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Recently mathematicians have studied some interesting relations betweenq-Genocchi numbers, q-Euler numbers, polynomials, Bernstein polynomials, and q-Bernstein polynomials. In this paper, we give some interesting identities of the twisted q-Genocchi numbers, polynomials, and q-Bernstein polynomials with weighted α.


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## Body

## 1. Introduction

Throughout this paper, letp be a fixed odd prime number. The symbols ℤp, ℚp, and ℂp denote the ring of p-adic integers, the field of p-adic rational numbers, and the completion of algebraic closure of ℚp. Let ℕ be the set of natural numbers and let ℤ+=ℕ∪{0}. As a well-known definition, the p-adic absolute value is given by |x|p=p-r, where x=prt/s with (t,p)=(s,p)=(t,s)=1. When one talks of q-extension, q is variously considered as an indeterminate, a complex number q∈ℂ, or a p-adic number q∈ℂp. In this paper we assume that q∈ℂp with |1-q|p<1.We assume thatUD(ℤp) is the space of the uniformly differentiable function on ℤp. For f∈UD(ℤp), Kim defined the fermionic p-adic q-integral on ℤp as follows:(1.1)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1f(x)(-q)x.
For n∈ℕ, let fn(x)=f(x+n) be translation. As a well known equation, by (1.1), we have (1.2)qn∫Zpf(x+n)dμ-q(x)=(-1)nI-q(f)+[2]q∑l=0n-1(-1)n-1-lqlf(l),
compared [1–4]. Throughout this paper we use the notation: (1.3)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q,(cf. [1–16]). limq→1[x]q=x for any x with |x|p≤1 in the present p-adic case. To investigate relation of the twisted q-Genocchi numbers and polynomials with weight α and the Bernstein polynomials with weight α, we will use useful property for [x]qα as follows;(1.4)[x]qα=1-[1-x]q-α,[1-x]q-α=1-[x]qα.The twistedq-Genocchi numbers and polynomials with weight α are defined by the generating function as follows, respectively:
(1.5)Gn,q,w(α)=n∫Zpϕw(x)[x]qαn-1dμ-q(x),(1.6)Gn,q,w(α)(x)=n∫Zpϕw(y)[y+x]qαn-1dμ-q(y).
In the special case, x=0, Gn,q,w(α)(0)=Gn,q,w(α) are called the nth twisted q-Genocchi numbers with weight α (see [9]).LetCpn={w∣wpn=1} be the cyclic group of order pn and let (1.7)Tp=lim⁡n→∞Cpn=⋃n≥1Cpn,
see [9, 12–15].Kim defined theq-Bernstein polynomials with weight α of degree n as follows: (1.8)Bn,k(α)(x)=(nk)[x]qαk[1-x]q-αn-k,wherex∈[0,1],n,k∈Z+,
compare [4, 7].In this paper, we investigate some properties for the twistedq-Genocchi numbers and polynomials with weight α. By using these properties, we give some interesting identities on the twisted q-Genocchi polynomials with weight α and q-Bernstein polynomials with weight α.


## 2. Some Identities on the Twistedq-Genocchi Polynomials with Weight α and q-Bernstein Polynomials with Weight α

From (1.8), we can derive the following recurrence formula for the twisted q-Genocchi numbers with weight α:
(2.1)G0,q,w(α)=0,qwGn,q,w(α)(1)+Gn,q,w(α)={[2]q,ifn=1,0,ifn>1,(2.2)G0,q,w(α)=0,qw(1+qαGq,w(α))n+qαGn,q,w(α)={qα[2]q,ifn=1,0,ifn>1,(2.3)qαxGn+1,q,w(α)(x)=([x]qα+qαxGq,w(α))n+1
with usual convention about replacing (Gq,w(α))n by Gn,q,w(α).By (1.5), we easily get (2.4)Gn,q,w(α)(x)=n[2]q(11-qα)n-1∑l=0n-1(n-1l)(-1)lqαxl11+wqαl+1.
By (2.4), we obtain the theorem below.Theorem 2.1.
Letn∈ℤ+. For w∈Tp, one has
(2.5)Gn,q,w(α)(x)=(-1)n-1w-1qα(1-n)Gn,q-1,w-1(α)(1-x).By (2.1), (2.2), and (2.3) we note that (2.6)Gn,q,w(α)=-qwGn,q,w(α)(1)=-nwqG1,q,w(α)+w2q2-α∑l=2n(nl)qαlGl,q,w(α)(1)=-nwqG1,q,w(α)+w2q2-2α∑l=2n(nl)qαl(1+qαGq,w(α))l=-nwqG1,q,w(α)+w2q2-2α([2]qα+q2αGq,w(α))n-nw2q2G1,q,w(α)=-nwqG1,q,w(α)+w2q2Gn,q,w(α)(2)-nw2q2G1,q,w(α).Therefore, by (2.6), we obtain the theorem below.Theorem 2.2.
Forn∈ℕ with n>1, one has
(2.7)Gn,q,w(α)(2)=w-2q-2Gn,q,w(α)+w-1q-1n[2]q1+qw+n[2]q1+qw.By (1.6) and Theorem 2.2, (2.8)Gn+1,q,w(α)(2)n+1=∫Zpϕw(y)[y+2]qαndμ-q(y)=1n+1((n+1)[2]q1+qw+(n+1)w-1q-1[2]q1+qw)+w-2q-2Gn+1,q,w(α)n+1=[2]q1+qw+w-1q-1[2]q1+qw+w-2q-2Gn+1,q,w(α)n+1.
Hence, we obtain the corollary below.Corollary 2.3.
Forn∈ℕ, one has
(2.9)∫Zpϕw(y)[y+2]qαndμ-q(y)=[2]q1+qw+w-1q-1[2]q1+qw+w-2q-2Gn+1,q,w(α)n+1.By fermionic integral onℤp,Theorems 2.1 and 2.2, we note that(2.10)∫Zpϕw(x)[1-x]q-αndμ-q(x)=(-1)nqαn∫Zpϕw(x)[x-1]qαndμ-q(x)=(-1)nqαnGn+1,q,w(α)(-1)n+1=w-1Gn+1,q-1,w-1(α)(2)n+1=w-1([2]q-11+q-1w-1+wq[2]q-11+q-1w-1+w2q2Gn+1,q-1,w-1(α)n+1)=[2]q1+qw+wq[2]q1+qw+wq2Gn+1,q-1,w-1(α)n+1.Therefore, we have the theorem below.Theorem 2.4.
Forn∈ℕ with n>1, one has
(2.11)∫Zpϕw(x)[1-x]q-αndμ-q(x)=[2]q1+qw+wq[2]q1+qw+wq2Gn+1,q-1,w-1(α)n+1.By (1.4), Theorem 2.4, we take the fermionic p-adic invariant integral on ℤp for one q-Bernstein polynomials as follows:
(2.12)∫Zpϕw(x)Bn,k(x,q)dμ-q(x)=∫Zpϕw(x)(nk)[x]qαk[1-x]q-αn-kdμ-q(x)=(nk)∫Zpϕw(x)[x]qαk(1-[x]qα)n-kdμ-q(x)=(nk)∑l=0n-k(n-kl)(-1)lGk+l+1,q,w(α)k+l+1.By symmetry ofq-Bernstein polynomials with weight α of degree n, we get the following formula;
(2.13)∫Zpϕw(x)Bn,k(x,q)dμ-q(x)=∫Zpϕw(x)(nk)[x]qαn-k[1-x]q-αkdμ-q(x)=∫Zpϕw(x)(nk)[1-x]q-αk(1-[1-x]q-α)n-kdμ-q(x)=(nk)∑l=0n-k(n-kl)(-1)n-k-l∫Zpϕw(x)[1-x]q-αn-ldμ-q(x)=(nk)∑l=0n-k(n-kl)(-1)n-k-l([2]q1+qw+wq[2]q1+qw+wq2Gn-l+1,q-1,w-1(α)n-l+1).Therefore, by (2.12) and (2.13), we have the theorem below.Theorem 2.5.
Forn∈ℕ with n>1, one has
(2.14)∑l=0n-k(n-kl)(-1)lGk+l+1,q,w(α)k+l+1=∑l=0n-k(n-kl)(-1)n-k-l([2]q1+qw+wq[2]q1+qw+wq2Gn-l+1,q-1,w-1(α)n-l+1).
Also, we note that
(2.15)∫Zpϕw(x)Bn,k(x,q)dμ-q(x)=(nk)∑l=0n-k(n-kl)(-1)lGk+l+1,q,w(α)k+l+1=(nk)∫Zpϕw(x)[1-x]q-αn-k[x]qαkdμ-q(x)=(nk)∫Zpϕw(x)[1-x]q-αn-k(1-[1-x]q-α)kdμ-q(x)=(nk)∑l=0k(kl)(-1)k-l∫Zpϕw(x)[1-x]q-αn-ldμ-q(x)=(nk)∑l=0k(kl)(-1)k-l([2]q1+qw+wq[2]q1+qw+wq2Gn-l+1,q-1,w-1(α)n-l+1).Therefore, we have the theorem below.Theorem 2.6.
Forn,k∈ℤ+ with n>k+1, one has
(2.16)∫Zpϕw(x)Bk,n(x,q)dμ-q(x)=(nk)∑l=0k(kl)(-1)k-l([2]q1+qw+wq[2]q1+qw+wq2Gn-l+1,q-1,w-1(α)n-l+1).By (2.11) and Theorem 2.6, we have the theorem below.Theorem 2.7.
Letn,k∈ℤ+ with n>k+1. Then one has
(2.17)∑l=0n-k(n-kl)(-1)lGk+l+1,q,w(α)k+l+1=∑l=0k(kl)(-1)k-l([2]q1+qw+wq[2]q1+qw+wq2Gn-l+1,q-1,w-1(α)n-l+1).Letn1,n2,k∈ℤ+ with n1+n2>2k+1. Then we get (2.18)∫Zpϕw(x)Bn1,k(α)(x,q)Bn2,k(α)(x,q)dμ-q(x)=(n1k)(n2k)∑l=02k(2kl)(-1)2k-l∫Zpϕw(x)[1-x]q-αn1+n2-ldμ-q(x)=(n1k)(n2k)∑l=02k(2kl)(-1)2k-l([2]q1+qw+wq[2]q1+qw+wq2Gn1+n2-l+1,q-1,w-1(α)n1+n2-l+1).Therefore, we obtain the theorem below.Theorem 2.8.
Forn1,n2,k∈ℤ+, one has
(2.19)∫Zpϕw(x)Bn1,k(α)(x,q)Bn2,k(α)(x,q)dμ-q(x)=(n1k)(n2k)∑l=02k(2kl)(-1)2k-l([2]q1+qw+wq[2]q1+qw+wq2Gn1+n2-l+1,q-1,w-1(α)n1+n2-l+1)={[2]q1+qw+wq[2]q1+qw+wq2Gn1+n2-l+1,q-1,w-1(α)n1+n2-l+1,ifk=0,wq2(n1k)(n2k)∑l=02k(2kl)(-1)2k-lGn1+n2-l+1,q-1,w-1n1+n2-l+1,ifk>0,By simple calculation, we easily see that(2.20)∫Zpϕw(x)Bn1,k(α)(x,q)Bn2,k(α)(x,q)dμ-q(x)=(n1k)(n2k)∑l=0n1+n2-2k(-1)l(n1+n2-2kl)∫Zpϕw(x)[x]qα2k+ldμ-q(x)=(n1k)(n2k)∑l=0n1+n2-2k(-1)l(n1+n2-2kl)G2k+l+1,q,w(α)2k+l+1,wheren1,n2,k∈Z+.
Therefore, by (2.20) and Theorem 2.8, we obtain the theorem below.Theorem 2.9.
Letn1,n2,k∈ℤ+ with n1+n2>2k+1. Then one has
(2.21)∑l=02k(2kl)(-1)2k-l([2]q1+qw+wq[2]q1+qw+wq2Gn1+n2-l+1,q-1,w-1(α)n1+n2-l+1)=∑l=0n1+n2-2k(-1)l(n1+n2-2kl)G2k+l+1,q,w(α)2k+l+1.Forn1,n2,…,ns,k∈ℤ+, n1+n2+⋯+ns>sk+1, and let ∑i=1sni=m, then by the symmetry of q-Bernstein polynomials with weight α, we see that (2.22)∫Zpϕw(x)∏i=1sBk,ni(α)(x,q)dμ-q(x)=∏i=1s(nik)∑l=0sk(skl)(-1)sk-l∫Zpϕw(x)[1-x]q-αm-ldμ-q(x)=∏i=1s(nik)∑l=0sk(skl)(-1)sk-l([2]q1+qw+wq[2]q1+qw+wq2Gm-l+1,q-1,w-1(α)m-l+1).Therefore, we have the theorem below.Theorem 2.10.
Forn1,n2,…,ns,k∈ℤ+ with n1+n2+⋯+ns>sk+1, one has
(2.23)∫Zpϕw(x)∏i=1sBk,ni(α)(x,q)dμ-q(x)=∏i=1s(nik)∑l=0sk(skl)(-1)sk-l([2]q1+qw+wq[2]q1+qw+wq2Gm-l+1,q-1,w-1(α)m-l+1),
where n1+⋯+ns=m.In the same manner as in (2.15), we can get the following relation:(2.24)∫Zpϕw(x)∏i=1sBk,ni(α)(x,q)dμ-q(x)=∏i=1s(nik)∫Zpϕw(x)[x]qαsk∑l=0m-sk(-1)l(m-skl)(-1)l[x]qαldμ-q(x)=∏i=1s(nik)∑l=0m-sk(-1)l(m-skl)Gsk+l+1,q,w(α)sk+l+1,
where n1,n2,…,ns,k∈ℤ+ with m=n1+n2+⋯+ns>sk+1.By Theorem2.10 and (2.13), we have the following corollary.Corollary 2.11.
Letm∈ℕ. For n1,n2,…,ns,k∈ℤ+ with n1+⋯+ns>mk+1, one has
(2.25)∑l=0sk(skl)(-1)sk-l([2]q1+qw+wq[2]q1+qw+wq2Gm-l+1,q-1,w-1(α)m-l+1)=∑l=0m-sk(-1)l(m-skl)Gsk+l+1,q,w(α)sk+l+1,
where n1+⋯+ns=m.



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*Source: 123483-2011-11-02.xml*