# Some Relations between Twisted (h,q)-Euler Numbers with Weight α and q-Bernstein Polynomials with Weight α

Authors: N. S. Jung; H. Y. Lee; C. S. Ryoo
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

By using fermionicp-adic q-integral on ℤp, we give some
interesting relationship between the twisted (h, q)-Euler numbers with weight α and the
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 integers, ℚp denotes the field 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={w∣wpn=1} be the cyclic group of order pn, and let
(1.1)Tp=lim⁡n→∞Cpn=Cp∞=∪n≥0Cpn,
(see [1–22]), be the locally constant space. For w∈Tp, we denote by ϕw:ℤp→ℂp the locally constant function x↦wx. The p-adic absolute value is defined by |x|p=1/pr, where x=prs/t(r∈ℚands,t∈ℤwith(s,t)=(p,s)=(p,t)=1). In this paper, we assume that q∈ℂp with |q-1|p<1 as an indeterminate. The q-number is defined by
(1.2)[x]q=1-qx1-q,
(see [1–22]). Note that limq→1[x]q=x. For
(1.3)f∈UD(Zp)={f∣f:Zp⟶Cpisuniformlydifferentiablefunction},
the fermionic p-adic q-integral on ℤp is defined by Kim as follows:
(1.4)I-q(f)=∫Zpf(x)dμ-q(x)=lim⁡N→∞1+q1+qpN∑x=0pN-1f(x)(-q)x,
(see [1–7]). From (1.4), we note that
(1.5)qnI-q(fn)=(-1)nI-q(f)+[2]q∑l=0n-1(-1)n-1-lqlf(l),
where fn(x)=f(x+n) for n∈ℕ.Fork,n∈ℤ+ and x∈[0,1], Kim defined q-Bernstein polynomials, which are different q-Bernstein polynomials of Phillips, as follows:
(1.6)Bk,n(x,q)=(nk)[x]qk[1-x]q-1n-k,
(see [5]). In [9], the p-adic extension of (1.6) is given by
(1.7)Bk,n(x,q)=(nk)[x]qk[1-x]q-1n-k,wherex∈Zp,n,k∈Z+.
For α∈ℤ, h∈ℤ, w∈Tp, and q∈ℂp with |1-q|p≤1, twisted (h,q)-Euler numbers En,q,w(h,α) with weight α are defined by
(1.8)En,q,w(h,α)=∫Zpϕw(x)qx(h-1)[x]qαndμ-q(x).
In the special case, x=0, En,q,w(h,α)(0)=En,q,w(h,α) are called the n-th twisted (h,q)-Euler numbers with weight α.In this paper, we investigate some relations between theq-Bernstein polynomials and the twisted (h,q)-Euler numbers with weight α. From these relations, we derive some interesting identities on the twisted (h,q)-Euler numbers and polynomials with weight α.


## 2. Twisted(h,q)-Euler Numbers and Polynomials with Weight α

By usingp-adic q-integral and (1.8), we obtain
(2.1)∫Zpϕw(x)qx(h-1)[x]qαndμ-q(x)=lim⁡N→∞1[pN]-q∑x=0pN-1[x]qαnwxqx(h-1)(-q)x=[2]q(11-qα)n∑l=0n(nl)(-1)l11+wqαl+h.
We set
(2.2)Fq,w(h,α)(t)=∑n=0∞En,q,w(h,α)tnn!.
By (2.1) and (2.2), we have
(2.3)Fq,w(h,α)(t)=∑n=0∞En,q,w(h,α)tnn!=[2]q∑n=0∞((11-qα)n∑l=0n(nl)(-1)l11+wqαl+h)tnn!=[2]q∑m=0∞(-1)mwmqhme[m]qαt.
Since [x+y]qα=[x]qα+qαx[y]qα, we obtain
(2.4)En,q,w(h,α)(x)=∫Zpϕw(y)qy(h-1)[y+x]qαtdμ-q(y)=∑l=0n(nl)qαxl[x]qαn-l∫Zpϕw(y)qy(h-1)[y]qαldμ-q(y)=∑l=0n(nl)qαxl[x]qαn-lEl,q,w(h,α).Therefore, we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+ and w∈Tp, we have
(2.5)En,q,w(h,α)(x)=[2]q∑m=0∞(-1)mwmqhm[x+m]qαn.
Furthermore,
(2.6)En,q,w(h,α)(x)=∑l=0n(nl)qαxl[x]qαn-lEl,q,w(h,α)=([x]qα+qαxEq,w(h,α))n,
with usual convention about replacing (Eq,w(h,α))n with En,q,w(h,α).
LetFq,w(h,α)(t,x)=∑n=0∞En,q,w(h,α)(x)tn/n!. Then we see that
(2.7)Fq,w(h,α)(t,x)=[2]q∑m=0∞(-1)mwmqmhe[x+m]qαt.
In the special case, x=0, let Fq,w(h,α)(t,0)=Fq,w(h,α)(t).
By (2.1), we get
(2.8)En,q-1,w-1(h,α)(1-x)=(-1)nwqαn+h-1En,q,w(h,α)(x).
From (2.3) and (2.7), we note that
(2.9)wqhFq,w(h,α)(t,1)+Fq,w(h,α)(t)=[2]q.
By (2.9), we get the following recurrence formula:
(2.10)E0,q,w(h,α)=[2]q1+qhw,qhwEn,q,w(h,α)(1)+En,q,w(h,α)=0ifn>0.By (2.10) and Theorem 2.1, we obtain the following theorem.Theorem 2.2.
Forn∈ℤ+ and w∈Tp, we have
(2.11)E0,q,w(h,α)=[2]q1+qhw,qhw(qαEq,w(h,α)+1)n+En,q,w(h,α)=0ifn>0,
with usual convention about replacing (Eq,w(h,α))n with En,q,w(h,α).
By (2.4), Theorem 2.1, and Theorem 2.2, we have
(2.12)q2hw2En,q,w(h,α)(2)-[2]q1+qhwq2hw2-[2]q1+qhwqhw=q2hw2∑l=0n(nl)qαl(qαEq,w(h,α)+1)l-[2]q1+qhwq2hw2-[2]q1+qhwqhw=q2hw2∑l=1n(nl)qαl(qαEq,w(h,α)+1)l-[2]q1+qhwqhw=-qhw∑l=1n(nl)qαlEl,q,w(h,α)-[2]q1+qhwqhw=-qhw∑l=0n(nl)qαlEl,q,w(h,α)=-qhwEn,q,w(h,α)(1)=En,q,w(h,α)ifn>0.Therefore, we obtain the following theorem.Theorem 2.3.
Forn∈ℕ, we have
(2.13)En,q,w(h,α)(2)=(1q2hw2)En,q,w(h,α)+[2]q1+qhw+(1qhw)[2]q1+qhw.
By (2.8), we see that
(2.14)qh-1w∫Zp[1-x]q-αnq(h-1)xwxdμ-q(x)=(-1)nqαn+h-1w∫Zp[x-1]qαnq(h-1)xwxdμ-q(x)=(-1)nqαn+h-1wEn,q,w(h,α)(-1)=En,q-1,w-1(h,α)(2).Therefore, we obtain the following theorem.Theorem 2.4.
Forn∈ℤ+, we have
(2.15)qh-1w∫Zp[1-x]q-αnq(h-1)xwxdμ-q(x)=En,q-1,w-1(h,α)(2).
Let n∈ℕ. By Theorems 2.3 and 2.4, we get
(2.16)qh-1w∫Zp[1-x]q-αnq(h-1)xwxdμ-q(x)=q2hw2En,q-1,w-1(h,α)+qh-1w([2]q1+qhw)+q2h-1w2([2]q1+qhw).
From (2.16), we have
(2.17)∫Zp[1-x]q-1nq(h-1)xwxdμ-q(x)=qh+1wEn,q-1,w-1(h,α)+([2]q1+qhw)+qhw([2]q1+qhw).Therefore, we obtain the following corollary.Corollary 2.5.
Forn∈ℕ, we have
(2.18)∫Zp[1-x]q-αnq(h-1)xwxdμ-q(x)=qh+1wEn,q-1,w-1(h,α)+[2]q.
Kim defined theq-Bernstein polynomials with weight α of degree n as below.
Forx∈ℤp, the p-adic q-Bernstein polynomials with weight α of degree n are given by
(2.19)Bk,n(α)(x,q)=(nk)[x]qαk[1-x]q-αn-k,wheren,k∈Z+.
compare [5, 10, 22] By (2.19), we get the symmetry of q-Bernstein polynomials as follows:
(2.20)Bk,n(α)(x,q)=Bn-k,n(α)(1-x,q-1),
see [8]. Thus, by Corollary 2.5, (2.19), and (2.20), we see that
(2.21)∫ZpBk,n(α)(x,q)q(h-1)xwxdμ-q(x)=∫ZpBn-k,n(α)(1-x,q-1)q(h-1)xwxdμ-q(x)=(nk)∑l=0k(kl)(-1)k+l∫Zp[1-x]q-αn-lq(h-1)xwxdμ-q(x)=(nk)∑l=0k(kl)(-1)k+l(qh+1wEn-l,q-1,w-1(h,α)+[2]q).
Forn,k∈ℤ+ with n>k, we have
(2.22)∫ZpBk,n(α)(x,q)q(h-1)xwxdμ-q(x)=(nk)∑l=0k(kl)(-1)k+l(qh+1wEn-l,q-1,w-1(h,α)+[2]q)={qh+1wEn,q-1,w-1(h,α)+[2]qifk=0,qh+1w(nk)∑l=0k(kl)(-1)k+lEn-l,q-1,w-1(h,α)ifk>0.
Let us take the fermionic q-integral on ℤp for the q-Bernstein polynomials with weight α of degree n as follows:
(2.23)∫ZpBk,n(α)(x,q)q(h-1)xwxdμ-q(x)=(nk)∫Zp[x]qk[1-x]q-αn-kq(h-1)xwxdμ-q(x)=(nk)∑l=0n-k(n-kl)(-1)lEl+k,q,w(h,α).Therefore, by (2.22) and (2.23), we obtain the following theorem.Theorem 2.6.
Letn,k∈ℤ+ with n>k. Then we have
(2.24)∫ZpBk,n(α)(x,q)q(h-1)xwxdμ-q(x)={qh+1wEn,q-1,w-1(h,α)+[2]qifk=0,qh+1w(nk)∑l=0k(kl)(-1)k+lEn-l,q-1,w-1(h,α)ifk>0.
Moreover,
(2.25)∑l=0n-k(n-kl)(-1)lEl+k,q,w(h,α)={qh+1wEn,q-1,w-1(h,α)+[2]qifk=0,qh+1w∑l=0k(kl)(-1)k+lEn-l,q-1,w-1(h,α)ifk>0.
Letn1,n2,k∈ℤ+ with n1+n2>2k. Then we get
(2.26)∫ZpBk,n1(α)(x,q)Bk,n2(α)(x,q)q(h-1)xwxdμ-q(x)=(n1k)(n2k)∑l=02k(2kl)(-1)l+2k∫Zp[1-x]q-αn1+n2-lq(h-1)xwxdμ-q(x)=(n1k)(n2k)∑l=02k(2kl)(-1)l+2k(qh+1wEn1+n2-l,q-1,w-1(h,α)+[2]q)={qh+1wEn1+n2,q-1,w-1(h,α)+[2]qifk=0,(n1k)(n2k)∑l=02k(2kl)(-1)2k+l(qh+1wEn1+n2-l,q-1,w-1(h,α)+[2]q)ifk≠0.Therefore, by (2.26), we obtain the following theorem.Theorem 2.7.
Forn1,n2,k∈ℤ+ with n1+n2>2k, we have
(2.27)∫ZpBk,n1(α)(x,q)Bk,n2(α)(x,q)q(h-1)xwxdμ-q(x)={qh+1wEn1+n2,q-1,w-1(h,α)+[2]qifk=0,qh+1w(n1k)(n2k)∑l=02k(2kl)(-1)2k+lEn1+n2-l,q-1,w-1(h,α)ifk≠0.
From the binomial theorem, we can derive the following equation:(2.28)∫ZpBk,n1(α)(x,q)Bk,n2(α)(x,q)q(h-1)xwxdμ-q(x)=(n1k)(n2k)∑l=0n1+n2-2k(-1)l(n1+n2-2kl)∫Zp[x]q2k+lq(h-1)xwxdμ-q(x)=(n1k)(n2k)∑l=0n1+n2-2k(-1)l(n1+n2-2kl)E2k+l,q,w(h,α).Thus, by (2.28) and Theorem 2.7, we obtain the following corollary.Corollary 2.8.
Letn1,n2,k∈ℤ+ with n1+n2>2k. Then we have
(2.29)∑l=0n1+n2-2k(-1)l(n1+n2-2kl)E2k+l,q(h,α)={qh+1wEn1+n2,q-α,w-1(h,α)+[2]qifk=0,qh+1w∑l=02k(2kl)(-1)2k+lEn1+n2-l,q-α,w-1(h,α)ifk>0.
Forx∈ℤp and s∈ℕ with s≥2, let n1,n2,…,ns,k∈ℤ+ with n1+⋯+ns>sk. Then we take the fermionic p-adic q-integral on ℤp for the q-Bernstein polynomials with weight α of degree n as follows:
(2.30)∫ZpBk,n1(α)(x,q)⋯Bk,ns(α)(x,q)︸s-timesq(h-1)xwxdμ-q(x)=(n1k)⋯(nsk)∫Zp[x]qsk[1-x]q-αn1+⋯+ns-skq(h-1)xwxdμ-q(x)=(n1k)⋯(nsk)∑l=0sk(skl)(-1)l+sk∫Zp[1-x]q-αn1+⋯+ns-lq(h-1)xwxdμ-q(x)=(n1k)⋯(nsk)∑l=0sk(skl)(-1)l+sk(qh+1wEn1+⋯+ns-l,q-1,w-1(h,α)+[2]q)={qh+1wEn1+⋯+ns,q-1,w-1(h,α)+[2]qifk=0,qh+1w(n1k)⋯(nsk)∑l=0sk(skl)(-1)l+skEn1+⋯+ns-l,q-1,w-1(h,α)ifk>0.Therefore, by (2.30), we obtain the following theorem.Theorem 2.9.
Fors∈ℕ with s≥2, let n1,n2,…,ns,k∈ℤ+ with n1+⋯+ns>sk. Then we get
(2.31)∫ZpBk,n1(α)(x,q)⋯Bk,ns(α)(x,q)︸s-timesq(h-1)xwxdμ-q(x)={qh+1wEn1+⋯+ns,q-1,w-1(h,α)+[2]qifk=0,qh+1w(n1k)⋯(nsk)∑l=0sk(skl)(-1)l+skEn1+⋯+ns-l,q-1,w-1(h,α)ifk>0.
By the definition ofq-Bernstein polynomials with weight α and the binomial theorem, we easily get
(2.32)∫ZpBk,n1(α)(x,q)⋯Bk,ns(α)(x,q)︸s-timesq(h-1)xwxdμ-q(x)=(n1k)⋯(nsk)∑l=0n1+⋯+ns-sk(-1)l(n1+⋯+ns-skl)∫Zp[x]qsk+lq(h-1)xwxdμ-q(x)=(n1k)⋯(nsk)∑l=0n1+⋯+ns-sk(-1)l(n1+⋯+ns-skl)Esk+l,q,w(h,α).Therefore, we have the following corollary.Corollary 2.10.
Forw∈Tp,s∈ℕ with s≥2, let n1,n2,…,ns,k∈ℤ+ with n1+⋯+ns>sk. Then we have
(2.33)∑l=0n1+⋯+ns-sk(-1)l(n1+⋯+ns-skl)Esk+l,q,w(h,α)={qh+1wEn1+⋯+ns,q-1,w-1(h,α)+[2]qifk=0,qh+1w∑l=0sk(skl)(-1)l+skEn1+⋯+ns-l,q-1,w-1(h,α)ifk>0.



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*Source: 176296-2011-10-23.xml*