# Onq-Euler Numbers Related to the Modified q-Bernstein Polynomials

Authors: Min-Soo Kim; Daeyeoul Kim; Taekyun Kim
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We considerq-Euler numbers, polynomials, and q-Stirling numbers of first and second kinds. Finally, we investigate some interesting properties of the modified q-Bernstein polynomials related to q-Euler numbers and
q-Stirling numbers by using fermionic p-adic integrals on ℤp.


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## Body

## 1. Introduction

LetC[0,1] be the set of continuous functions on [0,1]. The classical Bernstein polynomials of degree n for f∈C[0,1] are defined by(1.1)𝔹n(f)=∑k=0nf(kn)Bk,n(x),0≤x≤1,
where 𝔹n(f) is called the Bernstein operator and(1.2)Bk,n(x)=(nk)xk(x-1)n-k
are called the Bernstein basis polynomials (or the Bernstein polynomials of degree n) (see [1]). Recently, Acikgoz and Araci have studied the generating function for Bernstein polynomials (see [2, 3]). Their generating function for Bk,n(x) is given by(1.3)F(k)(t,x)=tke(1-x)txkk!=∑n=0∞Bk,n(x)tnn!,
where k=0,1,… and x∈[0,1]. Note that (1.4)Bk,n(x)={(nk)xk(1-x)n-k,ifn≥k,0,ifn<k,
for n=0,1,… (see [2, 3]).Letp be an odd prime number. Throughout this paper, ℤp,ℚp, and ℂp will denote the ring of p-adic rational integers, the field of p-adic rational numbers, and the completion of the algebraic closure of ℚp, respectively. Let vp be the normalized exponential valuation of ℂp with |p|p=p-1.Throughout this paper, we use the following notation:(1.5)[x]q=1-qx1-q,[x]-q=1-(-q)x1+q
(cf. [4–7]). Let ℕ be the natural numbers and ℤ+=ℕ∪{0}. Let UD(ℤp) be the space of uniformly differentiable function on ℤp.Letq∈ℂp with |1-q|p<p-1/(p-1) and x∈ℤp. Then q-Bernstein type operator for f∈UD(ℤp) is defined by (see [8, 9])(1.6)𝔹n,q(f)=∑k=0nf(kn)(nk)[x]qk[1-x]qn-k=∑k=0nf(kn)Bk,n(x,q),
for k,n∈ℤ+, where(1.7)Bk,n(x,q)=(nk)[x]qk[1-x]qn-k
is called the modified q-Bernstein polynomials of degree n. When we put q→1 in (1.7), [x]qk→xk,[1-x]qn-k→(1-x)n-k, and we obtain the classical Bernstein polynomial, defined by (1.2). We can deduce very easily from (1.7) that(1.8)Bk,n(x,q)=[1-x]qBk,n-1(x,q)+[x]qBk-1,n-1(x,q)
(see [8]). For 0≤k≤n, derivatives of the nth degree modified q-Bernstein polynomials are polynomials of degree n-1:(1.9)ddxBk,n(x,q)=n(qxBk-1,n-1(x,q)-q1-xBk,n-1(x,q))lnqq-1
(see [8]).The Bernstein polynomials can also be defined in many different ways. Thus, recently, many applications of these polynomials have been looked for by many authors. In the recent years, theq-Bernstein polynomials have been investigated and studied by many authors in many different ways (see [1, 8, 9] and references therein [10, 11]). In [11], Phillips gave many results concerning the q-integers and an account of the properties of q-Bernstein polynomials. He gave many applications of these polynomials on approximation theory. In [2, 3], Acikgoz and Araci have introduced several type Bernstein polynomials. The Acikgoz and Araci paper to announced in the conference is actually motivated to write this paper. In [1], Simsek and Acikgoz constructed a new generating function of the q-Bernstein type polynomials and established elementary properties of this function. In [8], Kim et al. proposed the modified q-Bernstein polynomials of degree n, which are different q-Bernstein polynomials of Phillips. In [9], Kim et al. investigated some interesting properties of the modified q-Bernstein polynomials of degree n related to q-Stirling numbers and Carlitz's q-Bernoulli numbers.In the present paper, we considerq-Euler numbers, polynomials, and q-Stirling numbers of first and second kinds. We also investigate some interesting properties of the modified q-Bernstein polynomials of degree n related to q-Euler numbers and q-Stirling numbers by using fermionic p-adic integrals on ℤp.


## 2.q-Euler Numbers and Polynomials Related to the Fermionic p-Adic Integrals on ℤp

ForN≥1, the fermionic q-extension μq of the p-adic Haar distribution μHaar,(2.1)μ-q(a+pNℤp)=(-q)a[pN]-q,
is known as a measure on ℤp, where a+pNℤp={x∈ℚp∣|x-a|p≤p-N} (cf. [4, 12]). We will write dμ-q(x) to remind ourselves that x is the variable of integration. Let UD(ℤp) be the space of uniformly differentiable function on ℤp. Then μ-q yields the fermionic p-adic q-integral of a function f∈UD(ℤp):(2.2)I-q(f)=∫Zpf(x)dμ-q(x)=limN→∞1+q1+qpN∑x=0pN-1f(x)(-q)x
(cf. [12–15]). Many interesting properties of (2.2) were studied by many authors (see [12, 13] and the references given there). Using (2.2), we have the fermionic p-adic invariant integral on ℤp as follows:(2.3)limq→-1Iq(f)=I-1(f)=∫Zpf(a)dμ-1(x).
For n∈ℕ, write fn(x)=f(x+n). We have(2.4)I-1(fn)=(-1)nI-1(f)+2∑l=0n-1(-1)n-l-1f(l).
This identity is obtained by Kim in [12] to derive interesting properties and relationships involving q-Euler numbers and polynomials. For n∈ℤ+, we note that(2.5)I-1([x]qn)=∫ℤp[x]qndμ-1(x)=En,q,
where En,q are the q-Euler numbers (see [16]). It is easy to see that E0,q=1. For n∈ℕ, we have(2.6)∑l=0n(nl)qlEl,q=∑l=0n(nl)qllimN→∞∑x=0pN-1[x]ql(-1)x=limN→∞∑x=0pN-1(-1)x(q[x]q+1)n=limN→∞∑x=0pN-1(-1)x[x+1]qn=-limN→∞∑x=0pN-1(-1)x([x]qn+[pN]qn)=-En,q.
From this formula, we have the following recurrence formula:(2.7)E0,q=1,(qE+1)n+En,q=0ifn∈N,
with the usual convention of replacing El by El,q. By the simple calculation of the fermionic p-adic invariant integral on ℤp, we see that(2.8)En,q=2(1-q)n∑l=0n(nl)(-1)l11+ql,
where (nl)=n!/l!(n-l)!=n(n-1)⋯(n-l+1)/l!. Now, by introducing the following equations:(2.9)[x]1/qn=qnq-nx[x]qn,q-nx=∑m=0∞(1-q)m(n+m-1m)[x]qm
into (2.5), we find that(2.10)En,1/q=qn∑m=0∞(1-q)m(n+m-1m)En+m,q.
This identity is a peculiarity of the p-adic q-Euler numbers, and the classical Euler numbers do not seem to have a similar relation. Let Fq(t) be the generating function of the q-Euler numbers. Then we obtain that(2.11)Fq(t)=∑n=0∞En,qtnn!=∑n=0∞2(1-q)n∑l=0n(-1)l(nl)11+qltnn!=2et/(1-q)∑k=0∞(-1)k(1-q)k11+qktkk!.
From (2.11), we note that(2.12)Fq(t)=2et/(1-q)∑n=0∞(-1)ne(-qn/(1-q))t=2∑n=0∞(-1)ne[n]qt.
It is well known that(2.13)I-1([x+y]n)=∫ℤp[x+y]ndμ-1(y)=En,q(x),
where En,q(x) are the q-Euler polynomials (see [16]). In the special case x=0, the numbers En,q(0)=En,q are referred to as the q-Euler numbers. Thus, we have(2.14)∫ℤp[x+y]ndμ-1(y)=∑k=0n(nk)[x]qn-kqkx∫ℤp[y]kdμ-1(y)=∑k=0n(nk)[x]qn-kqkxEk,q=(qxE+[x]q)n.
It is easily verified, using (2.12) and (2.13), that the q-Euler polynomials En,q(x) satisfy the following formula:(2.15)∑n=0∞En,q(x)tnn!=∫ℤpe[x+y]qtdμ-1(y)=∑n=0∞2(1-q)n∑l=0n(-1)l(nl)qlx1+qltnn!=2∑n=0∞(-1)ne[n+x]qt.
Using formula (2.15), when q tends to 1, we can readily derive the Euler polynomials, En(x), namely, (2.16)∫ℤpe(x+y)tdμ-1(y)=2extet+1=∑n=0∞En(x)tnn!
(see [12]). Note that En(0)=En are referred to as the nth Euler numbers. Comparing the coefficients of tn/n! on both sides of (2.15), we have(2.17)En,q(x)=2∑m=0∞(-1)m[m+x]qn=2(1-q)n∑l=0n(-1)l(nl)qlx1+ql.We refer to[n]q as a q-integer and note that [n]q is a continuous function of q. In an obvious way we also define a q-factorial, (2.18)[n]q!={[n]q[n-1]q⋯[1]q,n∈ℕ,1,n=0,
and a q-analogue of binomial coefficient,(2.19)(xn)q=[x]q![x-n]q![n]q!=[x]q[x-1]q⋯[x-n+1]q[n]q!
(cf. [14, 16]). Note that (2.20)limq→1(xn)q=(xn)=x(x-1)⋯(x-n+1)n!.
It readily follows from (2.19) that(2.21)(xn)q=(1-q)nq-(n2)[n]q!∑i=0nq(i2)(ni)q(-1)n+iq(n-i)x
(cf. [7, 16]). It can be readily seen that(2.22)qlx=([x]q(q-1)+1)l=∑m=0l(lm)(q-1)m[x]qm.
Thus, by (2.13) and (2.22), we have(2.23)∫ℤp(xn)qdμ-1(x)=(q-1)n[n]q!q(n2)∑i=0nq(i2)(ni)q(-1)i∑j=0n-i(n-ij)(q-1)jEj,q.
From now on, we use the following notation:(2.24)[x]q![x-k]q!=q-(k2)∑l=0ks1,q(k,l)[x]ql,k∈ℤ+,[x]qn=∑k=0nq(k2)s2,q(n,k)[x]q![x-k]q!,n∈ℤ+
(see [7]). From (2.24), and (2.22), we calculate the following consequence:(2.25)[x]qn=∑k=0nq(k2)s2,q(n,k)1(1-q)k∑l=0k(kl)qq(l2)(-1)lql(x-k+1)=∑k=0nq(k2)s2,q(n,k)1(1-q)k∑l=0k(kl)qq(l2)+l(1-k)(-1)l×∑m=0l(lm)(q-1)m[x]qm=∑k=0nq(k2)s2,q(n,k)1(1-q)k×∑m=0k(q-1)m(∑l=mk(kl)qq(l2)+l(1-k)(lm)(-1)l)[x]qm.
Therefore, we obtain the following theorem.Theorem 2.1.
Forn∈ℤ+,(2.26)En,q=∑k=0n∑m=0k∑l=mkq(k2)s2,q(n,k)(q-1)m-k(kl)qq(l2)+l(1-k)(lm)(-1)l+kEm,q.By (2.22) and simple calculation, we find that(2.27)∑m=0n(nm)(q-1)mEm,q=∫ℤpqnxdμ-1(x)=∑k=0n(q-1)kq(k2)(nk)q∫ℤp∏i=0k-1[x-i]qdμ-1(x)=∑k=0n(q-1)k(nk)q∑m=0ks1,q(k,m)∫ℤp[x]qmdμ-1(x)=∑m=0n(∑k=mn(q-1)k(nk)qs1,q(k,m))Em,q.
Therefore, we deduce the following theorem.Theorem 2.2.
Forn∈ℤ+,(2.28)∑m=0n(nm)(q-1)mEm,q=∑m=0n∑k=mn(q-1)k(nk)qs1,q(k,m)Em,q.Corollary 2.3.
Form,n∈ℤ+ with m≤n,(2.29)(nm)(q-1)m=∑k=mn(q-1)k(nk)qs1,q(k,m).By (2.17) and Corollary 2.3, we obtain the following corollary.Corollary 2.4.
Forn∈ℤ+,(2.30)En,q(x)=2(1-q)n∑l=0n∑k=ln(-1)l(q-1)k-l(nk)qs1,q(k,l)qlx1+ql.It is easy to see that(2.31)(nk)q=∑l0+⋯+lk=n-kq∑i=0kili
(cf. [7]). From (2.31) and Corollary 2.4, we can also derive the following interesting formula for q-Euler polynomials.Theorem 2.5.
Forn∈ℤ+,(2.32)En,q(x)=2∑l=0n∑k=ln∑l0+⋯+lk=n-kq∑i=0kili1(1-q)n+l-ks1,q(k,l)(-1)kqlx1+ql.These polynomials are related to the many branches of Mathematics, for example, combinatorics, number theory, and discrete probability distributions for finding higher-order moments (cf. [14–16]). By substituting x=0 into the above, we have (2.33)En,q=2∑l=0n∑k=ln∑l0+⋯+lk=n-kq∑i=0kili1(1-q)n+l-ks1,q(k,l)(-1)k11+ql,
where En,q is the q-Euler numbers.


## 3.q-Euler Numbers, q-Stirling Numbers, and q-Bernstein Polynomials Related to the Fermionic p-Adic Integrals on ℤp

First, we consider theq-extension of the generating function of Bernstein polynomials in (1.3).Forq∈ℂp with |1-q|p<p-1/(p-1), we obtain that(3.1)Fq(k)(t,x)=tke[1-x]qt[x]qkk!=[x]qk∑n=0∞(n+kk)[1-x]qntn+k(n+k)!=∑n=k∞(nk)[x]qk[1-x]qn-ktnn!=∑n=0∞Bk,n(x,q)tnn!,
which is the generating function of the modified q-Bernstein type polynomials (see [9]). Indeed, this generating function is also treated by Simsek and Acikgoz (see [1]). Note that limq→1Fq(k)(t,x)=F(k)(t,x). It is easy to show that(3.2)[1-x]qn-k=∑m=0∞∑l=0n-k(l+m-1m)(n-kl)(-1)l+mql[x]ql+m(q-1)m.From (1.6), (2.3), (2.15), and (3.2), we derive the following theorem.Theorem 3.1.
Fork,n∈ℤ+ with n≥k,(3.3)∫ℤpBk,n(x,q)(nk)dμ-1(y)=∑m=0∞∑l=0n-k(l+m-1m)(n-kl)(-1)l+mql(q-1)mEl+m+k,q,
where En,q are the q-Euler numbers.It is possible to write[x]qk as a linear combination of the modified q-Bernstein polynomials by using the degree evaluation formulae and mathematical induction. Therefore, we obtain the following theorem.Theorem 3.2 (see [8, Theorem 7]).
Fork,n∈ℤ+,i∈ℕ, and x∈[0,1],(3.4)∑k=i-1n(ki)(ni)Bk,n(x,q)=[x]qi([x]q+[1-x]q)n-i.Leti-1≤n. Then from (1.7), (3.2), and Theorem 3.2, we have(3.5)[x]qi=∑k=i-1n((ki)(nk)/(ni))[x]qk[1-x]qn-k[x]qn-i(1+([1-x]q/[x]q))n-k=∑m=0∞‍∑k=i-1n‍∑l=0m+n-k‍∑p=0∞(ki)(nk)(ni)(l+p-1p)(m+n-kl)×(n-i+m-1m)(-1)l+p+mql(q-1)p[x]qi-n-m+k+p+l.
Using (2.13) and (3.5), we obtain the following theorem.Theorem 3.3.
Fork,n∈ℤ+ and i∈ℕ with i-1≤n,(3.6)Ei,q=∑m=0∞‍∑k=i-1n‍∑l=0m+n-k∑p=0∞(ki)(nk)(ni)(l+p-1p)(m+n-kl)×(n-i+m-1m)(-1)l+p+mql(q-1)pEi-n-m+k+p+l,q.Theq-String numbers of the first kind is defined by(3.7)∏k=1n(1+[k]qz)=∑k=0nS1(n,k;q)zk,
and the q-String number of the second kind is also defined by(3.8)∏k=1n(1+[k]qz)-1=∑k=0nS2(n,k;q)zk
(see [9]). Therefore, we deduce the following theorem.Theorem 3.4 (see [9, Theorem 4]).
Fork,n∈Z+ and i∈ℕ,(3.9)∑k=i-1n((ki)/(ni))Bk,n(x,q)([x]q+[1-x]q)n-i=∑k=0i∑l=0kS1(k,l;q)S2(k,i-k;q)[x]ql.By Theorems3.2 and 3.4 and the definition of fermionic p-adic integrals on ℤp, we obtain the following theorem.Theorem 3.5.
Fork,n∈ℤ+ and i∈ℕ,(3.10)Ei,q=∑k=i-1n(ki)(ni)∫ℤpBk,n(x,q)([x]q+[1-x]q)n-idμ-1(x)=∑k=0i∑l=0kS1(k,l;q)S2(k,i-k;q)El,q,
where Ei,q is the q-Euler numbers.Leti-1≤n. It is easy to show that(3.11)[x]qi([x]q+[1-x]q)n-i=∑l=0n-i(n-il)[x]ql+i[1-x]qn-i-l=∑l=0n-i∑m=0n-i-l(n-il)(n-i-lm)(-1)mqm[x]qm+i+lq-mx=∑l=0n-i∑m=0n-i-l∑s=0∞(n-il)(n-i-lm)(m+s-1s)(-1)mqm(1-q)s[x]qm+i+l+s.
From (3.11) and Theorem 3.2, we have the following theorem.Theorem 3.6.
Fork,n∈ℤ+ and i∈ℕ,(3.12)∑k=i-1n(ki)(ni)∫ℤpBk,n(x,q)dμ-1(x)=∑l=0n-i∑m=0n-i-l∑s=0∞(n-il)(n-i-lm)(m+s-1s)×(-1)mqm(1-q)sEm+i+l+s,q,
where Ei,q are the q-Euler numbers.In the same manner, we can obtain the following theorem.Theorem 3.7.
Fork,n∈ℤ+ and i∈ℕ,(3.13)∫ℤpBk,n(x,q)dμ-1(x)=∑j=kn∑m=0∞(jk)(nj)(j-k+m-1m)(-1)j-k+mqj-k(q-1)mEm+j,q,
where Ei,q are the q-Euler numbers.


## 4. Further Remarks and Observations

Theq-binomial formulas are known as(4.1)(a;q)n=(1-a)(1-aq)⋯(1-aqn-1)=∑i=0n(ni)qq(i2)(-1)iai,1(a;q)n=1(1-a)(1-aq)⋯(1-aqn-1)=∑i=0∞(n+i-1i)qai.
For h∈ℤ,n∈ℤ+, and r∈ℕ, we introduce the extended higher-order q-Euler polynomials as follows [16]:(4.2)En,q(h,r)(x)=∫ℤp⋯∫ℤpq∑j=1r(h-j)xj[x+x1+⋯+xr]qndμ-1(x1)⋯dμ-1(xr).
Then,(4.3)En,q(h,r)(x)=2r(1-q)n∑l=0n(nl)(-1)lqlx(-qh-1+l;q-1)r=2r(1-q)n∑l=0n(nl)(-1)lqlx(-qh-r+l;q)r.
Let us now define the extended higher-order Nörlund type q-Euler polynomials as follows [16]:(4.4)En,q(h,-r)(x)=1(1-q)n∑l=0n(nl)(-1)lqlx∫ℤp⋯∫ℤpql(x1+⋯+xr)q∑j=1r(h-j)xjdμ-1(x1)⋯dμ-1(xr).
In the special case x=0,En,q(h,-r)=En,q(h,-r)(0) are called the extended higher-order Nörlund type q-Euler numbers. From (4.4), we note that(4.5)En,q(h,-r)(x)=12r(1-q)n∑l=0n(nl)(-1)lqlx(-qh-r+l;q)r=12r∑m=0rq(m2)q(h-r)m(rm)q[m+x]qn.
A simple manipulation shows that
(4.6)q(m2)(rm)q=q(m2)[r]q⋯[r-m+1]q[m]q!=1[m]q!∏k=0m-1([r]q-[k]q),∏k=0n-1(z-[k]q)=zn∏k=0n-1(1-[k]qz)=∑k=0nS1(n-1,k;q)(-1)kzn-k.
Formula (4.5) implies the following lemma.Lemma 4.1.
Forh∈ℤ,n∈ℤ+, and r∈ℕ,(4.7)En,q(h,-r)(x)=12r[m]q!∑m=0r∑k=0mq(h-r)mS1(m-1,k;q)(-1)k[r]qm-k[x+m]qn.From (2.22), we can easily see that(4.8)[x+m]qn=1(1-q)n∑j=0n∑l=0j(nj)(jl)(-1)j+l(1-q)lqmj[x]ql.
Using (2.13) and (4.8), we obtain the following lemma.Lemma 4.2.
Form,n∈ℤ+,(4.9)En,q(m)=1(1-q)n∑j=0n∑l=0j(nj)(jl)(-1)j+l(1-q)lqmjEl,q.By Lemma4.2, and the definition of fermionic p-adic integrals on ℤp, we obtain the following theorem.Theorem 4.3.
Forh∈ℤ,n∈ℤ+, and r∈ℕ,(4.10)∫ℤpEn,q(h,-r)(x)dμ-1(x)=2-r[m]q!∑m=0r‍∑k=0mq(h-r)mS1(m-1,k;q)(-1)k[r]qm-kEn,q(m)=12r[m]q!∑m=0r∑k=0mq(h-r)mS1(m-1,k;q)(-1)k[r]qm-k×1(1-q)n∑j=0n∑l=0j(nj)(jl)(-1)j+l(1-q)lqmjEl,q.Puth=0 in (4.4). We consider the following polynomials En,q(0,-r)(x):(4.11)En,q(0,-r)(x)=∑l=0n(1-q)-n(nl)(-1)lqlx∫ℤp⋯∫ℤpql(x1+⋯+xr)q-∑j=1rjxjdμ-1(x1)⋯dμ-1(xr).
Then, (4.12)En,q(0,-r)(x)=12r∑m=0r(rm)q(m2)-rm[m+x]qn.
A simple calculation of the fermionic p-adic invariant integral on ℤp shows that (4.13)∫ℤpEn,q(0,-r)(x)dμ-1(x)=12r∑m=0r(rm)q(m2)-rmEn,q(m).
Using Theorem 4.3, we can also prove that (4.14)∫ℤpEn,q(0,-r)(x)dμ-1(x)=2-r[m]q!∑m=0r∑k=0mq-rmS1(m-1,k;q)(-1)k[r]qm-kEn,q(m).
Therefore, we obtain the following theorem.Theorem 4.4.
Form∈ℤ+,r∈ℕ with m≤r,(4.15)(rm)q(m2)-rm=1[m]q!∑k=0mq-rmS1(m-1,k;q)(-1)k[r]qm-k.



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