Polynomials Associated with Bernstein Polynomials

doi:10.1155/2010/305018

Research Article

Some Identities of Bernoulli Numbers and

Polynomials Associated with Bernstein Polynomials

Min-Soo Kim,

Taekyun Kim,

Byungje Lee,

and Cheon-Seoung Ryoo

1Department of Mathematics, KAIST, 373-1 Guseong-dong, Yuseong-gu, Daejeon 305-701, Republic of Korea

2Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

3Department of Wireless Communications Engineering, Kwangwoon University, Seoul 139-701, Republic of Korea

4Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Correspondence should be addressed to Taekyun Kim,[email protected] Received 30 August 2010; Accepted 27 October 2010

Academic Editor: Istvan Gyori

Copyrightq2010 Min-Soo Kim et al. 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 investigate some interesting properties of the Bernstein polynomials related to the bosonicp- adic integrals onZp.

1. Introduction

Let C0,1 be the set of continuous functions on 0,1. Then the classical Bernstein polynomials of degreenforf∈C0,1are defined by

Bn

f ⁿ

k0

f k

n

Bk,nx, 0≤x≤1, 1.1

whereBnfis called the Bernstein operator and

Bk,nx n

k

x^kx−1^n−k 1.2

are called the Bernstein basis polynomials or the Bernstein polynomials of degree n.

Recently, Acikgoz and Araci have studied the generating function for Bernstein polynomials see1,2. Their generating function forBk,nxis given by

Fkt, x t^ke^1−xtx^k

k! ^∞

n0

Bk,nxtⁿ

n!, 1.3

wherek0,1, . . .andx∈0,1. Note that

Bk,nx

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎛

⎝n k

⎞

⎠x^k1−x^n−k, if n≥k,

0, if n < k

1.4

forn 0,1, . . . see1,2. In3, Simsek and Acikgoz defined generating function of the q-Bernstein-Type Polynomials,Ynk, x, qas follows:

Fk,qt, x t^ke^1−xqtx^k_q

k! ^∞

nk

Yn

k, x, qtⁿ

n!, 1.5

wherex_q 1−q^x/1−q. Observe that

qlim→1Yn

k, x, q

Bk,nx. 1.6

Hence by the above one can very easily see that

Fkt, x t^ke^1−xtx^k

k! ^∞

nk

Bk,nxtⁿ

n!. 1.7

Thus, we have arrived at the generating function in1,2and also in1.3as well.

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. Some researchers have studied the Bernstein polynomials in the area of approximation theory see1–7. In recent years, Acikgoz and Araci1,2have introduced several type Bernstein polynomials.

In the present paper, we introduce the Bernstein polynomials on the ring of p-adic integers Zp. We also investigate some interesting properties of the Bernstein polynomials related to the bosonicp-adic integrals on the ring ofp-adic integersZp.

2. Bernstein Polynomials Related to the Bosonic p -Adic Integrals on Z

Letpbe a fixed prime number. Throughout this paper,Zp,Qp, andCpwill denote the ring of p-adic integers, the field ofp-adic numbers, and the completion of the algebraic closure ofQp,

respectively. Letvpbe the normalized exponential valuation ofCpwith|p|_pp⁻¹. ForN≥1, the bosonic distributionµ1onZp

µ

ap^NZp

1

p^N 2.1

is known as thep-adic Haar distributionµHaar,whereap^NZp{x∈Qp | |x−a|_p ≤p^−N} cf.8. We will writedµ1xto remind ourselves thatxis the variable of integration. Let UDZpbe the space of uniformly differentiable function onZp. Thenµ1yields the fermionic p-adicq-integral of a functionf ∈UDZp

I1

f

Zp

fxdµ1x lim

N→ ∞

1 p^N

p^N−1 x0

fx 2.2

cf.8. Many interesting properties of2.2were studied by many authorscf.8,9and the references given there. Forn∈N, writefnx fxn. We have

I1

fn

I1

f ⁿ⁻¹

l0

fl. 2.3

This identity is to derives interesting relationships involving Bernoulli numbers and polynomials. Indeed, we note that

I1

xy_n

Zp

xy_n dµ1

y

Bnx, 2.4

whereBnxare the Bernoulli polynomialscf.8. From1.2, we have

Zp

Bk,nxdµ1x n

k _n−k

j0

n−k j

−1^n−k−jBn−j,

Zp

Bk,nxdµ1x

Zp

Bn−k,n1−xdµ1x

n k

_k

j0

k j

−1^k−jn−j

l0

n−j l

−1^lBl.

2.5

By2.5, we obtain the following proposition.

Proposition 2.1. Forn≥k,

n−k

j0

n−k j

−1^n−k−jBn−j^k

j0

k j

−1^k−jn−j

l0

n−j l

−1^lBl. 2.6

From2.4, we note that

Bn2−n B1 1ⁿ−n B1ⁿBn, n >1 2.7

with the usual convention of replacingBⁿbyBnandB1ⁿbyBn1. Thus, we have

Zp

xⁿdµ1x

Zp

x2ⁿdµ1x−n

−1ⁿ

Zp

x−1ⁿdµ1x−n

Zp

1−xⁿdµ1x−n

2.8

forn >1, since−1ⁿBnx Bn1−x. Therefore we obtain the following theorem.

Theorem 2.2. Forn >1,

Zp

1−xⁿdµ1x

Zp

xⁿdµ1x n. 2.9

Also we obtain

Zp

B_n−k,kxdµ1x

Zp

x^n−k1−x^kdµ1x

^n−k

l0

n−k l

−1^l

Zp

1−x^lkdµ1x

^n−k

l0

n−k l

−1^l

Zp

x^lkdµ1x lk

^n−k

l0

n−k l

−1^lBlklk.

2.10

Therefore we obtain the following result.

Corollary 2.3. Fork >1,

Zp

B_n−k,kxdµ1x ^n−k

l0

n−k l

−1^lB_lklk. 2.11

From the property of the Bernstein polynomials of degreen, we easily see that

Zp

Bk,nxBk,mxdµ1x n

k m

k _Zpx^2k1−x^nm−2kdµ1x

n k

m k

_nm−2k

l0

nm−2k l

−1^lB_2kl

Zp

Bk,nxBk,mxBk,sxdµ1x n

k m

k s

k _Zpx^3k1−x^nm−3kdµ1x

n k

m k

s k

_nms−3k

l0

nms−3k l

−1^lB3kl. 2.12

Continuing this process, we obtain the following theorem.

Theorem 2.4. The multiplication of the sequence of Bernstein polynomials

Bk,n1x, Bk,n2x, . . . , Bk,nsx 2.13

fors∈Nwith different degree underp-adic integral onZp, can be given as

Zp

Bk,n1xBk,n2x· · ·Bk,nsxdµ1x

n1

k n2

k

· · · ns

k

_{n1n2···n s−sk}

l0

n1n2· · ·ns−sk l

−1^lB_skl.

2.14

We put

B^m_k,nx Bk,nx× · · · × Bk,nx

m-times

. 2.15

Theorem 2.5. The multiplication of

B_k,n^m1

1x, B_k,n^m2

2x, . . . , B^m_k,n^s

sx 2.16

Bernstein polynomials with different degreesn1, n2, . . . , nsunderp-adic integral onZpcan be given as

Zp

B_k,n^m1₁xB^m_k,n²₂x· · ·B^m_k,n^s_sxdµ1x

n1

k m1

n2

k m2

· · · ns

k

msn1m1n2m2···nsms−m1···msk l0

−1^l

×

n1m1n2m2· · ·nsms−m1· · ·msk l

Bm1···mskl.

2.17

Theorem 2.6. The multiplication of B^m_k¹

1,n1x, B^m_k²

2,n2x, . . . , B^m_k^s

s,nsx 2.18

Bernstein polynomials with different degrees n1, n2, . . . , ns with different powers m1, m2, . . . , ms

underp-adic integral onZpcan be given as

Zp

B_k^m₁¹_,n1xB_k^m₂²_,n2x· · ·B_k^m_s^s_,nsxdµ1x

n1

k1

m1 n2

k2

m2

· · · ns

ks

msn1m1n2m2···nsms−k1m1···ksms l0

−1^l

×

n1m1n2m2· · ·nsms−k1m1· · ·ksms l

Bk1m1···ksmsl.

2.19

Problem. Find the Witt’s formula for the Bernstein polynomials inp-adic number field.

Acknowledgments

The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea NRF funded by the Ministry of Education, Science, and Technology 2010-0001654. The second author was supported by the research grant of Kwangwoon University in 2010.

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