A Note on the Modified q-Bernstein Polynomials

Volume 2010, Article ID 706483,12pages doi:10.1155/2010/706483

Review Article

A Note on the Modified q-Bernstein Polynomials

Taekyun Kim,

Lee-Chae Jang,

and Heungsu Yi

1Division of General Education, Kwangwoon University, Seoul 139-701, Republic of Korea

2Department of Mathematics and Computer Science, Konkuk University, Chungju 138-701, Republic of Korea

3Department of Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea

Correspondence should be addressed to Taekyun Kim,[email protected] Received 20 May 2010; Revised 11 July 2010; Accepted 14 July 2010 Academic Editor: Leonid Shaikhet

Copyrightq2010 Taekyun 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 propose the modifiedq-Bernstein polynomials of degree nwhich are different q-Bernstein polynomials of Phillips1997. From these modifiedq-Bernstein polynomials of degree n, we derive some recurrence formulae for the modifiedq-Bernstein polynomials.

1. Introduction

Let C0,1 denote the set of continuous function on 0,1. For f ∈ C0,1, Bernstein introduced the following well-known linear positive operators in1:

Bn

f:x :ⁿ

k0

f k

n n

k

x^k1−x^n−kn

k0

f k

n

Bk,nx, 1.1

whereⁿ_k nn−1· · ·n−k1/k!. HereBnf:xis called the Bernstein operator of ordern forf. Fork, n∈Z, the Bernstein polynomial of degreenis defined by

Bk,nx n

k

x^k1−x^n−k, 1.2

wherex∈0,1. For example,

B0,1x 1−x, B1,1x x,

B0,2x 1−x², B1,2x 2x1−x, B2,2x x², . . . . 1.3

Also,Bk,nx 0,fork > n, becauseⁿ_k 0.

Some people have studied the Bernstein polynomials in the area of approximation theorysee2through3. Note that fork∈Zandx∈0,1,

t^ke^1−xtx^k k! x^k

k!

t^k

∞ n0

1−xⁿtⁿ n!

x^k k!

∞ n0

1−xⁿn1· · ·nk nk! t^nk

^∞

nk

n k

x^k1−x^n−k tⁿ

n!

^∞

nk

Bk,nxtⁿ n!.

1.4

BecauseBk,0x Bk,1x · · ·Bk,k−1x 0, we obtain the generating function forBk,nx as follows:

F^kt, x: t^ke^1−xtx^k

k! ^∞

n0

Bk,nxtⁿ

n! 1.5

see4,5, wherek∈Zandx∈0,1. Notice that

Bk,nx

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎛

⎝n k

⎞

⎠x^k1−x^n−k ifn≥k,

0 ifn < k,

1.6

forn, k∈Zsee2.

Let 0< q <1. Define theq-number of xby

x_q: 1−q^x

1−q. 1.7

See2through3for details and related facts. Note that lim_q→1x_q x. In6, Phillips proposed a generalization of the classical Bernstein polynomials based onq-integers. In the last decade some new generalizations of well-known positive linear operators, based on q-integers were introduced and studied by several authorssee 1–13. Recently, Simsek

and Acikgoz have also studied theq-extension of Bernstein-type polynomials5. Theirq- Bernstein-type polynomials are given by

Yn

k;x:q

n k

−1^kk!

1−q_n−k ∞ m,l0

n−k

j0

kl−1 l

n−k k

×

−1^jq^lj1−xSm, k

xlnq_m m!

,

1.8

where Sm, k are the second-kind stirling number. In 5, we can find some interesting formulae related to q-extension of Bernstein polynomials which are different q-Bernstein polynomials of Phillips. In the conference of Jangjeon Mathematical Society which was held in IRANon Feb.2010, Acikgoz and Arci has introduced several-type Bernstein polynomials see2. The Acikgoz paper2announced in the conference is actually what motivated us to write this paper. In this paper, we considered theq-extension of Bernstein polynomials which were introduced by Acikgoz at the conference of Jangjeon Mathematical Society on Feb. 2010.

First, we consider theq-extension of the generating function of Bernstein polynomials in1.5.

Indeed, this generating function is also treated by Simsek and Acikgoz in a previous paper see5. From thisq-extension of the generating function for the Bernstein polynomials, we propose the modifiedq-Bernstein polynomials of degree nwhich are differentq-Bernstein polynomials of Phillips. By using the properties of the modifiedq-Bernstein polynomials, we obtain some recurrence formulae for the modifiedq-Bernstein polynomials of degreen.

2. The Modified q -Bernstein Polynomials

For 0< q <1, consider theq-extension of1.5as follows:

Fq^kt, x: t^ke^1−xqtx^k_q k!

x^k_q k!

∞ n0

1−xⁿ_q n! t^{nk ∞}

nk

n k

x^k_q1−x^n−k_q tⁿ n!,

2.1

wherek, n∈Zandx∈0,1. Note that limq→1Fq^kt, x F^kt, x. We define the modified q-Bernstein polynomials as follows:

Fq^kt, x t^ke^1−xqtx^k_q

k! ^∞

n0

Bk,n

x, qtⁿ

n!, 2.2

wherek, n∈Zandx∈0,1.

Remark. This generating function is also introduced by Simsek and Acikgoz in a previous papersee5.

By comparing the coefficients of2.1and2.2, we obtain the following theorem.

Theorem 2.1. Fork, n∈Zandx∈0,1,

Bk,n

x, q

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

⎛

⎝n k

⎞

⎠x^k_q1−x^n−k_q , ifn≥k

0, ifn < k.

2.3

For 0≤k≤n, we have 1−x_qBk,n−1

x, q

xqBk−1,n−1 x, q 1−x_q

n−1 k

x^k_q1−x^n−1−k_q x_q n−1

k−1

x^k−1_q 1−x^n−k_q

n−1

k

x^k_q1−x^n−k_q n−1

k−1

x^k_q1−x^n−k_q

n

k

x^k_q1−x^n−k_q ,

2.4

and the derivatives of the modifiedq-Bernstein polynomials of degreenare also polynomials of degreen−1, that is,

d dxBk,n

x, q

n k

kx^k−1_q 1−x^n−k_q lnq q−1q^x

n k

x^k_qn−k1−x^n−k−1_q −lnq

q−1

q^1−x

lnq q−1

n k

kx^k−1_q 1−x^n−k_q q^x− n

k

x^k_qn−k1−x^n−k−1_q q^1−x

n

q^xB_k−1,n−1 x, q

−q^1−xB_k,n−1

x, q lnq q−1.

2.5

Therefore, we obtain the following recurrence formulae.

Theorem 2.2recurrence formulae forBk,nx, q. Fork, n∈Zand forx∈0,1,

1−x_qBk,n−1 x, q

xqBk−1,n−1 x, q

Bk,n

x, q , d

dxBk,n

x, q n

q^xBk−1,n−1 x, q

−q^1−xBk,n−1

x, qlnq q−1.

2.6

Letfbe a continuous function on0,1. Then the modifiedq-Bernstein operator of order nforfis defined by

Bn,q

f:x :ⁿ

k0

f k

n

Bk,n

x, q

, 2.7

where 0≤x≤1,n∈Z. We get fromTheorem 2.1and2.7that forfx x,

Bn,q

f:x ⁿ

k0

f k

n n

k

x^k_q1−x^n−k_q

x_q

1−1−x_qx_q

q−1ⁿ⁻¹ f

x_q

1 1−qxq1−x_qn−1 .

2.8

We also see fromTheorem 2.1that

Bn,q1 :x ⁿ

k0

Bk,n

x, q

ⁿ

k0

n k

x^k_q1−x^n−k_q

ⁿ

k0

n k

x^k_q

1−q^1−xx_qn−k

1 1−q

x_q1−x_qn

.

2.9

The modified q-Bernstein polynomials are symmetric polynomials in the following sense:

Bn−k,n

1−x, q

n n−k

1−x^n−k_q x^k_q Bk,n

x, q

. 2.10

Therefore, we get the following theorem.

Theorem 2.3. Fork, n∈Zandx∈0,1,

Bn−k,n

1−x, q Bk,n

x, q , Bn,q1 :x

1 1−q

x_q1−x_qn

. 2.11

Forζ∈C,x∈0,1and forn∈Z, consider

n!

2πi

C

x_qζ_k

k! e^1−xqζ dζ

ζⁿ¹, 2.12

where C is a circle around the origin and integration is in the positive direction. We see from the definition of the modifiedq-Bernstein polynomials and the basic theory of complex analysis including Laurent series that

C

x_qζ_k

k! e^1−xqζ dζ ζ^{n1 ∞}

m0

C

Bk,m

x, q ζ^m m!

dζ ζⁿ¹ 2πi

Bk,n

x, q n!

. 2.13

We get from2.12and2.13that

n!

2πi

C

x_qζk

k! e^1−xqζ dζ ζⁿ¹ Bk,n

x, q

, 2.14

C

x_qζk

k! e^1−xqζ dζ

ζⁿ¹ x^k_q k!

∞ m0

1−x^m_q m!

C

ζ^m−n−1kdζ

2πi

⎛

⎝x^k_q1−x^n−k_q k!n−k!

⎞

⎠.

2.15

We also get from2.12and2.15that

n!

2πi

C

x_qζk

k! e^1−xqζ dζ ζⁿ¹

n k

x^k_q1−x^n−k_q . 2.16

Therefore, we see from2.14and2.16that

Bk,n

x, q

n k

x^k_q1−x^n−k_q . 2.17

Note that

n−k n

Bk,n

x, q

k1 n

Bk1,n

x, q

n−1!

k!n−k−1!x^k_q1−x^n−k_q n−1!

k!n−k−1!x^k1_q 1−x^n−k−1_q

1−x_q xq

B_k,n−1

x, q

1 xq

1−q^1−x Bk,n−1

x, q

1 1−q

x_q1−x_q B_k,n−1

x, q .

2.18

Therefore, we can write the modified q-Bernstein polynomials as a linear combination of polynomials of higher order as follows.

Theorem 2.4. Fork, n∈Zandx∈0,1, n1−k

n1

Bk,n1 x, q

k1

n1

Bk1,n1 x, q

1

1−q

x_q1−x_q Bk,n

x, q . 2.19

We easily see from2.17that forn, k∈N,

n−k1 k

x_q 1−x_q

Bk−1,n

x, q

n−k1 k

x_q 1−x_q

n k−1

x^k−1_q 1−x^n−k1_q

n!

k!n−k!x^k_q1−x^n−k_q Bk,n

x, q .

2.20

Thus, the following corollary holds.

Corollary 2.5. Forn, k∈Nandx∈0,1,

n−k1 k

x_q 1−x_q

Bk−1,n

x, q Bk,n

x, q

. 2.21

Note from the definition of the modifiedq-Bernstein polynomials and the binomial theorem that fork, n∈Z,

Bk,n

x, q

n k

x^k_q1−x^n−k_q

n

k

x^k_q

1−q^1−xx_qn−k

n

k

x^k_q^n−k

l0

n−k l

−1^lq^l1−xx^l_q

^n−k

l0

kl k

n kl

−1^lq^l1−xx^lk_q

ⁿ

jk

n k

n j

−1^j−kq^1−xj−kx^jq.

2.22

Therefore, we showed that the following theorem holds.

Theorem 2.6. For k, n∈Zandx∈0,1,

Bk,n

x, q ⁿ

jk

j k

n j

−1^j−kq^1−xj−kx^jq. 2.23

It is possible to write x^k_q as a linear combination of the modified q-Bernstein polynomials by using the degree evaluation formulae and mathematical induction. We easily see from the property of the modifiedq-Bernstein polynomials that

n k1

k n

Bk,n

x, q ⁿ

k1

n−1 k−1

x^k_q1−x^n−k_q

ⁿ⁻¹

k0

n−1 k

x^k1_q 1−x^n−1−k_q

xq

x_q 1−x_qn−1 ,

2.24

and that

n k2

_k

2

ⁿ₂Bk,n

x, q ⁿ

k2

n−2 k−2

x^k_q1−x^n−k_q

ⁿ⁻²

k0

n−2 k

x^k2_q 1−x^n−2−k_q

x²_q

x_q 1−x_qn−2 .

2.25

Continuing this process, we obtain

n kj

k j

_n

j

Bk,n

x, q x^jq

x_q 1−x_qn−j

, 2.26

forj∈N. Therefore, we obtain the following theorem.

Theorem 2.7. Forn, j∈Zandx∈0,1,

1

1−x_q xq

_n−j n kj

k j

_n

j

Bk,n

x, q

x^jq. 2.27

Fork∈N, the Bernoulli polynomial of orderkis defined by t

e^t−1 _k

e^xt t

e^t−1

× · · · × t

e^t−1

k-times

e^xt∞

n0

B^kn xtⁿ

n!, 2.28

and Bn^k Bn^k0 are called thenth Bernoulli numbers of orderk. It is well known that the second kind stirling number is defined by

e^t−1_k k! :^∞

n0

Sn, ktⁿ

n!, 2.29

fork∈N. We note from2.2that x_qtk

e^1−xqt

k! x^k_q e^t−1_k k!

t e^t−1

_k e^1−xqt

x^kq

_∞

m0

Sm, kt^m m!

_∞

n0

B^kn

1−x_qtⁿ n!

x^kq

∞ l0

⎛

⎜⎝^l

n0

Bn^k

1−x_q

Sl−n, kl!

n!l−n!

⎞

⎟⎠t^l l!.

2.30

We have from2.2and2.30that

Bk,l

x, q x^kq

l n0

l n

B^kn

1−x_q

Sl−n, k, 2.31

andBk,0x, q Bk,1x, q · · ·B_k,k−1x, q 0.

Remark. The Equations 2.30 and 2.31 are already known by Simsek and Acikgoz in a previous paper5, page 7.

LetΔbe the shift difference operator defined byΔfx fx1−fx. We see from the iterative method that

Δⁿf0 ⁿ

k0

n k

−1^n−kfk, 2.32

forn∈N. We get from2.29and2.32that ∞

n0

Sn, ktⁿ n! 1

k!

k l0

k l

−1^k−le^lt

^∞

n0

1 k!

k l0

k l

−1^k−llⁿ tⁿ

n!

^∞

n0

Δ^k0ⁿ k!

tⁿ n!.

2.33

By comparing the coefficients on both sides above, we have

Sn, k Δ^k0ⁿ

k! , 2.34

forn, k∈Z. Thus, we get from2.31and2.34that

Bk,l

x, q

x^k_q^l

n0

l n

Bn^k

1−x_qΔ^k0^l−n

k! . 2.35

LetEhx hx1be the shift operator. Then theq-difference operator is defined by Δⁿ_q Πⁿ⁻¹_j0

E−q^jI

, 2.36

whereIis an identity operatorsee7through11. Forf ∈C0,1andn∈N, we have

Δⁿ_qf0 ⁿ

k0

n k

q

−1^kqⁿ²fn−k, 2.37

whereⁿ_kqis the Gaussian binomial coefficient defined by x

k

q

x_qx−1_q· · ·x−k1_q

k_q! . 2.38

LetFqtbe the generating function of theq-extension of the second kind stirling number as follows:

Fqt: q^−k2 k_q!

k j0

−1^k−j k

j

q

q^k−j2 e^jqt∞

n0

S

n, k:qtⁿ

n!. 2.39

We have from2.39that

S

n, k:q

q^−k2 k_q!

k j0

−1^jq^2j k

j

q

k−j_n

q q^−k2

k_q!Δ^k_q0ⁿ, 2.40

wherek_q! k_qk−1_q· · ·2_q1_q. It is not difficult to see that

xⁿ_q ⁿ

k0

q^k2 x

k

q

k_q!S

n, k:q

. 2.41

See also7through11for details and related facts for above. Then, we get from2.41and Theorem 2.7that

j k0

q^k2 x

k

q

k_q!S j, k:q

1

1−x_q x_qn−j n kj

k j

_n

j

Bk,n

x, q

. 2.42

Therefore, this completes the proof of the following theorem.

Theorem 2.8. Forn, j∈Zandx∈0,1,

1

1−x_q xq

_n−j n kj

k j

_n

j

Bk,n

x, q

j k0

q^k2 x

k

q

k_q!S j, k:q

. 2.43

Acknowledgments

The authors express their sincere gratitude to the referees for their valuable suggestions and comments. This paper is supported in part by the Research Grant of Kwangwoon University in 2010.

References

1 S. Bernstein, “Demonstration du theoreme de Weierstrass, fondee sur le calcul des probabilities,”

Communications of the Kharkov Mathematical Society, vol. 2, no. 13, pp. 1–2, 1912-1913.

2 M. Acikgoz and S. Araci, “A study on the integral of the product of several type Bernstein polynomials,” IST Transaction of Applied Mathematics-Modelling and Simulation. In press.

3 N. K. Govil and V. Gupta, “Convergence ofq-Meyer-K ¨onig-Zeller-Durrmeyer operators,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 181–189, 2009.

4 M. Acikgoz and S. Araci, “On the generating function of the Bernstein polynomials,” in Proceedings of the 8th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM ’10), AIP, Rhodes, Greece, March 2010.

5 Y. Simsek and M. Acikgoz, “A new generating function ofq-Bernstein-type polynomials and their interpolation function,” Abstract and Applied Analysis, vol. 2010, Article ID 769095, 12 pages, 2010.

6 G. M. Phillips, “Bernstein polynomials based on theq-integers,” Annals of Numerical Mathematics, vol.

4, no. 1–4, pp. 511–518, 1997.

7 T. Kim, “q-Volkenborn integration,” Russian Journal of Mathematical Physics, vol. 9, no. 3, pp. 288–299, 2002.

8 T. Kim, “New approach toq-Euler polynomials of higher order,” Russian Journal of Mathematical Physics, vol. 17, no. 2, pp. 218–225, 2010.

9 T. Kim, “Barnes-type multiple q-zeta functions and q-Euler polynomials,” Journal of Physics A:

Mathematical and Theoretical, vol. 43, no. 25, Article ID 255201, 11 pages, 2010.

10 T. Kim, “Note on the Eulerq-zeta functions,” Journal of Number Theory, vol. 129, no. 7, pp. 1798–1804, 2009.

11 T. Kim, S. D. Kim, and D.-W. Park, “On uniform differentiability andq-Mahler expansions,” Advanced Studies in Contemporary Mathematics, vol. 4, no. 1, pp. 35–41, 2001.

12 S. Zorlu, H. Aktuglu, and M. A. ¨Ozarslan, “An estimation to the solution of an initial value problem viaq-Bernstein polynomials,” Journal of Computational Analysis and Applications, vol. 12, no. 3, pp.

637–645, 2010.

13 V. Gupta and C. Cristina, “Statistical approximation properties ofq-Baskakov-Kantorovich opera- tors,” Central European Journal of Mathematics. In press.