On p-Adic Analogue of q-Bernstein Polynomials and Related Integrals

(1)

Volume 2010, Article ID 179430,9pages doi:10.1155/2010/179430

Research Article

On p-Adic Analogue of q-Bernstein Polynomials and Related Integrals

T. Kim,

¹

J. Choi,

¹

Y. H. Kim,

¹

and L. C. Jang

²

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

2Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, Republic of Korea

Correspondence should be addressed to T. Kim,[email protected] Received 17 September 2010; Accepted 22 December 2010 Academic Editor: Binggen Zhang

Copyrightq2010 T. 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.

Recently, Kim’s workin pressintroducedq-Bernstein polynomials which are diﬀerent Phillips’

q-Bernstein polynomials introduced in the work by Phillips, 1996; 1997. The purpose of this paper is to study some properties of several type Kim’sq-Bernstein polynomials to express the p-adicq-integral of these polynomials onZpassociated with Carlitz’sq-Bernoulli numbers and polynomials. Finally, we also derive some relations on thep-adic q-integral of the products of several type Kim’sq-Bernstein polynomials and the powers of them onZp.

1. Introduction

LetC0,1denote the set of continuous functions on0,1. For 0< q <1 andf∈C0,1, Kim introduced theq-extension of Bernstein linear operator of ordernforfas follows:

Bn,q

f|x ⁿ

k0

f k

n n

k

x^k_q1−x^n−k_1/q ⁿ

k0

f k

n

Bk,n

x, q

, 1.1

wherex_q 1−q^x/1−q see1. HereBn,qf |xis called Kim’sq-Bernstein operator of ordernforf. Fork, n∈ZN∪ {0},Bk,nx, q ⁿkx^k_q1−x^n−k_1/q are called the Kim’s q-Bernstein polynomials of degreensee2–6.

(2)

In7, Carlitz defined a set of numbersξkξkqinductively by

ξ01,

qξ1k−ξk

⎧⎨

⎩

1 ifk1,

0 ifk >1, 1.2

with the usual convention of replacingξ^kbyξk. These numbers areq-analogues of ordinary Bernoulli numbersBk, but they do not remain finite forq1. So he modified the definition as follows:

β0,q1, q

qβ1k−βk,q

⎧⎨

⎩

1 ifk1,

0 ifk >1, 1.3

with the usual convention of replacingβ^kbyβk,qsee7. These numbersβn,qare called the nth Carlitzq-Bernoulli numbers. And Carlitz’sq-Bernoulli polynomials are defined by

βk,qx

q^xβ x_qk

^k

i0

k i

βi,qq^ixx^k−i_q . 1.4

Asq → 1, we haveβk,q → Bk andβk,qx → Bkx, where Bk andBkxare the ordinary Bernoulli numbers and polynomials, respectively.

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

be the normalized exponential valuation ofCpsuch that|p|pp^−ν^p^p1/p.

Letqbe regarded as either a complex numberq ∈ Cor ap-adic numberq ∈ Cp. If q∈C, we assume|q|<1, and ifq∈Cp, we normally assume|1−q|p<1.

We say thatf is a uniformly diﬀerentiable function at a pointa∈Zpand denote this property byf∈UDZpif the diﬀerence quotientFfx, y fx−fy/x−yhas a limit faasx, y → a, a see1,3,8–13.

Forf∈UDZp, let us begin with the expression 1

p^N

q

0≤x<p^N

q^xfx

0≤x<p^N

fxμq

xp^NZp

, 1.5

representing aq-analogue of the Riemann sums forf see11. The integral off onZpis defined as the limit asn → ∞of the sums if exists. Thep-adicq-integral on a function f∈UDZpis defined by

Iq

f

Zp

fxdμqx lim

N→ ∞

1 p^N

q p^N−1

x0

fxq^x, 1.6

see11.

(3)

As was shown in 3, Carlitz’s q-Bernoulli numbers can be represented by p-adic q-integral onZpas follows:

Zp

x^m_qdμqx βm,q, form∈Z. 1.7

Also, Carlitz’sq-Bernoulli polynomialsβk,qxcan be represented

βm,qx

Zp

xy_m

qdμq

y

, form∈Z, 1.8

see3.

In this paper, we consider thep-adic analogue of Kim’sq-Bernstein polynomials onZp

and give some properties of the several type Kim’sq-Bernstein polynomials to represent the p-adicq-integral onZpof these polynomials. Finally, we derive some relations on thep-adic q-integral of the products of several type Kim’sq-Bernstein polynomials and the powers of them onZp.

2. q -Bernstein Polynomials Associated with p -Adic q -Integral on Z

p

In this section, we assume thatq∈Cpwith|1−q|p<1.

From1.5,1.7and1.8, we note that

Zp

1−xx1ⁿ_1/qdμ1/qx1 qⁿ q−1_n−1

n l0

n l

−1^lq^lx l1 q^l1−1,

Zp

xx1ⁿ_qdμqx1

1

1−q_n−1 n

l0

n l

−1^lq^lx l1 1−q^l1.

2.1

By2.1, we get

−1ⁿqⁿ

Zp

xx1ⁿ_qdμqx1

Zp

1−xx1ⁿ_1/qdμ1/qx1. 2.2

Therefore, we obtain the following theorem.

Theorem 2.1. Forn∈Z, one has

Zp

1−xx1ⁿ_1/qdμ1/qx1 −1ⁿqⁿ

Zp

xx1ⁿ_qdμqx1. 2.3

(4)

By the definition of Carlitz’sq-Bernoulli numbers and polynomials, we get q²βn,q2−n1q²qq

qβ1_n

βn,q ifn >1. 2.4

Thus, we have the following proposition.

Proposition 2.2. Forn∈Nwithn >1, one has

βn,q2 1

q²βn,qn1−1

q. 2.5

It is easy to show that

1−xⁿ_1/q

1−x_qn

−1ⁿqⁿx−1ⁿ_q. 2.6

Hence, we have

Zp

1−xⁿ_1/qdμqx −1ⁿqⁿ

Zp

x−1ⁿ_qdμqx. 2.7

By1.8, we get

Zp

1−xⁿ_1/qdμqx −1ⁿqⁿβn,q−1. 2.8

ByTheorem 2.1and2.8, we see that

Zp

1−xⁿ_1/qdμqx −1ⁿqⁿβn,q−1 βn,1/q2. 2.9

From2.9andProposition 2.2, we have

Zp

1−xⁿ_1/qdμqx βn,1/q2 q²βn,1/qn1−q. 2.10

By1.7and2.10, we obtain the following theorem.

Theorem 2.3. Forn∈Nwithn >1, one has

Zp

1−xⁿ_1/qdμqx q²

Zp

xⁿ_1/qdμ1/qx n1−q. 2.11

(5)

Taking thep-adicq-integral onZpfor one Kim’sq-Bernstein polynomials, we get

Zp

Bk,n

x, q

dμqx n

k Zp

x^k_q1−x^n−k_1/qdμqx

n k

_n−k

l0

n−k l

−1^l

Zp

x^kl_q dμqx

n k

_n−k

l0

n−k l

−1^lβ_kl,q,

2.12

and, by theq-symmetric property ofBk,nx, q, we see that

Zp

Bk,n

x, q

dμqx

Zp

B_n−k,n

1−x,1 q

dμqx

n k

_k

l0

k l

−1^kl

Zp

1−x^n−l_1/qdμqx.

2.13

Forn > k1, byTheorem 2.3and2.13, one has

Zp

Bk,n

x, q

dμqx n

k _k

l0

−1^kl k

l

n−l1−qq²

Zp

x^n−l_1/qdμ1/qx

n

k _k

l0

−1^kl k

l

n−l1−qq²βn−l,1/q

.

2.14

Letm, n, k ∈Zwithmn >2k1. Then thep-adicq-integral for the multiplication of two Kim’sq-Bernstein polynomials onZpcan be given by the following relation:

Zp

Bk,n

x, q Bk,m

x, q

dμqx n

k m

k Zp

x^2k_q 1−x^nm−2k_1/q dμqx

n k

m k

_2k

l0

2k l

−1^l2kq

Zp

1−x^nm−l_1/q dμqx.

2.15

(6)

ByTheorem 2.3and2.15, we get

Zp

Bk,n

x, q Bk,m

x, q dμqx

n k

m k

_2k

l0

2k l

−1^l2k

nm−l1−qq²

Zp

x^nm−l_1/q dμ1/qx

n

k m

k _2k

l0

2k l

−1^l2k

nm−l1−qq²β_nm−l,1/q .

2.16

By the simple calculation, we easily get

Zp

Bk,n

x, q Bk,m

x, q

dμqx n

k m

k Zp

x^2k_q 1−x^nm−2k_1/q dμqx

n k

m k

_nm−2k

l0

nm−2k l

−1^l

Zp

x^l2k_q dμqx

n k

m k

_nm−2k

l0

nm−2k l

−1^lβl2k,q.

2.17

Continuing this process, we obtain

Zp

_s

i1

Bk,ni

x, q

dμqx _s

i1

ni

k Zp

x^sk_q 1−xⁿ_1/q¹^···n^s^−skdμqx

_s

i1

ni

k _sk

l0

sk l

−1^skl

Zp

1−xⁿ_1/q¹^···n^s^−ldμqx.

2.18 Lets∈ Nandn1, . . . , ns,k ∈Z withn1n2· · ·ns > sk1. ByTheorem 2.3and 2.18, we get

Zp

_s

i1

Bk,ni

x, q dμqx

_s

i1

ni

k _sk

l0

sk l

−1^skl _s

i1

ni−l1−qq²

Zp

xⁿ_1/q¹^···n^s^−ldμ1/qx

_s

i1

ni

k _sk

l0

sk l

−1^skl _s

i1

ni−l1−qq²βn1···ns−l,1/q

.

2.19

(7)

From the definition of binomial coeﬃcient, we note that

Zp

_s

i1

Bk,ni

x, q dμqx

_s

i1

ni

k Zp

x^sk_q 1−xⁿ_1/q¹^···n^s^−skdμqx

_s

i1

ni

k _n

1···ns−sk l0

n1· · ·ns−sk l

−1^l

Zp

x^skl_q dμqx

_s

i1

ni

k _n

1···ns−sk l0

n1· · ·ns−sk l

−1^lβskl,q,

2.20

wheres∈Nandn1, . . . , ns,k∈Z.

Theorem 2.4. IFors∈Nandn1, . . . , ns,k∈Nwithn1n2· · ·ns> sk1, one has

Zp

_s

i1

Bk,ni

x, q dμqx

_s

i1

ni

k _sk

l0

sk l

−1^skl _s

i1

.

2.21

IIFors∈Nandn1, . . . , ns,k∈Z, one has

Zp

_s

i1

Bk,ni

x, q

dμqx _s

i1

ni

k _n

1···ns−sk l0

n1· · ·ns−sk l

−1^lβskl,q. 2.22

ByTheorem 2.4, we obtain the following corollary.

Corollary 2.5. Fors∈Nandn1, . . . , ns,k∈Nwithn1n2· · ·ns> sk1, one has

sk l0

sk l

−1^skl _s

i1

ⁿ¹^···n^s^−sk

l0

n1· · ·ns−sk l

−1^lβskl,q.

2.23

(8)

Lets∈Nandm1, . . . , ms,n1, . . . , ns,k∈Zwithm1n1· · ·msns>m1· · ·msk1.

Then one has

Zp

_s

i1

B^m_k,nⁱ

i

x, q

dμqx _s

i1

ni

k

mi_ks i1mi

l0

⎛

⎜⎝k s

i1

mi

l

⎞

⎟⎠−1^k^sⁱ¹^mⁱ^−l

×

Zp

1−xq^sⁱ¹ⁿⁱ^mⁱ^−ldμqx

_s

i1

ni

k

mi_k^s

i1mi

l0

⎛

⎜⎝k s

i1

mi

l

⎞

⎟⎠−1^k^sⁱ¹^mⁱ^−l

× _s

i1

mini−l1

−qq²

Zp

x_1/q^sⁱ¹ⁿⁱ^mⁱ^−ldμ1/qx

_s

i1

ni

k

mi_k^s

i1mi

l0

⎛

⎜⎝k s

i1

mi

l

⎞

⎟⎠−1^k^sⁱ¹^mⁱ^−l

× _s

i1

mini−l1

−qq²βn1m1···nsms−l,1/q

.

2.24

From the definition of binomial coeﬃcient, one has

Zp

_s

i1

B_k,n^mⁱ

i

x, q dμqx

_s

i1

ni

k

mi^s

i1nimi−ks i1mi

l0

⎛

⎜⎝ s

i1

nimi−k s

i1

mi

l

⎞

⎟⎠−1^l

×

Zp

x^m_q ¹^···m^s^kldμqx

_s

i1

ni

k

mis

i1nimi−ks i1mi

l0

⎛

⎜⎝ s

i1

nimi−k s

i1

mi

l

⎞

⎟⎠

×−1^lβ_m₁_···m_s_kl,q.

2.25

(9)

Theorem 2.6. Fors∈Nandm1, . . . , ms,n1, . . . , ns,k∈Zwithm1n1· · ·msns>m1· · · msk1, one has

ks i1mi

l0

⎛

⎜⎝k s

i1

mi

l

⎞

⎟⎠−1^k^sⁱ¹^mⁱ^−l _s

i1

mini−l1

−qq²βn1m1···nsms−l,1/q

s

i1nimi−ks i1mi

l0

⎛

⎜⎝ s

i1

nimi−k s

i1

mi

l

⎞

⎟⎠−1^lβm1···mskl,q.

2.26

Acknowledgment

This paper was supported by the research grant of Kwangwoon University in 2010.

References

1 T. Kim, “A note onq-Bernstein polynomials,” Russian Journal of Mathematical Physics. In press.

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, vol. 1, no. 1, pp. 10–

14, 2010.

3 T. Kim, “On aq-analogue of thep-adic log gamma functions and related integrals,” Journal of Number Theory, vol. 76, no. 2, pp. 320–329, 1999.

4 G. M. Phillips, “On generalized Bernstein polynomials,” in Numerical Analysis, pp. 263–269, World Scientific, River Edge, NJ, USA, 1996.

5 G. M. Phillips, “Bernstein polynomials based on theq-integers,” Annals of Numerical Analysis, vol. 4, pp. 511–514, 1997.

6 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.

7 L. Carlitz, “q-Bernoulli numbers and polynomials,” Duke Mathematical Journal, vol. 15, pp. 987–1000, 1948.

8 M. Cenkci, V. Kurt, S. H. Rim, and Y. Simsek, “On i, q Bernoulli and Euler numbers,” Applied Mathematics Letters, vol. 21, no. 7, pp. 706–711, 2008.

9 L.-C. Jang, “A newq-analogue of Bernoulli polynomials associated withp-adicq-integrals,” Abstract and Applied Analysis, vol. 2008, Article ID 295307, 6 pages, 2008.

10 T. Kim, “Barnes-type multipleq-zeta functions andq-Euler polynomials,” Journal of Physics A, vol. 43, no. 25, Article ID 255201, 11 pages, 2010.

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

12 T. Kim, L.-C. Jang, and H. Yi, “A note on the modifiedq-Bernstein polynomials,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 706483, 12 pages, 2010.

13 B. A. Kupershmidt, “Reflection symmetries of q-Bernoulli polynomials,” Journal of Nonlinear Mathematical Physics, vol. 12, supplement 1, pp. 412–422, 2005.