A Study on the Adic Integral Representation on Associated with the Weighted Bernstein and Bernoulli Polynomials

(1)

Volume 2011, Article ID 513821,8pages doi:10.1155/2011/513821

Research Article

A Study on the

p

-Adic

q

-Integral Representation on

p

Associated with the Weighted

q

-Bernstein and

q

-Bernoulli Polynomials

T. Kim,

1

_{A. Bayad,}

2

_{and Y.-H. Kim}

1

1_{Division of General Education-Mathematics, Kwangwoon University, Seoul 139-701, Republic of Korea} 2_{Département de Mathématiques, Université d’Evry Val d’Essonne, Boulevard François Mitterrand,}

91025 Evry Cedex, France

Correspondence should be addressed to A. Bayad,[email protected]

Received 6 December 2010; Accepted 15 January 2011 Academic Editor: Vijay Gupta

Copyrightq2011 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.

We investigate some interesting properties of the weightedq-Bernstein polynomials related to the weightedq-Bernoulli numbers and polynomials by usingp-adicq-integral on p.

1. Introduction and Preliminaries

Letpbe a fixed prime number. Throughout this paper, p,p, andp will denote the ring ofp-adic integers, the field ofp-adic rational numbers, and the completion of the algebraic closure ofp, respectively. Let be the set of natural numbers, and let ∪ {0}. Letνp be the normalized exponential valuation ofp with|p|

p p−νpp 1/p. Letqbe regarded

as either a complex numberq ∈ or ap-adic number q ∈ p. Ifq ∈ , then we always assume|q|<1. Ifq∈p, we assume that|1−q|

p <1. In this paper, we define theq-number

asxq 1−qx/1−q see1–13.

LetC0,1be the set of continuous functions on0,1. Forα ∈ andn, k ∈ , the weightedq-Bernstein operator of ordernforf∈C0,1is defined by

α n,q

f |x

n

k0

f

k n

_n

k

xkqα1−xn_q−α−k

n

k0

f

k n

Bα_k,nx, q. 1.1

(2)

Let UD pbe the space of uniformly diﬀerentiable functions on p. Forf ∈UD p,

thep-adicq-integral on p, which is called the bosonicq-integral on p, is defined by

Iq

f

p

fxdμqx lim N→ ∞

1

pN q

pN₋₁

x0

fxqx, 1.2

see10.

The Carlitz’sq-Bernoulli numbers are defined by

β0,q1, q

qβ1k−βk,q

1, ifk1,

0, ifk >1, 1.3

with the usual convention about replacingβk_by_β

k,qsee3,9,10. In3, Carlitz also defined

the expansion of Carlitz’sq-Bernoulli numbers as follows:

βh 0,q

h hq

, qh_qβh₁n₋_βh n,q

1, ifn1,

0, ifn >1, 1.4

with the usual convention about replacingβhn_by_βh n,q.

The weightedq-Bernoulli numbers are constructed in previous paper6as follows: forα∈,

β₀α_,q 1, qqαβα1n−βαn,q ⎧ ⎨ ⎩

α αq

, ifn1,

0, ifn >1,

1.5

with the usual convention about replacing βαn by βn,qα. Let fnx fx n. By the

definition1.2ofp-adicq-integral on p, we easily get

qIq

f1

q lim

N→ ∞

1

pN q

pN₋₁

x0

fx1qx,

lim

N→ ∞

1

pN q

pN₋₁

x0

fxqx _lim N→ ∞

fpN_qpN

−f0

pN q

p

fxdμqx

q−1f0 q−1 logqf

₀_,

1.6

Continuing this process, we obtain easily the relation

qn

p

fnxdμqx−

p

fxdμqx

q−1

n−1

l0

qlfl q−1 logq

n−1

l0

qlfl, 1.7

wheren∈andf

(3)

Then by1.2, applying to the functionx → xnqα, we can see that

βn,qα

p

xnqαdμqx −nα

αq

∞

m0

qmαmmnqα−1

1−q

∞

m0

qmmnqα. 1.8

The weightedq-Bernoulli polynomials are also defined by the generating function as follows:

Fqαt, x −tα

αq

∞

m0

qmαmemxqαt₁−_q

∞

m0

qmemxqαt

∞

n0

βαn,qxt n

n!, 1.9

see6. Thus, we note that

βαn,qx n

l0

n l

xnqα−lqαlxβα_l,q

−nα

α_q

∞

m0

qmαmmxnqα−1

1−q

∞

m0

qmmxnqα.

1.10

From1.2and the previous equalities, we obtain the Witt’s formula for the weighted q-Bernoulli polynomials as follows:

βαn,qx

p

xyn_qαdμq

y

n

l0

n l

qαlxxnq−αl p

yl_qαdμq

y. 1.11

By using1.2and the weightedq-Bernoulli polynomials, we easily get

qnβαm,qn−βαm,q

q−1

n−1

l0

qllmqα

mα αq

n−1

l0

qαlllmqα−1, 1.12

wheren, α∈andm∈ see6.

In this paper, we consider the weightedq-Bernstein polynomials to express the bosonic q-integral on pand investigate some properties of the weighted q-Bernstein polynomials

associated with the weighted q-Bernoulli polynomials by using the expression of p-adicq -integral on pof those polynomials.

2. Weighted

q

-Bernstein Polynomials and

q

-Bernoulli Polynomials

In this section, we assume thatα∈ andq∈p with|1−q|

p<1.

Now we consider thep-adic weightedq-Bernstein operator as follows:

α n,q

f|xfx

n

k0

f

k n

_n

k

xkqα1−xn_q−−αk

n

k0

f

k n

(4)

Thep-adicq-Bernstein polynomials with weightαof degreenare given by

Bα_k,nx, q

n k

xkqα1−xn_q−α−k, 2.2

wherex∈ p,α∈, andn, k ∈ see6,7. Note thatB

α

k,nx, q B α

n−k,n1−x,1/q. That

is, the weightedq-Bernstein polynomials are symmetric.

From the definition of the weightedq-Bernoulli polynomials, we have

βα_n,q−11−x −1

n_qαn_βα

n,qx. 2.3

By the definition ofp-adicq-integral on p, we get

p

1−xnq−αdμqx qαn−1n

p

−1xnqαdμqx

p

1−x_qα n

dμqx.

2.4

From2.3and2.4, we have

p

1−xn_q−αdμqx n

l0

n

l

−1l_βα

l,q qαn−1 n_βα

n,q−1 βαn,q2. 2.5

Therefore, we obtain the following lemma.

Lemma 2.1. Forn∈ , one has

p

1−xn_q−αdμqx n

l0

n

l

−1l_βα l,q q

αn₋₁n_βα

n,q−1 βn,qα2,

βα_n,q−11−x −1

n_qαn_βα n,qx.

2.6

By2.2,2.3, and2.4, we get

q2βn,qα2 nα

αq

q1αq2−qβαn,q, if n >1. 2.7

Thus, we have

βαn,q2

1 q2β

α n,qnα

αq

qα−11− 1

q, if n >1. 2.8

(5)

Proposition 2.2. Forn∈withn >1, one has

βαn,q2

1 q2β

α n,q nα

αq

qα−11−1

q. 2.9

By usingProposition 2.2andLemma 2.1, we obtain the following corollary.

Corollary 2.3. Forn∈withn >1, one has

p

1−xnq−αdμqx q2β_n,qα−1 nα αq

1−q, 2.10

p

1−xn_q−αdμqx nα

αq

1−qq2 p

xn_q−αdμq−1x

p

1−x_qα

n

dμqx.

2.11

Taking the bosonicq-integral on pfor one weightedq-Bernstein polynomials in2.1,

we have

p

Bα_k,nx, qdμqx

n k

p

xkqα1−x_qn−α−kdμqx

n k

_n₋_k

l0

n−k l

−1l

p

xklqα dμqx

n k

n−k

l0

n−k l

−1l_βα kl,q.

2.12

By the symmetry ofq-Bernstein polynomials, we get

p

B_k,nαx, qdμqx

p

B_nα₋_k,n

1−x,1 q

dμqx

n k

_k

l0

k l

−1kl p

1−xnq−α−ldμqx.

(6)

Forn > k1, by2.11and2.13, we have

p

B_k,nαx, qdμqx

n k

_k

l0

k l

−1kl

nα αq

1−qq2

p

xnq−α−ldμq−1x

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

nα αq

1−qq2βα_n,q−1, ifk0,

⎛

⎝n

k

⎞ ⎠_q2k

l0 ⎛ ⎝k

l

⎞

⎠₋₁kl_βα

n−l,q−1_, ifk >0.

2.14

By comparing the coeﬃcients on the both sides of 2.12 and 2.14, we obtain the

following theorem.

Theorem 2.4. Forn, k∈ withn > k1, one has

n−k

l0

n−k l

−1l_βα kl,qq2

k

l0

k l

−1klβα_n₋_l,q−1, ifk /0. 2.15

In particular, whenk0, one has

nα αq

1−qq2_βα n,q−1

n

l0

n

l

−1l_βα

l,q. 2.16

Letm, n, k∈ withmn >2k1. Then we see that

p

Bα_k,nx, qB_k,mαx, qdμqx

n k

m k

p

x2qkα1−x_qnm−α −2kdμqx

n k

m k

₂_k

l0

2k

l

−1l2k

p

1−xnmq−α −ldμqx.

n

k

m

k

₂_k

l0

2k

l

−1l2k

nα αq

1−qq2

p

xnmq−α −ldμq−1x

n k

m k

₂_k

l0

2k

l

−1l2k

nα αq

1−qq2βα_nm₋_l,q−1

.

2.17

(7)

Theorem 2.5. Form, n, k∈ withmn >2k1, one has

p

Bα_k,nx, qB_k,mαx, qdμqx ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

nα αq

1−qq2_βα

nm,q−1, ifk0,

⎛

⎝n

k

⎞ ⎠

⎛

⎝m

k

⎞ ⎠_q22k

l0 ⎛ ⎝2k

l

⎞

⎠₋₁l2k_βα

nm−l,q−1, ifk /0.

2.18

Form, n, k∈ , we have

p

Bα_k,nx, qB_k,mαx, qdμqx

n

k

m

k

p

x2qkα1−x_qnm−α −2kdμqx

n k

m k

_nm₋₂_k

l0

nm−2k l

−1l

p

x2qklα dμqx

n k

m k

_nm₋₂_k

l0

nm−2k l

−1l_βα l2k,q.

2.19

Therefore, by2.18and2.19, we obtain the following theorem.

Theorem 2.6. Form, n, k∈ withmn >2k1, one has

nα

α_q 1−qq

2_βα nm−l,q−1

nm

l0

nm l

−1l_βα

l,q . 2.20

Furthermore, for_{k /}0, one has

nm−2k

l0

nm−2k l

−1l_βα l2k,qq

22k

l0

2k

l

−1l2k_βα

nm−l,q−1. 2.21

By the induction hypothesis, we obtain the following theorem.

Theorem 2.7. Fors∈ andk, n1, . . . , ns∈ withn1n2· · ·ns> sk1, one has

p

_s

i1

B_k,nα_ix, q

dμqx ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

nα αq

1−qq2β_nα

1···ns,q−1, if k0, ⎛

⎝s

i1 ⎛ ⎝ni

k

⎞ ⎠

⎞ ⎠sk

l0 ⎛ ⎝sk

l

⎞

⎠₋₁lsk_βα

(8)

Fors∈, letk, n1, . . . , ns∈ withn1n2· · ·ns> sk1. Then we show that

p

_s

i1

Bα_k,n_ix, q

dμqx

_s

i1

ni

k

_n

1···ns−sk

l0

n1· · ·ns−sk

l

−1l

βα_lsk,q.

2.23

Therefore, byTheorem 2.7and2.23, we obtain the following theorem.

Theorem 2.8. Fors∈, letk, n1, . . . , ns∈ withn1n2· · ·ns> sk1. Then one sees that fork0

n1···ns

l0

n1· · ·ns

l

−1l_βα l,q

nα αq

1−qq2βα_n

1···ns,q−1. 2.24

For_{k /}0, one has

sk

l0

sk l

−1lskβα_n

1···ns−l,q−1

n1···ns−sk

l0

n1· · ·ns−sk

l

−1l

βα_lsk,q. 2.25

References

1 M. Acikgoz and Y. Simsek, “On multiple interpolation functions of the N ¨orlund-type q-Euler polynomials,”Abstract and Applied Analysis, vol. 2009, Article ID 382574, 14 pages, 2009.

2 A. Bayad, J. Choi, T. Kim, Y.-H. Kim, and L. C. Jang, “q-extension of Bernstein polynomials with weightedα;β,”Journal of Computational and Applied Mathematics. In press.

3 L. Carlitz, “Expansions ofq-Bernoulli numbers,”Duke Mathematical Journal, vol. 25, pp. 355–364, 1958.

4 A. S. Hegazi and M. Mansour, “A note onq-Bernoulli numbers and polynomials,”Journal of Nonlinear Mathematical Physics, vol. 13, no. 1, pp. 9–18, 2006.

5 L.-C. Jang, W.-J. Kim, and Y. Simsek, “A study on thep-adic integral representation on passociated

with Bernstein and Bernoulli polynomials,” Advances in Diﬀerence Equations, vol. 2010, Article ID 163217, 6 pages, 2010.

6 T. Kim, “On the weightedq-Bernoulli numbers and polynomials,”http://arxiv.org/abs/1011.5305.

7 T. Kim, “A note onq-Bernstein polynomials,”Russian Journal of Mathematical Physics, vol. 18, no. 1, 2011.

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

9 T. Kim, “q-Bernoulli numbers and polynomials associated with Gaussian binomial coeﬃcients,”

Russian Journal of Mathematical Physics, vol. 15, no. 1, pp. 51–57, 2008.

10 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.

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

12 H. Ozden, I. N. Cangul, and Y. Simsek, “Remarks onq-Bernoulli numbers associated with Daehee numbers,”Advanced Studies in Contemporary Mathematics, vol. 18, no. 1, pp. 41–48, 2009.