On the Fermionic p -adic Integral Representation of Bernstein Polynomials Associated

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Volume 2010, Article ID 864247,12pages doi:10.1155/2010/864247

Research Article

On the Fermionic p -adic Integral Representation of Bernstein Polynomials Associated

with Euler Numbers and Polynomials

T. Kim,

¹

J. Choi,

¹

Y. H. Kim,

¹

and C. S. Ryoo

²

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

2Department of Mathematics, Hannam University, Daejeon 306-791, Republic of Korea

Correspondence should be addressed to T. Kim,[email protected] Received 30 August 2010; Accepted 3 December 2010

Academic Editor: Paolo E. Ricci

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.

The purpose of this paper is to give some properties of several Bernstein type polynomials to represent the fermionicp-adic integral on p. From these properties, we derive some interesting identities on the Euler numbers and polynomials.

1. Introduction

Throughout this paper, letpbe an odd prime number. The symbol, p,p, andpdenote the ring ofp-adic integers, the field ofp-adic rational numbers, the complex number field and the completion of algebraic closure ofp, respectively.

Let be the set of natural numbers and ∪ {0}. Let νp be the normalized exponential valuation of _p with |p|p p^−ν^p^p 1/p. Note that p {x | |x|p ≤ 1}

lim_←

N /p^N p.

When one talks of q-extension, q is variously considered as an indeterminate, a complex numberq ∈ , or p-adic numberq ∈ p. Ifq ∈ , we normally assume|q| < 1, and ifq∈p, we always assume|1−q|p<1.

We say that f is uniformly diﬀerentiable function at a point a ∈ p and write f ∈ UD p, if the diﬀerence quotient Ffx, y fx−fy/x−yhas a limit fa asx, y → a, a. Forf ∈UD p, the fermionicp-adicq-integral on pis defined as

I−q f

p

fxdμ−qx lim

N→ ∞

1q 1q^p^N

p^N−1 x0

fx

−q_x

, 1.1

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see1. In the special caseq1 in1.1, the integral

I−1 f

p

fxdμ−1x, 1.2

is called the fermionicp-adic invariant integral on psee2. From1.2, we note I−1

f1

−I₋₁ f

2f0, 1.3

wheref1x fx1.

Moreover, forn∈, letfnx fxn. Then we note that

I−1 fn

−1ⁿI−1 f

2

n−1

l0

−1^n−1−lfl. 1.4

It is well known that the Euler polynomials are defined by 2

e^t1e^xt^∞

n0Enxtⁿ

n!, 1.5

see1–15. In the special case,x0, andEn0 Enare called thenth Euler numbers.

Letfx e^tx. Then, by1.3,1.4, and1.5, we see that

p

e^xytdμ−1 y

2

e^t1e^xt^∞

n0

Enxtⁿ

n!. 1.6

LetC0,1denote the set of continuous functions on0,1. Forf ∈C0,1, Bernstein introduced the following well-known linear positive operator in the field of real numbers:

n

f:x ⁿ

k0

f k

n n

k x^k1−x^n−kⁿ

k0

f k

n

Bk,nx, 1.7

whereⁿ_k nn−1· · ·n−k1/k!n!/k!n−k!see3,4,7,10,11,14. Here,nf:x is called the Bernstein operator of ordernforf.

Fork, n∈ , the Bernstein polynomial of degreenis defined by

Bk,nx n

k x^k1−x^n−k, forx∈0,1. 1.8

For example,B0,1x 1−x,B1,1x x,B0,2x 1−x²,B1,2x 2x−2x²,B2,2x x², . . ., andBk,nx 0 forn < k,Bk,nx Bn−k,n1−x.

In this paper, we study the properties of Bernstein polynomials in thep-adic number field. For f ∈ UD p, we give some properties of several type Bernstein polynomials

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to represent the fermionicp-adic invariant integral on p. From those properties, we derive some interesting identities on the Euler polynomials.

2. Fermionic p-adic Integral Representation of Bernstein Polynomials

By1.5and1.6, we see that

2

e^t1e^1−xt^∞

n0

En1−xtⁿ

n!. 2.1

We also have that

2

e^t1e^1−xt 2

1e^−te^−xt^∞

n0

Enx−1ⁿ

n! tⁿ. 2.2

From2.1and2.2, we note thatEn1−x −1ⁿEnx. It is easy to show that

En2 2−ⁿ

l0

n

l El2En, forn >0. 2.3

By1.5,1.6,2.1,2.2, and2.3, we see that forn >0,

p

1−xⁿdμ₋₁x −1ⁿ

p

x−1ⁿdμ₋₁x

p

x2ⁿdμ₋₁x

2

p

xⁿdμ−1x.

2.4

Therefore, we obtain the following theorem.

Theorem 2.1. Forn∈, one has

p

1−xⁿdμ−1x 2

p

xⁿdμ−1x. 2.5

Theorem 2.1is important to derive our main result in this paper.

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Taking the fermionicp-adic integral on pfor one Bernstein polynomial in1.8, we get

p

Bk,nxdμ−1x

p

n

k x^k1−x^n−kdμ−1x

n k

n−k

j0

n−k

j −1^n−k−j

p

x^n−jdμ−1x

n k

n−k

j0

n−k

j −1^n−k−jE_n−j

n k

n−k

j0

n−k

j −1^jEkj.

2.6

Therefore, we obtain the following proposition.

Proposition 2.2. Fork, n∈ , one is

p

Bk,nxdμ₋₁x n

k

n−k

j0

n−k

j −1^jEkj. 2.7

It is known thatBk,nx Bn−k,n1−x. Thus, one has

p

Bk,nxdμ−1x

p

Bn−k,n1−xdμ−1x

n n−k

k j0

k

j −1^k−j

p

1−x^n−jdμ−1x.

2.8

By2.8andTheorem 2.1, we see that forn > k,

p

Bk,nxdμ−1x n

k k j0

k

j −1^k−j

2

p

x^n−jdμ−1x

n k

k j0

k

j −1^k−j

2En−j

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2En ifk0,

⎛

⎝n k

⎞

⎠^k

j0

⎛

⎝k j

⎞

⎠−1^k−jEn−j ifk >0.

2.9

From2.9, we obtain the following theorem.

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Theorem 2.3. Forn, k∈ withn > k, we have

p

Bk,nxdμ−1x

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2En ifk0,

⎛

⎝n k

⎞

⎠^k

j0

⎛

⎝k j

⎞

⎠−1^k−jEn−j ifk >0.

2.10

ByProposition 2.2andTheorem 2.3, we obtain the following corollary.

Corollary 2.4. Forn, k∈ withn > k, we have

n−k

j0

n−k

j −1^jEkj

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2En if k0,

k j0

⎛

⎝k j

⎞

⎠−1^k−jEn−j if k >0.

2.11

Form, n, k∈ withmn >2k, fermionicp-adic invariant integral for multiplication of two Bernstein polynomials on pcan be given by the following relation:

p

Bk,nxBk,mxdμ−1x

p

n

k x^k1−x^n−k m

k x^k1−x^m−kdμ−1x

n k

m k

p

x^2k1−x^nm−2kdμ−1x

n k

m k

2k j0

2k

j −1^j2k

p

1−x^nm−jdμ−1x

n

k m

k 2k

j0

2k

j −1^j2k

2

p

x^nm−jdμ−1x

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2Enm ifk0,

⎛

⎝n k

⎞

⎠

⎛

⎝m k

⎞

⎠^2k

j0

⎛

⎝2k j

⎞

⎠−1^j2kEnm−j ifk >0.

2.12

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

p

Bk,nxBk,mxdμ−1x

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2Enm ifk0,

⎛

⎝n k

⎞

⎠

⎛

⎝m k

⎞

⎠^2k

j0

⎛

⎝2k j

⎞

⎠−1^j2kEnm−j ifk >0. 2.13

Form, n, k∈ , one has

p

Bk,nxBk,mxdμ−1x n

k m

k

p

x^2k1−x^nm−2kdμ−1x

n

k m

k

nm−2k

j0

nm−2k

j −1^j

p

x^j2kdμ−1x

n

k m

k

nm−2k

j0

nm−2k

j −1^jEj2k.

2.14

Thus, we obtain the following proposition.

Proposition 2.6. Form, n, k∈ , one has

p

Bk,nxBk,mxdμ−1x n

k m

k

nm−2k

j0

nm−2k

j −1^jEj2k. 2.15

ByTheorem 2.5andProposition 2.6, we obtain the following corollary.

Corollary 2.7. Form, n, k∈ withmn >2k, one has

nm−2k

j0

nm−2k

j −1^jE_j2k

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2Enm ifk0,

2k j0

⎛

⎝2k j

⎞

⎠−1^j2kEnm−j ifk >0.

2.16

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In the same manner, multiplication of three Bernstein polynomials can be given by the following relation:

p

Bk,nxBk,mxBk,sxdμ−1x

n k

m k

s k

nms−3k

j0

nms−3k

j −1^j

p

x^j3kdμ−1x

n

k m

k s k

nms−3k

j0

nms−3k

j −1^jEj3k,

2.17

wherem, n, s, k∈ withmns >3k.

Form, n, s, k∈ withmns >3k, by the symmetry of Bernstein polynomals, we see that

p

n k

m k

s k

3k j0

3k

j −1^3k−j

p

1−x^nms−jdμ−1x

n

k m

k s k

3k j0

3k

j −1^3k−j

2

p

x^nms−jμ−1x

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2Enms ifk0,

⎛

⎝n k

⎞

⎠

⎛

⎝m k

⎞

⎠

⎛

⎝s k

⎞

⎠^3k

j0

⎛

⎝3k j

⎞

⎠−1^3k−jEnms−j ifk >0.

2.18

Theorem 2.8. Form, n, s, k∈ withmns >3k, one has

p

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2Enms ifk0,

⎛

⎝n k

⎞

⎠

⎛

⎝m k

⎞

⎠

⎛

⎝s k

⎞

⎠^3k

j0

⎛

⎝3k j

⎞

⎠−1^3k−jEnms−j ifk >0.

2.19

By2.17andTheorem 2.8, we obtain the following corollary.

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Corollary 2.9. Form, n, s, k∈ withmns >3k, one has

nms−3k

j0

nms−3k

j −1^jEj3k

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2Enms ifk0,

3k j0

⎛

⎝3k j

⎞

⎠−1^3k−jEnms−j ifk >0.

2.20

Using the above theorems and mathematical induction, we obtain the following theorem.

Theorem 2.10. Lets∈. Forn1, n2, . . . , ns, k∈ withn1n2· · ·ns> sk, the multiplication of the sequence of Bernstein polynomialsBk,n1x, . . . , Bk,nsxwith diﬀerent degrees under fermionic p-adic invariant integral on pcan be given as

p

s i1

Bk,nix dμ−1x

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2En1n2···ns if k0,

⎛

⎝^s

i1

⎛

⎝ni

k

⎞

⎠

⎞

⎠^sk

j0

⎛

⎝sk j

⎞

⎠−1^sk−jEn1n2···ns−j if k >0.

2.21

We also easily see that

p

_s

i1

Bk,nix dμ−1x _s

i1

ni

k _n

1···ns−sk j0

n1· · ·ns−sk

j −1^jEjsk. 2.22

ByTheorem 2.10and2.22, we obtain the following corollary.

Corollary 2.11. Lets∈. Forn1, n2, . . . , ns, k∈ withn1n2· · ·ns> sk, one has

n1···ns−sk j0

n1· · ·ns−sk

j −1^jEjsk

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2En1n2···ns ifk0, sk

j0

⎛

⎝sk j

⎞

⎠−1^sk−jEn1n2···ns−j ifk >0.

2.23

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Letm1, . . . , ms, n1, . . . , ns, k ∈ with m1n1· · ·msns > m1 · · ·msk. By the definition ofB^m_k,n^s_sx, we easily get

p

s i1

B^m_k,nⁱ_ix dμ−1x

_s

i1

ni

k

mi k_s

i1mi

j0

−1^k^sⁱ¹^mⁱ^−j

p

1−x^sⁱ¹ⁿⁱ^mⁱ^−jdμ−1x

_s

i1

ni

k

mi ks i1mi

j0

⎛

⎜⎝k s

i1

mi

j

⎞

⎟⎠−1^k^sⁱ¹^mⁱ^−j

2E^s_i1nimi−j

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

2Em1n1···msns if k0,

⎛

⎝^s

i1

⎛

⎝ni

k

⎞

⎠

mi⎞

⎠^k

s i1mi

j0

⎛

⎜⎝k s

i1

mi

j

⎞

⎟⎠−1^k^sⁱ¹^mⁱ^−jE^s_i1nimi−j if k >0.

2.24

Theorem 2.12. Lets∈. Form1, . . . , ms, n1, . . . , ns, k∈ withm1n1· · ·msns>m1· · · msk, one has

p

_s

i1

B^m_k,nⁱ_ix dμ−1x

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

2Em1n1···msns ifk0,

⎛

⎝^s

i1

⎛

⎝ni

k

⎞

⎠

mi⎞

⎠^k

_s

i1mi

j0

⎛

⎜⎝k s

i1

mi

j

⎞

⎟⎠−1^k^sⁱ¹^mⁱ^−jE^s_i1nimi−j ifk >0.

2.25

By simple calculation, we easily get

p

_s

i1

B_k,n^mⁱ

ix dμ−1x

_s

i1

ni

k

mi s

i1nimi−ks i1mi

j0

⎛

⎜⎝ s

i1

nimi−k s

i1

mi

j

⎞

⎟⎠−1^jEks i1mi−j,

2.26

wherem1, . . . , ms, n1, . . . , ns, k∈ fors ∈. ByTheorem 2.12and2.26, we obtain the following corollary.

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Corollary 2.13. Lets∈. Form1, . . . , ms, n1, . . . , ns, k∈ withm1n1· · ·msns>m1· · · msk, one has

_s

i1nimi−k_s

i1mi

j0

⎛

⎜⎝ s

i1

nimi−k s

i1

mi

j

⎞

⎟⎠−1^jEk_s

i1mi−j

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

2Em1n1···msns ifk0,

k_s

i1mi

j0

⎛

⎜⎝k s

i1

mi

j

⎞

⎟⎠−1^k^sⁱ¹^mⁱ^−jE^s_i1nimi−j ifk >0.

2.27

The fermionic p-adic invariant integral of multiplication of n 1 Bernstein polynomials, the nth degree Bernstein polynomials Bi,nx with i 0,1, . . . , n and with multiplicitym0, m1, . . . , mnon _p, respectively, can be given by

p

_n

i0

B^m_i,nⁱx dμ−1x _n

i0

n i

mi

p

xⁿⁱ¹^imⁱ1−xⁿⁿⁱ⁰^mⁱ⁻ⁿⁱ¹^imⁱdμ−1x

_n

i1ⁿ_i^mⁱ _nn

i0mi

_n

i1imi

p

Bⁿ_i1imi, nn

i0mixdμ−1x,

2.28

wherem0, m1, . . . , mn∈ withn∈ .

Assume thatnm0nm1· · ·nmn> m12m2· · ·nmn. Then one has

p

_n

i0B^m_i,nⁱx dμ−1x

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

2Enm0nm1···nmn if

n i1

imi0,

⎛

⎝ⁿ

i0

⎛

⎝n i

⎞

⎠

mi⎞

⎠

n i1mi

j0

⎛

⎜⎝ n

i1

imi

j

⎞

⎟⎠−1ⁿⁱ¹^imⁱ^−jEnn i0mi−n

i1imi if

n i1

imi>0.

2.29

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Theorem 2.14. Letn∈ .

iForm0, m1, . . . , mn∈ withn_n

i0mi>_n

i1imi, one has

p

_n

i0

B^m_i,nⁱx dμ₋₁x

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

n i1imi0,

⎛

⎝ⁿ

i0

⎛

⎝n i

⎞

⎠

mi⎞

⎠

_n

i1mi

j0

⎛

⎜⎝ n

i1

imi

j

⎞

⎟⎠−1ⁿⁱ¹^imⁱ^−jEn_n

i0mi−_n

i1imi if

n i1imi>0.

2.30

iiForm0, m1, . . . , mn∈ , one has

p

_n

i0

B_i,n^mⁱx dμ−1x _n

i0

n i

mi nn i0mi−n

i1imi

j0

⎛

⎜⎝n n i0

mi−ⁿ

i1

imi

j

⎞

⎟⎠−1^jEⁿ_i1imij. 2.31

ByTheorem 2.14, we obtain the following corollary.

Corollary 2.15. Forn, m0, m1, . . . , mn ∈ withn_n

i0mi>_n

i1 imi, one has

n_n

i0mi−_n

i1imi

j0

⎛

⎜⎝n n

i0

mi−ⁿ

i1

imi

j

⎞

⎟⎠−1^jEⁿ_i1imij

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

n i1

imi0,

n i1mi

j0

⎛

⎜⎝ n

i1

imi

j

⎞

⎟⎠−1ⁿⁱ¹^imⁱ^−jEnn i0mi−n

i1imi if

n i1

imi>0.

2.32

References

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

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

3 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.

4 S. Bernstein, “Démonstration du théorème de Weierstrass, fondée sur le calcul des probabilities,”

Communications of the Kharkov Mathematical Society, vol. 13, pp. 1–2, 1912.

(12)

5 T. Kim, J. Choi, and Y. H. Kim, “On extended carlitz’s type q-Euler numbers and polynomials,”

Advanced Studies in Contemporary Mathematics, vol. 20, no. 4, pp. 499–505, 2010.

6 N. K. Govil and V. Gupta, “Convergence ofq-Meyer-K ¨onig-Zeller-Durrmeyer operators,” Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 97–108, 2009.

7 V. Gupta, T. Kim, J. Choi, and Y.-H. Kim, “Generating function forq-Bernstein,q-Meyer-K ¨onig-Zeller andq-beta basis,” Automation Computers Applied Mathematics, vol. 19, pp. 7–11, 2010.

8 T. Kim, “q-extension of the Euler formula and trigonometric functions,” Russian Journal of Mathematical Physics, vol. 14, no. 3, pp. 275–278, 2007.

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, J. Choi, and Y.-H. Kim, “Some identities on theq-Bernstein polynomials,q-Stirling numbers andq-Bernoulli numbers,” Advanced Studies in Contemporary Mathematics, vol. 20, no. 3, pp. 335–341, 2010.

11 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.

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

13 V. Kurt, “A further symmetric relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials,” Applied Mathematical Sciences, vol. 3, no. 53–56, pp. 2757–2764, 2009.

14 I. N. Cangul, V. Kurt, H. Ozden, and Y. Simsek, “On the higher-order w-q-Genocchi numbers,”

Advanced Studies in Contemporary Mathematics, vol. 19, no. 1, pp. 39–57, 2009.

15 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.