On the Fermionic adic Integral Representation of Bernstein Polynomials Associated with Euler Numbers and Polynomials

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,

_{J. Choi,}

_{Y. H. Kim,}

_{and C. S. Ryoo}

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

2_{Department 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

−νpp _{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

1qpN

pN₋₁

x0

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

f2f0, 1.3

wheref1x fx1.

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

I−1

fn

−1n

I−1

f2

n−1

l0

−1n−1−l

fl. 1.4

It is well known that the Euler polynomials are defined by

2

et₁e xt∞

n0 Enxt

n

n!, 1.5

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

Letfx etx_{. Then, by1.3,1.4, and1.5, we see that}

p

exytdμ−1

y 2

et₁e xt∞

n0 Enxt

n

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

k_1−xn−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

k_1−xn−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.

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

et₁e

1−xt∞ n0

En1−xt n

n!. 2.1

We also have that

2

et₁e

1−xt 2

1e−te− xt∞

n0

Enx−1 n

n! t

n_{. 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

xn_dμ

−1x. 2.5

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

p

Bk,nxdμ−1x

p

n

k x

k_1−xn−k_dμ

−1x

n

k

n−k

j0

n−k

j −1

n−k−j

p

xn−jdμ−1x

n

k

n−k

j0

n−k

j −1

n−k−j_E n−j

n

k

n−k

j0

n−k

j −1

j_E kj.

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

j

Ekj. 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−xn−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

xn−jdμ−1x

n k

k

j0

k

j −1

k−j₂

En−j

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2En ifk0,

⎛ ⎝n

k

⎞ ⎠k

j0 ⎛ ⎝k

j

⎞ ⎠₋₁k−j

En−j ifk >0.

2.9

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

p

Bk,nxdμ−1x ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2En ifk0,

⎛ ⎝n

k

⎞ ⎠k

j0 ⎛ ⎝k

j

⎞ ⎠₋₁k−j

En−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

j_E kj

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2En if k0,

k

j0 ⎛ ⎝k

j

⎞ ⎠₋₁k−j

En−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

k_1−xn−k

m

k x

k_1−xm−k

dμ−1x

n k

m k

p

x2k1−xnm−2kdμ−1x

n k

m k

2k

j0

2k

j −1

j2k

p

1−xnm−jdμ−1x

n k

m k

2k

j0

2k

j −1

j2k

2

p

xnm−jdμ−1x

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2Enm ifk0,

⎛ ⎝n

k

⎞ ⎠

⎛

⎝m

k

⎞ ⎠2k

j0 ⎛ ⎝2k

j

⎞ ⎠₋₁j2k

Enm−j ifk >0.

Therefore, we obtain the following theorem.

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

⎞ ⎠₋₁j2k

Enm−j ifk >0. 2.13

Form, n, k∈ , one has

p

Bk,nxBk,mxdμ−1x

n k

m k

p

x2k1−xnm−2kdμ−1x

n k

m k

nm−2k

j0

nm−2k

j −1

j

p

xj2kdμ−1x

n k

m k

nm−2k

j0

nm−2k

j −1

j_E j2k.

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

j_E

j2k. 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

j_E j2k

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2Enm ifk0,

2k

j0 ⎛ ⎝2k

j

⎞

⎠₋₁j2k_E

nm−j ifk >0.

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

xj3kdμ−1x

n k

m k

s k

nms−3k

j0

nms−3k

j −1

j

Ej3k,

2.17

wherem, n, s, k∈ withmns >3k.

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

see that

p

Bk,nxBk,mxBk,sxdμ−1x

n k

m k

s k

3k

j0

3k

j −1

3k−j

p

1−xnms−jdμ−1x

n k

m k

s k

3k

j0

3k

j −1

3k−j

2

p

xnms−jμ−1x

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2Enms ifk0,

⎛ ⎝n

k

⎞ ⎠

⎛

⎝m

k

⎞ ⎠

⎛ ⎝s

k

⎞ ⎠3k

j0 ⎛ ⎝3k

j

⎞ ⎠₋₁3k−j

Enms−j ifk >0.

2.18

Therefore, we obtain the following theorem.

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

p

Bk,nxBk,mxBk,sxdμ−1x

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2Enms ifk0,

⎛ ⎝n

k

⎞ ⎠

⎛

⎝m

k

⎞ ⎠

⎛ ⎝s

k

⎞ ⎠3k

j0 ⎛ ⎝3k

j

⎞ ⎠₋₁3k−j

Enms−j ifk >0.

2.19

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

nms−3k

j0

nms−3k

j −1

j

Ej3k

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2Enms ifk0,

3k

j0 ⎛ ⎝3k

j

⎞

⎠₋₁3k−j

Enms−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

⎞

⎠₋₁sk−j

En1n2···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

j

Ejsk. 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

j_E jsk

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

2En1n2···ns ifk0,

sk

j0 ⎛ ⎝sk

j

⎞ ⎠₋₁sk−j

En1n2···ns−j ifk >0.

Letm1, . . . , ms, n1, . . . , ns, k ∈ with m1n1· · ·msns > m1 · · ·msk. By the

definition ofBms

k,nsx, we easily get

p

s

i1 Bmi

k,nix dμ−1x

_s

i1

ni

k

mi _ks i1mi

j0

−1ksi1mi−j

p

1−xsi1nimi−j_dμ −1x

_s

i1

ni

k

mi _ks_i

1mi

j0 ⎛ ⎜ ⎝k

s

i1 mi

j

⎞ ⎟ ⎠−1ks

i1mi−j_2Es i1nimi−j

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2Em1n1···msns if k0, ⎛

⎝s

i1 ⎛ ⎝ni

k

⎞ ⎠

mi⎞ ⎠k

s i1mi

j0 ⎛ ⎜ ⎝k

s

i1 mi

j

⎞ ⎟

⎠−1ks

i1mi−j_Es

i1nimi−j if k >0.

2.24

Therefore, we obtain the following theorem.

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

msk, one has

p

_s

i1 Bmi

k,nix dμ−1x

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2Em1n1···msns ifk0, ⎛

⎝s

i1 ⎛ ⎝ni

k

⎞ ⎠

mi⎞ ⎠k

s i1mi

j0 ⎛ ⎜ ⎝k

s

i1 mi

j

⎞ ⎟

⎠−1ksi1mi−j_Es

i1nimi−j ifk >0.

2.25

By simple calculation, we easily get

p

_s

i1 Bmi

k,nix dμ−1x

_s

i1

ni

k

mi s_i

1nimi−ksi1mi

j0

⎛ ⎜ ⎝

s

i1

nimi−k s

i1 mi

j

⎞ ⎟ ⎠−1j_E

ks_i1mi−j,

2.26

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

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

msk, one has

s

i1nimi−ksi1mi

j0

⎛ ⎜ ⎝

s

i1

nimi−k s

i1 mi

j

⎞ ⎟ ⎠−1j_E

ks i1mi−j

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2Em1n1···msns ifk0,

ks i1mi

j0 ⎛ ⎜ ⎝k

s

i1 mi

j

⎞ ⎟

⎠−1ksi1mi−j_Es

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 Bmi

i,nx dμ−1x

_n

i0

n i

mi

p

xni1imi1−_xnni0mi−ni1imi_dμ −1x

_n

i1nimi

_nn

i0mi

n i1imi

p

Bn

i1imi, nni0mixdμ−1x,

2.28

wherem0, m1, . . . , mn∈ withn∈ .

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

p

_n

i0 Bmi

i,nx dμ−1x

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2Enm0nm1···nmn if

n

i1

imi0,

⎛ ⎝n

i0 ⎛ ⎝n

i

⎞ ⎠

mi⎞ ⎠

n i1mi

j0 ⎛ ⎜ ⎝

n

i1 imi

j

⎞ ⎟

⎠−1ni1imi−j_E

nn

i0mi−ni1imi if

n

i1

imi>0.

2.29

Theorem 2.14. Letn∈ .

iForm0, m1, . . . , mn∈ withn

_n

i0mi>

_n

i1imi, one has

p

_n

i0 Bmi

i,nx dμ−1x

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2Enm0nm1···nmn if

n

i1

imi0,

⎛ ⎝n

i0 ⎛ ⎝n

i

⎞ ⎠

mi⎞ ⎠

n i1mi

j0 ⎛ ⎜ ⎝

n

i1 imi

j

⎞ ⎟

⎠−1ni1imi−j_E

nn

i0mi−ni1imi if

n

i1

imi>0.

2.30

iiForm0, m1, . . . , mn∈ , one has

p

_n

i0 Bmi

i,nx dμ−1x

_n

i0

n i

mi nn

i0mi−ni1imi

j0

⎛ ⎜ ⎝n

n

i0 mi−

n

i1

imi

j

⎞ ⎟ ⎠−1j

En i1imij.

2.31

ByTheorem 2.14, we obtain the following corollary.

Corollary 2.15. Forn, m0, m1, . . . , mn ∈ withnin0mi>ni1 imi, one has

nn

i0mi−ni1imi

j0

⎛ ⎜ ⎝n

n

i0 mi−

n

i1 imi

j

⎞ ⎟

⎠−1j_En i1imij

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2Enm0nm1···nmn if

n

i1

imi0,

n i1mi

j0 ⎛ ⎜ ⎝

n

i1 imi

j

⎞ ⎟

⎠−1ni1imi−j_E

nni0mi−ni1imi if

n

i1

imi>0.

2.32

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