A NOTE ON DEGENERATE BERNSTEIN AND DEGENERATE EULER POLYNOMIALS

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TAEKYUN KIM AND DAE SAN KIM

Abstract. In this paper, we investigate the recently introduced degener-ate Bernstein polynomials and operators and derive some of their proper-ties. Also, we give some properties of the degenerate Euler numbers and polynomials and their connection with the degenerate Euler polynomials.

1. Introduction

LetC[0,1] denote the set of continuous function on [0,1]. For f ∈ C[0,1], Bernstein introduced the following well known linear operator (see [3,20]):

Bn(f|x) =

n X

k=0

f(k

n)

_n

k

xk(1−x)n−k

= n X

k=0

f(k

n)

_n

k

Bk,n(x), (see [3,8−27]),

(1.1)

where

Bk,n(x) = _n

k

xk(1−x)n−k, (n, k∈_Z≥0) (1.2)

are called Bernstein polynomials of degreenor Bernstein basis polynomials. Here_Bn, (n≥0), is called Bernstein operator of ordern.

A Bernoulli trial involves performing an experiment once and noting whether a particular event A occurs. The outcome of Bernoulli trial is said to be ”Success” if A occurs and a ”failure” otherwise. The probabilityPn(k) of k successes in

nindependent Bernoulli trials with the probability of success pis given by the binomial probability law

Pn(k) = _n

k

pk(1−p)n−k,fork= 0,1,2,3,· · ·,

2010Mathematics Subject Classification. 11B83, 11C08, 05A19 .

Key words and phrases. degenerate Bernstein polynomials, degenerate Bernstein operators, degenerate Euler polynomials.

1

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From the definition of Bernstein polynomials, we note that Bernstein basis is the probability mass function of binomial distribution. The Bernstein polynomi-als of degreencan be defined by blending together two Bernstein polynomials of degreen−1. That is, thek-th Bernstein polynomial of degreencan be written as

Bk,n(x) = (1−x)Bk,n−1(x) +xBk−1,n−1(x), (k, n∈N). (1.3)

For example, B0,1(x) = 1−x, B1,1(x) = x, B0,2(x) = (1−x)2, B1,2(x) = 2x(1−x), B2,2(x) =x2, B0,3(x) = (1−x)3, B1,3(x) = 3x(1−x)2, B2,3(x) = 3x2(1−x),B3,3(x) =x3,· · ·.

Thus, we note that

xk =x(xk−1) =x

n X

i=k−1 i k−1

n k−1

Bi,n−1(x) = n X

i=k i−1 k−1

n−1 k−1

xBi−1,n−1(x)

= n−1 X

i=k−1 i k−1

n k−1

i

nBi,n(x) =

n−1 X

i=k−1 i k n k

Bi,n(x).

(1.4)

Forλ∈ R, L. Carlitz introduced the degenerate Euler poynomials given by

the generating function

2

(1 +λt)λ1 + 1

(1 +λt)xλ =

∞

X

n=0

En,λ(x)t n

n!, (see [4]). (1.5)

Whenx= 0, En,λ =En,λ(0) are called the degenerate Euler numbers. It is easy to show that limλ→0En,λ(x) = En(x), whereEn(x) are the Euler polyno-mials given by

2

et_{+ 1}e xt₌

∞

X

n=0

En(x)t n

n!, (see [1−7]). (1.6)

Forn≥0, we define theλ-product as follows:

(x)0,λ= 1, (x)n,λ=x(x−λ)(x−2λ)· · ·(x−(n−1)λ), (n≥1), (see [9]). (1.7)

Note that limλ→0(x)n,λ=xn, (n≥1).

Recently, the degenerate Bernstein polynomials of degreenare introduced as

Bk,n(x|λ) =

n k

(x)k,λ(1−x)n−k,λ, (x∈[0,1], n, k≥0), (see [9]). (1.8)

From (1.8), we note that the generating function forBk,n(x|λ) is given by

1

k!(x)k,λt

k_{(1 +}_λt₎1−x

λ =

∞

X

n=k

Bk,n(x|λ)

tn

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By (1.9), we easily get limλ→0Bk,n(x|λ) =Bk,n(x), (n, k≥0).

In this paper, we investigate the recently introduced degenerate Bernstein polynomials and operators and derive some of their properties (see [9]). Also, we give some properties of the degenerate Euler numbers and polynomials and their connection with the degenerate Euler polynomials.

2. Degenerate Bernstein polynomials and operators

Letf(x) be a continuous function on [0,1]. Then the degenerate Bernstein operator of ordernis defined as

Bn,λ(f|x) =

n X

k=0

f(k

n)

_n

k

(x)k,λ(1−x)n−k,λ= n X

k=0

f(k

n)Bk,n(x|λ), (2.1)

wherex∈[0,1] andn, k∈_Z≥0.From (2.1), we note that

Bn,λ(1|x) = n X

k=0 _n

k

(x)k,λ(1−x)n−k,λ. (2.2)

Now, we observe that

∞

X

n=0 (1)n,λt

n

n! = (1 +λt) 1

λ = (1 +λt) x

λ(1 +λt)

1−x λ

= ∞

X

l=0 (x)l,λt

l

l! ! _∞

X

m=0

(1−x)m,λt m

m! !

= ∞

X

n=0 n X

l=0 _n

l

(x)l,λ(1−x)n−l,λ !

tn

n!.

(2.3)

By comparing the coefficients on the both sides of (2.3), we get

(1)n,λ= n X

l=0 _n

l

(x)l,λ(1−x)n−l,λ. (2.4)

From (2.2) and (2.4), we have

Bn,λ(1|x) = n X

k=0 _n

k

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Also, we get from (2.1) that forf(x) =x,

Bn,λ(x|x) =

n X

k=0

k n

n k

(x)k,λ(1−x)n−k,λ

= n X

k=1 _n₋₁

k−1

= n−1 X

k=0 _n₋₁

k

(x)k,λ(1−x)n−1−k,λ(x−kλ)

=x(1)n−1,λ−(n−1)λ n−1 X

k=0 _n

−1

k

(x)k,λ(1−x)n−1−k,λ

k n−1

=x(1)n−1,λ−(n−1)λBn−1,λ(x|x).

(2.6)

From (2.6), we can derive the following equation (2.7).

Bn,λ(x|x) =x(1)n−1,λ−(n−1)λ{x(1)n−2,λ−(n−2)λBn−2,λ(x|x)}

=x(1)n−1,λ−x(n−1)λ(1)n−2,λ+ (−1)2(n−1)(n−2)λ2Bn−2,λ(x|x)

=x(1)n−1,λ−x(n−1)λ(1)n−2,λ

+ (−1)2(n−1)(n−2)λ2{x(1)n−3,λ−(n−3)λBn−3,λ(x|x)}

=x(1)n−1,λ−x(n−1)λ(1)n−2,λ+ (−1)2(n−1)(n−2)λ2x(1)n−3,λ

+ (−1)3(n−1)(n−2)(n−3)λ3Bn−3,λ(x|x)

=· · ·

=x

n−1 X

k=0

(−1)kλk(n−1)k(1)n−1−k,λ,

(2.7)

where (x)k =x(x−1)· · ·(x−k+ 1),(k≥1), (x)0= 1. Therefore, we obtain the following theorem.

Theorem 2.1. Forn≥0, we have

Bn,λ(f|0) =f(0)(1)n,λ, Bn,λ(f|1) =f(1)(1)n,λ,

and

Bn,λ(1|x) = (1)n,λ, Bn,λ(x|x) =x n−1 X

k=0

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Letf, g be continuous functions defined on [0,1]. Then we note that

Bn,λ(αf+βg|x) =αBn,λ(f|x) +βBn,λ(g|x), (n≥0), (2.8)

whereα, βare constants.

So, the degenerate Bernstein operator is linear. From (1.8), we note that

B0,1(x|λ) = 1−x, B1,1(x|λ) =x, B0,2(x|λ) = (1−x)2−λ(1−x),

B1,2(x|λ) = 2x(1−x), B2,2(x|λ) =x2−λx.

It is easy to show that

∞

X

n=0

(1−x)n,λ

tn

n! = (1 +λt) 1−x

λ = (1 +λt)

1

λ(1 +λt)− x λ

= ∞

X

l=0 (1)l,λ

tl

l! ! ∞

X

m=0

(−x)m,λ

tm

m! !

= ∞

X

n=0 n X

l=0 _n

l

(1)n−l,λ(−1)l(x)l,−λ !

tn

n!.

Thus we get

(1−x)n,λ= n X

l=0 _n

l

(1)n−l,λ(−1)l(x)l,−λ, (n≥0). (2.9)

From (2.1), we have

Bn,λ(f|x) =

n X

k=0

f(k

n)Bk,n(x|λ) =

n X

k=0

f(k

n)

_n

k

= n X

k=0

f(k

n)

_n

k

(x)k,λ

n−k X

j=0 _n₋_k

j

(−1)j(1)n−k−j,λ(x)j,−λ.

(2.10)

_n

k

_n₋_k

j

=

_n

k+j

_k₊_j

k

, (n, k≥0), (2.11)

and

(x)j,−λ=

(x)k+j,−λ

(x+ (j+k−1)λ)k,λ. (2.12)

Letk+j=m. Then, by (2.11), we get

_n

k

_n −k

j

=

_n

m

_m

k

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Bn,λ(f|x) =

n X m=0 n m

(x)m,−λ m X k=0 m k

(−1)m−k(1)n−m,λ

(x)k,λ

(x+ (m−1)λ)k,λf(

k n).

(2.13)

Therefore, by (2.13), we obtain the following theorem.

Theorem 2.2. Forf ∈C[0,1]andn∈_Z≥0, we have

Bn,λ(f|x) =

n X m=0 _n m

(x)m,−λ m X k=0 _m k

(−1)m−k(1)n−m,λ

(x)k,λ (x+ (m−1)λ)k,λ

f(k

n).

From (1.8), (2.9) and (2.12), we note that

Bk,n(x|λ) =

n k

= _n

k

(x)k,λ

n−k X

i=0

_n₋_k

i

(−1)i(x)i,−λ(1)n−k−i,λ

= n−k X

i=0 (−1)i

_n

k

_n₋_k

i

(x)k+i,−λ

(x)k,λ

(x+ (k+i−1)λ)k,λ(1)n−k−i,λ

= n X

i=k

(−1)i−k

n k

n−k i−k

(x)i,−λ

(x)k,λ

(x+ (i−1)λ)k,λ(1)n−i,λ

= n X

i=k

(−1)i−k _n

i

_i

k

(x)i,−λ

(x)k,λ

(x+ (i−1)λ)k,λ(1)n−i,λ.

(2.14)

Therefore, by (2.14), we obtain the following theorem.

Theorem 2.3. Forn, k∈Z≥0 andx∈[0,1], we have

Bk,n(x|λ) = n X

i=k

(−1)i−k _n

i

_i

k

(x)i,−λ

(x)k,λ (x+ (i−1)λ)k,λ

(1)n−i,λ.

From (1.8) and (2.5), it is to see that

(x−kλ)i,λ= 1 (1)n−i,λ

n X

k=i k i n

i

Bk,n(x|λ), (2.15)

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Indeed,

n X

k=1

k

nBk,n(x|λ) =

n X

k=1 _n

−1

k−1

= n−1 X

k=0 _n₋₁

k

(x)k+1,λ(1−x)n−1−k,λ

= (x−kλ)(1)n−1,λ.

In the same manner, we can show that

n X

k=i k i n i

Bk,n(x|λ) = (x−kλ)i,λ(1)n−i,λ.

3. Degenerate Euler polynomials associated with degenerate Bernstein polynomials

From (1.5), we note that

2 = ∞

X

l=0 El,λ

tl l!

!

(1 +λt)λ1 _{+ 1}

=

∞

X

l=0 El,λ

tl l!

! _∞ X

m=0 (1)m,λt

m

m!+ 1 !

= ∞

X

n=0 n X

l=0 _n

l

El,λ(1)n−l,λ !

tn

n! + ∞

X

n=0 En,λ

tn

n!

= ∞

X

n=0 n X

l=0 _n

l

(1)n−l,λEl,λ+En,λ !

tn

n!.

(3.1)

By comparing the coefficients on both sides of (3.1), we obtain the following theorem.

Theorem 3.1. Forn≥0, we have n

X

l=0 _n

l

(1)n−l,λEl,λ+En,λ= (

2, ifn= 0,

0, ifn >0.

From Theorem 4, we note that

E0,λ= 1, En,λ=− n X

l=0 _n

l

(1)n−l,λEl,λ =− 1 2

n−1 X

l=0 _n

l

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By (1.5), we get

∞

X

n=0

En,λ(1−x)t n

n! =

2

(1 +λt)λ1 + 1

(1 +λt)1−λx ₌ 2

(1 + +λt)−1λ + 1

(1 +λt)−xλ

= ∞

X

n=0

En,−λ(x)(−1)n

tn

n!.

(3.2)

By comparing the coefficients on the sides of (3.2), we have

En,λ(1−x) = (−1)nEn,−(x), (n≥0). (3.3)

In particular, if we takex=−1, we get

En,λ(2) = (−1)nEn,−λ(−1), (n≥0). (3.4)

Therefore, we obtain the following theorem.

Theorem 3.2. Forn≥0, we have

En,λ=− n X

l=0 _n

l

(1)n−l,λEl,λ, En,λ(1−x) = (−1)nEn,−λ(x).

In particular

En,λ(2) = (−1)nEn,−λ(−1), (n≥0).

From (1.5), we have

∞

X

n=0 En,λ(2)

tn

n! =

2

(1 +λt)λ1 + 1

(1 +λt)λ2 = 2

(1 +λt)λ1 + 1

(1 +λt)1λ((1 +λt)

1

λ + 1−1)

= 2(1 +λt)λ1 − 2

(1 +λt)λ1 + 1

(1 +λt)λ1

= ∞

X

n=0

2(1)n,λ− n X

l=0 _n

l

(1)n−l,λEl,λ !

tn

n!.

(3.5)

Thus, by (3.5), we get

En,λ(2) = 2(1)n,λ− n X

l=0 _n

l

(1)n−l,λEl,λ= 2(1)n,λ+En,λ, (n≥0).

Corollary 3.3. Forn≥0, we have

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En,λ(x) = n X

l=0

n l

El,λ(x)n−l,λ, (n≥0).

2

(1 +λt)1λ + 1

(1 +λt)xλ ₌ 2

(1 +λt)λ1 + 1

((1 +λt)1λ _{+ 1}−_{1)(1 +}_λt₎ x−1

λ

= 2(1 +λt)x−λ1 − 2

(1 +λt)1λ + 1

(1 +λt)x−λ1

= 2(1 +λt)x−λ1 −2(1 +λt) x−2

λ + 2

(1 +λt)λ1 + 1

(1 +λt)x−λ2.

(3.6)

Continuing the process in (3.6), we have

∞

X

n=0

En,λ(x)t n

n! = ∞

X

n=0 2

k X

i=1

(x−i)n,λ(−1)i−1 !

tn n!

+ (−1)k 2 (1 +λt)1λ + 1

(1 +λt)x−λk_.

(3.7)

Therefore, by (1.5) and (3.7), we obtain the following theorem.

Theorem 3.4. Forn≥0, k≥1, we have

En,λ(x) = 2 k X

i=1

(x−i)n,λ(−1)i−1+ (−1)kEn,λ(x−k).

By (1.9), we get

(x)k,λ(1 +λt) 1−x

λ =k!

tk ∞

X

n=k

Bk,n(x|λ)

tn

n! = ∞

X

n=0

Bk,n+k(x|λ) 1 n+k

n

tn

n!. (3.8)

On the other hand

(x)k,λ(1 +λt) 1−x

λ =

∞

X

n=0

(x)k,λ(1−x)n,λ

tn

n!. (3.9)

(x)k,λ(1−x)n,λ= 1 n+k

n

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(x)k,λ(1 +λt) 1−x

λ =(x)k,λ

2

(1 +λt)λ1 + 1

(1 +λt)1−λx((1 +λt)

1

λ + 1)

=(x)k,λ 2

₂

(1 +λt)λ1 + 1

(1 +λt)2−λx+ 2

(1 +λt)λ1 + 1

(1 +λt)1−λx

=(x)k,λ 2

∞

X

n=0

(En,λ(2−x) +En,λ(1−x))

tn

n! !

.

(3.11)

By (3.9) and (3.11), we get

(x)k,λ(1−x)n,λ= (x)k,λ

2 (En,λ(2−x) +En,λ(1−x)), (n≥0). (3.12) Therefore, by (3.10) and (3.12), we obtain the following theorem.

Theorem 3.5. Forn, k≥0, we have Bk,n+k(x|λ) =

1 2(x)k,λ

_n₊_k

k

(En,λ(2−x) +En,λ(1−x)).

References

1. A. Bayad, T. Kim, B. Lee, S.-H. Rim,Some identities on Bernstein polynomials associated withq-Euler polynomials, Abstr. Appl. Anal. 2011, Art. ID 294715, 10 pp.

2. A. Bayad, T. Kim,Identities involving values of Bernstein,q–Bernoulli, andq–Euler poly-nomials,Russ. J. Math. Phys. 18 (2011), no. 2, 133-143.

3. S. Bernstein,Demonstration du theoreme de Weierstrass basee sur le calcul des probabilites, Commun.Soc. Math. Kharkow(2) 13(1912-1913), 1–2.

4. L. Carlitz,Degenerate Stirling, Bernoulli and Eulerian numbers,Utilitas Math. 15 (1979), 51-88.

5. L.-C. Jang, T. Kim, Y.-H. Kim, B. Lee,Some new identities on the twisted Carlitz’s q

–Bernoulli numbers andq–Bernstein polynomials,J. Inequal. Appl. 2011, 2011:52, 6 pp. 6. M.-S. Kim, D. Kim, T. Kim, On q–Euler numbers related to the modified q-Bernstein

polynomials,Abstr. Appl. Anal. 2010, Art. ID 952384, 15 pp.

7. M.-S. Kim, T. Kim, B. Lee, C.-S. Ryoo,Some identities of Bernoulli numbers and polyno-mials associated with Bernstein polynopolyno-mials,Adv. Difference Equ. 2010, Art. ID 305018, 7 pp.

8. T. Kim, Y.-H. Kim, C. S. Ryoo,Some identities on the weighted q–Euler numbers and

q-Bernstein polynomials, J. Inequal. Appl. 2011, 2011:64, 7 pp.

9. T. Kim, D. S. Kim, Degenerate Bernstein Polynomials,arXiv:1806.06379 [math.NT]. 10. T. Kim, J. Choi, Y.–H. Kim,On thek–dimensional generalization ofq-Bernstein

polyno-mials,Proc. Jangjeon Math. Soc. 14 (2011), no. 2, 199-207.

11. T. Kim, J. Choi, Y.H. Kim, C. S. Ryoo,On the fermionicp–adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials,J. Inequal. Appl. 2010, Art. ID 864247, 12 pp.

(11)

13. T. Kim,A note onq-Bernstein polynomials,Russ. J. Math. Phys. 18 (2011), no. 1, 73-82. 14. T. Kim, Some identities on the q–integral representation of the product of several q

-Bernstein–type polynomials,Abstr. Appl. Anal. 2011, Art. ID 634675, 11 pp.

15. T. Kim, J. Choi, Y. H. Kim,Some identities on theq–Bernstein polynomials,q–Stirling numbers andq–Bernoulli numbers,Adv. Stud. Contemp. Math. (Kyungshang) 20 (2010), no. 3, 335-341.

16. T. Kim, A. Bayad, Y. H. Kim,A study on the p–adicq–integral representation on Zp associated with the weighted q–Bernstein andq-Bernoulli polynomials, J. Inequal. Appl. 2011, Art. ID 513821, 8 pp.

17. T. Kim, L.-C. Jang, H. Yi,A note on the modifiedq-Bernstein polynomials,Discrete Dyn. Nat. Soc. 2010, Art. ID 706483, 12 pp.

18. T. Kim, J. Choi, Y. H. Kim, L. C. Jang,On p–adic analogue ofq–Bernstein polynomials and related integrals,Discrete Dyn. Nat. Soc. 2010, Art. ID 179430, 9 pp.

19. T. Kim, B. Lee, J. Choi, Y. H. Kim, S. H. Rim,On theq–Euler numbers and weighted

q-Bernstein polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 21 (2011), no. 1, 13-18.

20. T. Kim,Identities on the weightedq–Euler numbers andq-Bernstein polynomials, Adv. Stud. Contemp. Math. (Kyungshang) 22 (2012), no. 1, 7-12.

21. T. Kim,Some formulae for theq–Bernstein polynomials andq–deformed binomial distri-butions,J. Comput. Anal. Appl. 14 (2012), no. 5, 917-933.

22. T. Kim, C. S. Ryoo, H. Yi,A note onq–Bernoulli numbers andq-Bernstein polynomials, Ars Combin. 104 (2012), 437-447.

23. T. Kim,A study on theq–Euler numbers and the fermionicq–integral of the product of several typeq–Bernstein polynomials onZp,Adv. Stud. Contemp. Math. (Kyungshang) 23

(2013), no. 1, 5-11.

24. T. Kim, B. Lee, S.-H. Lee, S.-H. Rim,A note onq–Bernstein polynomials associated with

p–adic integral onZp J. Comput. Anal. Appl. 15 (2013), no. 3, 584-592.

25. J.-W. Park, H. K. Pak, S.-H. Rim, T. Kim, S.-H. Lee,A note on theq–Bernoulli numbers andq–Bernstein polynomials,J. Comput. Anal. Appl. 15 (2013), no. 4, 722-729.

26. Y. Simsek, M. Gunay,On Bernstein type polynomials and their applications.Adv. Differ-ence Equ. 2015, 2015:79, 11 pp.

27. T. Tun¸c, E. S¸im¸sek,Some approximation properties of Szasz–Mirakyan–Bernstein opera-tors,Eur. J. Pure Appl. Math. 7 (2014), no. 4, 419-428.

Department of Mathematics, Kwangwoon University, Seoul 139-701, South Korea E-mail address: [email protected]