A note on degenerate Bernstein polynomials

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R E S E A R C H

Open Access

A note on degenerate Bernstein polynomials

Taekyun Kim

1

_{, Dae San Kim}

2

_{, Gwan-Woo Jang}

1

_{and Jongkyum Kwon}

3*

*_{Correspondence:} mathkjk26@gnu.ac.kr 3_{Department of Mathematics}

Education and ERI, Gyeongsang National University, Jinju, Republic of Korea

Full list of author information is available at the end of the article

Abstract

Recently, degenerate Bernstein polynomials have been introduced by Kim and Kim. In this paper, we investigate some properties and identities for the degenerate Bernstein polynomials associated with special numbers and polynomials including degenerate Bernstein polynomials and central factorial numbers of the second kind.

MSC: 11B75; 11B83

Keywords: Degenerate Bernstein polynomials; Degenerate Bernoulli polynomials; Central factorial numbers of the second kind

1 Introduction

Bernstein polynomials were ﬁrst used by Bernstein in a constructive proof for the Stone– Weierstrass approximation theorem (see [2,6,21]). With the advent of computer graphics, Bernstein polynomials, restricted to the interval [0, 1], became important in the form of Bézier curves (see [6]). The Bernstein polynomials are the mathematical basis for Bézier curves, which are frequently used in the mathematical ﬁeld of numerical analysis (see [6,

24]). The study of degenerate versions of special numbers and polynomials began with the papers by Carlitz (see [3,4]). Kim and his research colleagues have been studying various degenerate numbers and polynomials by means of generating functions, Fourier series, combinatorial methods, umbral calculus,p-adic analysis, and diﬀerential equations (see [10,11,13,17–19]).

As a degenerate version of Bernstein polynomials, the degenerate Bernstein polynomi-als were introduced recently (see (1.9)). Here we will study for the degenerate Bernstein polynomials some fundamental properties and identities associated with special numbers and polynomials including degenerate Bernoulli polynomials and central factorial num-bers of the second kind. Also, in the last section we will consider a matrix representation for those polynomials. For some recent works related to the present paper, the reader may want to see [14,20,22,25,27,29]. The rest of this section is devoted to reviewing what we need in the following sections.

Fork,n∈Z≥0, the Bernstein polynomials of degreenare deﬁned by

Bk,n(x) =

n k

xk(1 –x)n–k (see [1,2,6,21,26,28]), (1.1)

wherex∈[0, 1].

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Forλ∈R, the degenerate Bernoulli polynomials of orderrare deﬁned by the generating function

t

(1 +λt)λ1_{– 1}

r

(1 +λt)xλ₌

∞

n=0

β_n(r_,)_λ(x)t

n

n!. (1.2)

Whenr= 1,βn,λ(x) =β_n(1)_,λ(x) (n≥0) are called the degenerate Bernoulli polynomials (see

[3,4]). Further,βn,λ=βn,λ(0) are called the degenerate Bernoulli numbers.

The falling factorial sequences are deﬁned by

(x)0= 1, (x)n=x(x– 1)(x– 2)· · ·(x–n+ 1) (n≥1) (see [15,23]). (1.3)

Theλ-analogue of the falling factorial sequences are given by

(x)0,λ= 1, (x)n,λ=x(x–λ)(x– 2λ)· · ·

x– (n– 1)λ (n≥1) (see [7,15,16]).

(1.4)

Note thatlimλ→1(x)n,λ= (x)n,limλ→0(x)n,λ=xn.

It is known that the degenerate exponential function is deﬁned by

(1 +λt)xλ₌

∞

n=0 (x)n,λ

tn

n! (see [16]). (1.5)

Theλ-binomial coeﬃcients are given by

x n

λ

=(x)n,λ

n! =

x(x–λ)(x– 2λ)· · ·(x– (n– 1)λ)

n! (see [8,16]). (1.6)

From (1.6), we have

x+y

n

λ

=

n

l=0

x l

λ

y n–l

λ

(see [8,12]), (1.7)

which is equivalent to

(x+y)n,λ=

n

l=0

n

l

(x)l,λ(y)n–l,λ (see [12,14]). (1.8)

Recently, Kim and Kim [12, 14] introduced the degenerate Bernstein polynomials of degreen,Bk,n(x|λ) (n,k≥0), which are given by

(x)k,λ k! t

k_{(1 +}_λ_t₎1–x_λ ₌

∞

n=k

Bk,n(x|λ)

tn

n! (see [12]), (1.9)

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From (1.9), we note that

Bk,n(x|λ) =

n k

(x)k,λ(1 –x)n–k,λ (n≥k≥0) (see [12]). (1.10)

Thus, by (1.10), we easily get

(x)i,λ=

1 (1 –λi)n–i,λ

n

k=i

_k

i

n

i

Bk,n(x|λ) (see [12]), (1.11)

wherei,n∈Nwithi≤n,x∈[0, 1].

As is well known, the Stirling numbers of the second kind are deﬁned by

xn=

n

l=0

S2(n,l)(x)l (n≥0) (see [7,12]). (1.12)

In [8], the degenerate Stirling numbers of the second kind are given by

(x)n,λ=

n

l=0

S2,λ(n,l)(x)l (n≥0). (1.13)

Note thatlimλ→0S2,λ(n,l) =S2(n,l).

The degenerate Bernstein polynomials have been introduced recently by Kim and Kim. In this paper, we investigate some properties and identities for the degenerate Bernstein polynomials associated with special numbers and polynomials including degenerate Bern-stein polynomials and central factorial numbers of the second kind.

2 Some fundamental properties of the degenerate Bernstein polynomials

First, we observe that

1 –x– (n–k– 1)λBk,n–1(x|λ) +

x– (k– 1)λBk–1,n–1(x|λ) =

n– 1

k

(x)k,λ(1 –x)n–k,λ+

n– 1

k– 1

(x)k,λ(1 –x)n–k,λ

=

n– 1

k

+

n– 1

k– 1

(x)k,λ(1 –x)n–k,λ

=

n k

(x)k,λ(1 –x)n–k,λ=Bk,n(x|λ) (n,k∈N). (2.1)

Thus, by (2.1), we get the next theorem which already appeared in [12].

Theorem 2.1 For n,k∈N,we have

1 –x– (n–k– 1)λBk,n–1(x|λ) +

x– (k– 1)λBk–1,n–1(x|λ)

=Bk,n(x|λ). (2.2)

By (1.8) and (1.10), we easily get

k

i=0

Bi,k(x|λ) = k

i=0

k

i

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wherekis a nonnegative integer.

From (1.10), we have

(x–iλ)Bi,n(x|λ) = (x–iλ)

n

i

(x)i,λ(1 –x)n–i,λ

=

n

i

(x)i+1,λ(1 –x)n–i,λ

=

_n

i

n+1

i+1

n+ 1

i+ 1

(x)i+1,λ(1 –x)n+1–(i+1),λ

=

_n

i

_n₊₁

i+1

Bi+1,n+1(x|λ). (2.4)

Hence, by (2.2) and (2.4), we get the following theorem.

Theorem 2.2 For k≥0,we have

k

i=0

Bi,k(x|λ) = k

i=0

k

i

(x)i,λ(1 –x)k–i,λ= (x+ 1 –x)k,λ= (1)k,λ,

(x–iλ)Bi,n(x|λ) =

i+ 1

n+ 1Bi+1,n+1(x|λ).

On the other hand, we have

1 –x– (n–i)λBi,n(x|λ)

=

n i

(x)i,λ(1 –x)n–i,λ

1 –x– (n–i)λ

=

n i

(x)i,λ(1 –x)n+1–i,λ= _n

i

_n₊₁

i

n+ 1

i

(x)i,λ(1 –x)n+1–i,λ

=

n i

_n₊₁

i

Bi,n+1(x|λ) (i,n∈N). (2.5)

Thus, by (2.5), we get the next result.

Theorem 2.3 For0≤i≤n+ 1,we have

1 –x– (n–i)λBi,n(x|λ) =

n+ 1 –i

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We observe that

1

n i

Bi,n(x|λ) +

1

n i+1

Bi+1,n(x|λ)

= (x)i,λ(1 –x)n–i,λ+ (x)i+1,λ(1 –x)n–i–1,λ

= (x)i,λ(1 –x)n–i–1,λ

1 –x– (n–i– 1)λ+x–iλ

= (x)i,λ(1 –x)n–i–1,λ

1 – (n– 1)λ

=1 – (n_n_–1– 1)λ

i

n– 1

i

(x)i,λ(1 –x)n–i–1,λ

=1 – (n_n_–1– 1)λ

i

Bi,n–1(x|λ). (2.6)

Hence, by (2.6), we obtain the following identity which appeared already in [12].

1 – (n– 1)λBi,n–1(x|λ) =

n– 1

i

Bi,n(x|λ) n

i

+Bi+1,n_n(x|λ)

i+1

=n–i

n Bi,n(x|λ) + i+ 1

n Bi+1,n(x|λ) (n∈N,i≥0). (2.7)

Also, from (1.1) and (1.10), we have

Bk,n(x|λ)

Bk,n(x)

=

k–1

l=0

1 –λl

x n–k–1

l=0

1 –λ l

1 –x

. (2.8)

Hence, by (2.7) and (2.8), we get the following theorem.

Theorem 2.4 For n,k,i∈Nwith i≤n and k≤n,we have

1 – (n– 1)λBi,n–1(x|λ) =

n–i

n Bi,n(x|λ) + i+ 1

n Bi+1,n(x|λ), Bk,n(x|λ)

Bk,n(x)

=

k–1

l=0

1 –λl

x n–k–1

l=0

1 –λ l

1 –x

.

3 Some identities for degenerate Bernstein polynomials associated with special numbers and polynomials

Here in this section, we are going to derive some identities associated with special num-bers and polynomials including the degenerate Bernoulli polynomials and central factorial numbers of the second kind.

From (1.2), we note that

t

(1 +λt)λ1 – 1 =

∞

n=0

βn,λ tn

n! (3.1)

and

βn,λ(x) =

n

l=0

n

l

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Thus, by (3.1), we get

t=

_∞

l=0

βl,λ tl

l!

_∞

m=0 (1)m,λ

tm

m!– 1

=

∞

n=0

_n

l=0

n

l

(1)n–l,λβl,λ–βn,λ

tn

n!. (3.2)

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

n

l=0

n

l

(1)n–l,λβl,λ–βn,λ= ⎧ ⎨ ⎩

1, ifn= 1,

0, ifn> 1, β0,λ= 1 (see [3,4]). (3.3)

Whenn= 1, we have

β1,λ(1) =

1

l=0

1

l

(1)1–l,λβl,λ= 1 +β1,λ.

By (1.2), we easily get

∞

n=0

βn,λ(1 –x) tn

n!=

t

(1 +λt)1λ_{– 1}

(1 +λt)1–xλ

= –t

(1 + (–λ)(–t))–λ1– 1

1 + (–λ)(–t)– x λ

=

∞

n=0

βn,–λ(x)(–1)n tn

n!. (3.4)

Comparing the coeﬃcients on both sides of (3.4), we have

βn,λ(1 –x) = (–1)nβn,–λ(x) (n≥0).

Takingx= –1,βn,λ(2) = (–1)nβn,–λ(–1) (n≥0). We observe that

∞

n=0

βn,λ(2) tn

n! =

t

(1 +λt)1λ_{– 1}

(1 +λt)2λ

= t

(1 +λt)1λ– 1

(1 +λt)1λ_{– 1 + 1}_{(1 +}_λ_t₎1λ

=t(1 +λt)1λ₊ t (1 +λt)1λ– 1

(1 +λt)λ1

=t

∞

n=0 (1)n,λ

tn n!+

∞

n=0

βn,λ(1) tn n!

=

∞

n=0

n(1)n–1,λ+βn,λ(1) tn

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By (3.5), we get

βn,λ(2) =n(1)n–1,λ+βn,λ(1) (n≥1). (3.6)

It is not diﬃcult to show that

t

(1 +λt)λ1 _{– 1}

(1 +λt)xλ

= t

(1 +λt)1λ_{– 1}

(1 +λt)x–1λ _{(1 +}_λ_t₎ 1

λ_{– 1 + 1}

=t(1 +λt)x–1λ + t (1 +λt)λ1 _{– 1}

(1 +λt)x–1λ

=t(1 +λt)x–1λ ₊_t_{(1 +}_λ_t₎ x–2

λ ₊ t

(1 +λt)1λ_{– 1}

(1 +λt)x–2λ

=t(1 +λt)x–1λ +_t(1 +_λ_t) x–2

λ +_t(1 +_λ_t) x–3

λ + t

(1 +λt)λ1 _{– 1}

×(1 +λt)x–3λ _. _(3.7)

Continuing this process, we have

∞

n=0

βn,λ(x) tn

n! =t

k

i=1

(1 +λt)x–iλ ₊ t (1 +λt)1λ_{– 1}

(1 +λt)x–kλ

=t

k

i=1

∞

n=0

(x–i)n,λ tn

n!+

∞

n=0

βn,λ(x–k) tn

n!

=

∞

n=1

_k

i=1

n(x–i)n–1,λ

tn

n!+

∞

n=0

βn,λ(x–k) tn

n!. (3.8)

Therefore, by comparing the coeﬃcients on both sides of (3.8), we obtain the following proposition.

Proposition 1 For k∈Nand n≥0,we have

βn,λ(x) =

k

i=1

n(x–i)n–1,λ+βn,λ(x–k).

In particular,

βn,λ(k) =

k

i=1

(8)

From (1.9), we have

(x)k,λ(1 +λt)

1–x λ ₌k!

tk

1

k!(x)k,λt

k_{(1 +}_λ_t₎1–x_λ

=k!

tk

∞

n=k

Bk,n(x|λ)

tn

n!=k!

∞

n=0

Bk,n+k(x|λ)

n! (n+k)!

tn

n!

=

∞

n=0 1

_n₊_k

k

Bk,n+k(x|λ)

tn

n!. (3.9)

On the other hand,

(x)k,λ(1 +λt)

1–x λ =

∞

n=0

(x)k,λ(1 –x)n,λ tn

n!. (3.10)

From (3.9) and (3.10), we have

(x)k,λ(1 –x)n,λ=

1

n+k k

Bk,n+k(x|λ) (n,k≥0). (3.11)

Note here that (3.11) also follows from (1.10) by replacingnbyn+k. From (1.2), we note that

(x)k,λ(1 +λt)

1–x λ ₌(x)k,λ

t

(1 +λt)1λ– 1

(1 +λt)1–xλ _{(1 +}_λ_t₎1λ_{– 1}

=(x)k,λ

t

(1 +λt)1λ– 1

(1 +λt)2–xλ – t (1 +λt)λ1 – 1

(1 +λt)1–xλ

=(x)k,λ

t

_∞

n=1

βn,λ(2 –x) –βn,λ(1 –x)

tn

n!

= (x)k,λ

∞

n=0

βn+1,λ(2 –x) –βn+1,λ(1 –x) n+ 1

tn

n!. (3.12)

By (3.12), we get

(x)k,λ(1 –x)n,λ= (x)k,λ

βn+1,λ(2 –x) –βn+1,λ(1 –x)

n+ 1 , (3.13)

wherek∈Nandn≥0.

From (1.10) and (3.13), we have

Bk,n+k(x|λ) =

n+k

k

(x)k,λ

βn+1,λ(2 –x) –βn+1,λ(1 –x)

n+ 1 .

Theorem 3.1 For k∈Nand n≥0,we have

Bk,n+k(x|λ) =

n+k

k

(x)k,λ

βn+1,λ(2 –x) –βn+1,λ(1 –x)

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As is well known, the central factorial numbers of the second kind are deﬁned by the generating function

1

k!

e12t_–_e–12tk₌

∞

n=k

T(n,k)t

n

n! (k≥0) (see [5,9]). (3.14)

Recently, the degenerate central factorial numbersTλ(n,k) of the second kind were in-troduced by the generating function

1

k!

(1 +λt)2λ1 _{– (1 +}_λ_t₎–2λ1 k₌

∞

n=k

Tλ(n,k)t

n

n! (see [16]). (3.15)

By (3.15), we get

∞

n=k

lim λ→0Tλ(n,k)

tn

n! =λlim→0 1

k!

(1 +λt)2λ1 – (1 +_λ_t)– 1 2λk

= 1

k!

e12t_–_e–12tk₌

∞

n=k

T(n,k)t

n

n!. (3.16)

Thus, by (3.16), we get

lim

λ→0Tλ(n,k) =T(n,k) (n,k≥0) (see [9]).

Now, we observe that

(x)k,λ k! t

k_{(1 +}_λ_t₎1–x_λ ₌(x)k,λ k!

(1 +λt)2λ1 _{– (1 +}_λ_t₎–2λ1k ×

t

(1 +λt)1λ_{– 1}

k

(1 +λt) 1–x+k₂

λ

= (x)k,λ

_∞

m=k

Tλ(m,k)t

m

m!

_∞

l=0

β_l(_,k_λ)

1 –x+k 2

tl l!

=

∞

n=k

(x)k,λ

n

m=k

n m

Tλ(m,k)β_n(k_–)_m_,_λ

1 –x+k 2

tn

n!. (3.17)

By combining the right-hand side of (1.9) with (3.17), we obtain the following theorem.

Theorem 3.2 For n,k∈N∪ {0}with n≥k,we have

Bk,n(x|λ) = (x)k,λ

n

m=k

n m

Tλ(m,k)β_n(k_–)_m_,_λ

1 –x+k 2

.

4 A matrix representation for degenerate Bernstein polynomials

Forλ∈R, let

Pn,λ=

pλ(x)|pλ(x) =

n

i=0

(x)i,λCi,λ∈R[x]

.

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ForBλ(x)∈Pn,λ, we note thatBλ(x) can be written as a linear combination of degenerate

Bernstein basis functions:

Bλ(x) =C0,λB0,n(x|λ) +C1,λB1,n(x|λ) +C2,λB2,n(x|λ) +· · ·

+Cn,λBn,n(x|λ), (4.1)

where the constantsCi,λdepend onλfori= 0, 1, 2, . . . ,n.

Equation (4.1) can be written as the dot product of two vectors in the following:

Bλ(x) =

B0,n(x|λ) B1,n(x|λ) · · · Bn,n(x|λ)

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

C0,λ C1,λ

.. .

Cn,λ ⎞ ⎟ ⎟ ⎟ ⎟

⎠. (4.2)

Now, we can convert (4.2) to

Bλ(x) =

1 x · · · xn

×

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

b0,0(λ) 0 0 0 · · · 0 0

b1,0(λ) b1,1(λ) b1,2(λ) b1,3(λ) · · · b1,n–1(λ) b1,n(λ)

b2,0(λ) b2,1(λ) b2,2(λ) b2,3(λ) · · · b2,n–1(λ) b2,n(λ)

..

. ...

bn–1,0(λ) bn–1,1(λ) bn–1,2(λ) bn–1,3(λ) · · · bn–1,n–1(λ) bn–1,n(λ)

bn,0(λ) bn,1(λ) bn,2(λ) bn,3(λ) · · · bn,n–1(λ) bn,n(λ)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

×

⎛ ⎜ ⎜ ⎜ ⎜ ⎝

C0,λ C1,λ

.. .

Cn,λ ⎞ ⎟ ⎟ ⎟ ⎟ ⎠,

wherebi,j(λ) are the coeﬃcients of the power basis that are used to determine the

respec-tive degenerate Bernstein polynomials. For example, by (1.1), we get

B0,2(x|λ) =

2 0

(x)0,λ(1 –x)2,λ= (1 –x)(1 –x–λ) = (1 –x)2–λ(1 –x)

=x2+ (λ– 2)x+ 1 –λ,

B1,2(x|λ) =

2 1

(x)1,λ(1 –x)1,λ= 2x(1 –x) = 2x– 2x2,

B2,2(x|λ) =

2 2

(11)

In the quadratic case (n= 2),Bλ(x) can be represented in terms of matrices by

Bλ(x) =

1 x x2 ⎛_⎜

⎝

1 –λ 0 0

λ– 2 2 –λ

1 –2 1

⎞ ⎟ ⎠

⎛ ⎜ ⎝

C0,λ C1,λ C2,λ

⎞ ⎟ ⎠.

5 Conclusions

In Sect.2, we investigated some fundamental properties for the degenerate Bernstein poly-nomials. In Sect.3, we derived some identities for the degenerate Bernstein polynomials associated with special numbers and polynomials including degenerate Bernoulli poly-nomials and central factorial numbers of the second kind. In many applications, a matrix formulation for the Bernstein polynomials is useful. So, in Sect.4, we studied some further properties of the matrix representation for degenerate Bernstein polynomials.

Acknowledgements

This paper was dedicated to the renowned mathematician Professor Gradimir V. Milovanovi´c on the occasion of his 70th anniversary. Also, we would like to thank the referees whose suggestions and comments helped improve the original manuscript greatly.

Funding

This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (No. 2017R1E1A1A03070882).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Conceptualization, TK; Formal analysis, DSK and TK; Investigation, DSK and TK; Methodology, DSK and TK; Project administration, JK and TK; Supervision, DSK and TK; Publication fee payment, JK; Writing-original draft, TK; Writing-review and editing, DSK. All authors contributed equally to the manuscript, read, and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Kwangwoon University, Seoul, Republic of Korea.}2_{Department of Mathematics, Sogang}

University, Seoul, Republic of Korea.3_{Department of Mathematics Education and ERI, Gyeongsang National University,}

Jinju, Republic of Korea.

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Received: 5 February 2019 Accepted: 15 April 2019

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