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.
Similar content being viewed by others
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, 17,18,19]).
As a degenerate version of Bernstein polynomials, the degenerate Bernstein polynomials 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 numbers 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.
For \(k,n \in \mathbb{Z}_{\geq 0}\), the Bernstein polynomials of degree n are defined by
where \(x \in [0,1]\).
For \(\lambda \in \mathbb{R}\), the degenerate Bernoulli polynomials of order r are defined by the generating function
When \(r=1\), \(\beta _{n,\lambda }(x)=\beta _{n,\lambda }^{(1)}(x)\) \((n \geq 0)\) are called the degenerate Bernoulli polynomials (see [3, 4]). Further, \(\beta _{n,\lambda }=\beta _{n,\lambda }(0)\) are called the degenerate Bernoulli numbers.
The falling factorial sequences are defined by
The λ-analogue of the falling factorial sequences are given by
Note that \(\lim_{\lambda \rightarrow 1}(x)_{n,\lambda }=(x)_{n}\), \(\lim_{\lambda \rightarrow 0}(x)_{n,\lambda }=x^{n}\).
It is known that the degenerate exponential function is defined by
The λ-binomial coefficients are given by
From (1.6), we have
which is equivalent to
Recently, Kim and Kim [12, 14] introduced the degenerate Bernstein polynomials of degree n, \(B_{k,n}(x|\lambda )\) \((n,k \geq 0)\), which are given by
where k is a nonnegative integer.
From (1.9), we note that
Thus, by (1.10), we easily get
where \(i,n \in \mathbb{N}\) with \(i \leq n\), \(x \in [0,1]\).
As is well known, the Stirling numbers of the second kind are defined by
In [8], the degenerate Stirling numbers of the second kind are given by
Note that \(\lim_{\lambda \rightarrow 0}S_{2,\lambda }(n,l)=S_{2}(n,l)\).
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 Bernstein polynomials and central factorial numbers of the second kind.
2 Some fundamental properties of the degenerate Bernstein polynomials
First, we observe that
Thus, by (2.1), we get the next theorem which already appeared in [12].
Theorem 2.1
For \(n,k \in \mathbb{N}\), we have
By (1.8) and (1.10), we easily get
where k is a nonnegative integer.
From (1.10), we have
Hence, by (2.2) and (2.4), we get the following theorem.
Theorem 2.2
For \(k \geq 0\), we have
On the other hand, we have
Thus, by (2.5), we get the next result.
Theorem 2.3
For \(0 \leq i \leq n+1\), we have
We observe that
Hence, by (2.6), we obtain the following identity which appeared already in [12].
Also, from (1.1) and (1.10), we have
Hence, by (2.7) and (2.8), we get the following theorem.
Theorem 2.4
For \(n,k,i \in \mathbb{N}\) with \(i \leq n\) and \(k \leq n\), we have
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 numbers and polynomials including the degenerate Bernoulli polynomials and central factorial numbers of the second kind.
From (1.2), we note that
and
Thus, by (3.1), we get
By comparing the coefficients on both sides of (3.2), we get
When \(n=1\), we have
By (1.2), we easily get
Comparing the coefficients on both sides of (3.4), we have
Taking \(x=-1\), \(\beta _{n,\lambda }(2)=(-1)^{n}\beta _{n,-\lambda }(-1)\) \((n \geq 0)\). We observe that
By (3.5), we get
It is not difficult to show that
Continuing this process, we have
Therefore, by comparing the coefficients on both sides of (3.8), we obtain the following proposition.
Proposition 1
For \(k \in \mathbb{N}\) and \(n \geq 0\), we have
In particular,
From (1.9), we have
On the other hand,
From (3.9) and (3.10), we have
Note here that (3.11) also follows from (1.10) by replacing n by \(n+k\).
From (1.2), we note that
By (3.12), we get
where \(k \in \mathbb{N}\) and \(n \geq 0\).
From (1.10) and (3.13), we have
Theorem 3.1
For \(k \in \mathbb{N}\) and \(n \geq 0\), we have
As is well known, the central factorial numbers of the second kind are defined by the generating function
Recently, the degenerate central factorial numbers \(T_{\lambda }(n,k)\) of the second kind were introduced by the generating function
By (3.15), we get
Thus, by (3.16), we get
Now, we observe that
By combining the right-hand side of (1.9) with (3.17), we obtain the following theorem.
Theorem 3.2
For \(n,k \in \mathbb{N}\cup \{0\}\) with \(n \geq k\), we have
4 A matrix representation for degenerate Bernstein polynomials
For \(\lambda \in \mathbb{R}\), let
Then \(\mathbb{P}_{n,\lambda }\) is the \(n+1\)-dimensional vector space over \(\mathbb{R}\).
For \(B_{\lambda }(x) \in \mathbb{P}_{n,\lambda }\), we note that \(B_{\lambda }(x)\) can be written as a linear combination of degenerate Bernstein basis functions:
where the constants \(C_{i,\lambda }\) depend on λ for \(i=0,1,2,\ldots ,n\).
Equation (4.1) can be written as the dot product of two vectors in the following:
Now, we can convert (4.2) to
where \(b_{i,j}(\lambda )\) are the coefficients of the power basis that are used to determine the respective degenerate Bernstein polynomials.
For example, by (1.1), we get
In the quadratic case \((n=2)\), \(B_{\lambda }(x)\) can be represented in terms of matrices by
5 Conclusions
In Sect. 2, we investigated some fundamental properties for the degenerate Bernstein polynomials. In Sect. 3, we derived some identities for the degenerate Bernstein polynomials associated with special numbers and polynomials including degenerate Bernoulli polynomials 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.
References
Acikgoz, M., Araci, S.: On the generating function of the Bernstein polynomials. AIP Conf. Proc. CP1281, 1141–1143 (2010)
Bernstein, S.N.: Bernstein’s Démonstration du théorème de Weierstrass. Comm. Soc. Math. Charkow Ser. 2 t. 13, 1–2 (1912–1913)
Carlitz, L.: A degenerate Staudt–Clausen theorem. Arch. Math. (Basel) 7, 28–33 (1956)
Carlitz, L.: Degenerate Stirling, Bernoulli and Eulerian numbers. Util. Math. 15, 51–88 (1979)
Comtet, L.: The Art of Finite and Infinite Expansions. Reidel, Dordrecht (1974) Translated from the French by J.W. Nienhuys
Farouki, R.T.: The Bernstein polynomials basis: a centennial retrospective. Comput. Aided Geom. Des. 29(6), 379–419 (2012)
Kim, D.S., Kwon, J., Dolgy, D.V., Kim, T.: On central Fubini polynomials associated with central factorial numbers of the second kind. Proc. Jangjeon Math. Soc. 21(4), 589–598 (2018)
Kim, T.: A note on degenerate Stirling polynomials of the second kind. Proc. Jangjeon Math. Soc. 20(3), 319–331 (2017)
Kim, T.: A note on central factorial numbers. Proc. Jangjeon Math. Soc. 21(4), 575–588 (2018)
Kim, T., Kim, D.S.: Identities involving degenerate Euler numbers and polynomials arising from non-linear differential equations. J. Nonlinear Sci. Appl. 9, 2086–2098 (2016)
Kim, T., Kim, D.S.: Identities of symmetry for degenerate Euler polynomials and alternating generalized falling factorial sums. Iran. J. Sci. Technol., Trans. A, Sci. 41, 939–949 (2017)
Kim, T., Kim, D.S.: Degenerate Bernstein polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. (2018). https://doi.org/10.1007/s13398-018-0594-9
Kim, T., Kim, D.S.: Identities for degenerate Bernoulli polynomials and Korobov polynomials of the first kind. Sci. China Math. (2018). https://doi.org/10.1007/s11425-018-9338-5
Kim, T., Kim, D.S.: Some identities on degenerate Bernstein and degenerate Euler polynomials. Mathematics 7(1), Article ID 47 (2019). https://doi.org/10.3390/math7010047
Kim, T., Kim, D. S.: Extended stirling numbers of the first kind associated with Daehee numbers and polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113(2), 1159–1171 (2019)
Kim, T., Kim, D.S.: Degenerate central factorial numbers of the second kind. Submitted
Kim, T., Kim, D.S., Jang, G.-W.: Differential equations associated with degenerate Cauchy numbers. Iran. J. Sci. Technol., Trans. A, Sci. (2018). https://doi.org/10.1007/s40995-018-0531-y
Kim, T., Kim, D.S., Jang, G.-W., Jang, L.-C.: Degenerate ordered Bell numbers and polynomials associated with umbral calculus. J. Nonlinear Sci. Appl. 10, 5142–5155 (2017)
Kim, T., Yao, Y., Kim, D.S., Jang, G.-W.: Degenerate r-Stirling numbers and r-Bell polynomials. Russ. J. Math. Phys. 25(1), 44–58 (2018)
Kurt, V.: Some relation between the Bernstein polynomials and second kind Bernoulli polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 23(1), 43–48 (2013)
Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea, New York (1986)
Ostrovska, S.: On the q-Bernstein polynomials. Adv. Stud. Contemp. Math. (Kyungshang) 11(2), 193–204 (2005)
Roman, S.: The Umbral Calculus. Pure and Applied Mathematics, vol. 111. Academic Press, New York (1984)
Simsek, B., Yardimci, A.: Using Bezier curves in medical applications. Filomat 30(4), 937–943 (2016)
Simsek, Y.: A new class of polynomials associated with Bernstein and beta polynomials. Math. Methods Appl. Sci. 37(5), 676–685 (2014)
Simsek, Y.: Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers. Math. Methods Appl. Sci. 38(14), 3007–3021 (2015)
Simsek, Y.: New families of special numbers for computing negative order Euler numbers and related numbers and polynomials. Appl. Anal. Discrete Math. 12(1), 1–35 (2018)
Simsek, Y.: Generating functions for the Bernstein type polynomials: a new approach to deriving identities and applications for the polynomials. Hacet. J. Math. Stat. 43(1), 1–14 (2101)
Simsek, Y., Bayad, A., Lokesha, V.: q-Bernstein polynomials related to q-Frobenius-Euler polynomials, l-functions, and q-Stirling numbers. Math. Methods Appl. Sci. 35(8), 877–884 (2012)
Acknowledgements
This paper was dedicated to the renowned mathematician Professor Gradimir V. Milovanović 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).
Author information
Authors and Affiliations
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.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Kim, T., Kim, D.S., Jang, GW. et al. A note on degenerate Bernstein polynomials. J Inequal Appl 2019, 129 (2019). https://doi.org/10.1186/s13660-019-2071-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-019-2071-1