Abstract
Here we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials, which are motivated by Simsek’s recent work ‘Generating functions for unification of the multidimensional Bernstein polynomials and their applications’ (Simsek in Filomat 30(7):1683–1689, 2016, Math Methods Appl Sci 1–12, 2018) and Carlitz’s degenerate Bernoulli polynomials. We derived their generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.
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1 Introduction
For \(\lambda \in \mathbb {R}\), the degenerate Bernoulli polynomials of order k are defined by Carlitz as
Note that \(\lim _{\lambda \rightarrow 0} \beta _{n,\lambda }^{(k)} (x) = B_n^{(k)}(x)\) are the ordinary Bernoulli polynomials of order k given by
It is known that the Stirling numbers of the second kind are defined as
where \((x)_l = x(x-1) \cdots (x-l+1)\), \((l \ge 1),\)\((x)_0=1\).
For \(\lambda \in \mathbb {R}\), the \((x)_{n,\lambda }\) is defined as
In [8,9,10], \({x \atopwithdelims ()n}_\lambda \) is defined as
Thus, by (1.4), we get
From (1.5), we note that
The degenerate Stirling numbers of the second kind are defined by
By (1.7), we easily get
In this paper, we use the following notation.
The Bernstein polynomials of degree n is defined by
Let C[0, 1] be the space of continuous functions on [0, 1]. The Bernstein operator of order n for f is given by
where \(n \in \mathbb {N} \cup \{0\}\) and \(f \in C[0,1]\), (see [3, 6, 15]).
A Bernoulli trial involves performing a random experiment 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 probability \(P_n(k)\) of k successes in n independent Bernoulli trials is given by the binomial probability law:
From the definition of Bernstein polynomials we note that Bernstein basis is probability mass of binomial distribution with parameter \((n, x=p)\).
Here we would like to mention that in [18] the author studies the so-called Bernstein type polynomials, which are different from our degenerate Bernstein polynomials, and derives many interesting results on those polynomials.
Let us assume that the probability of success in an experiment is p. We wondered if we can say the probability of success in the ninth trial is still p after failing eight times in a ten trial experiment. Because there’s a psychological burden to be successful.
It seems plausible that the probability is less than p. This speculation motivated the study of the degenerate Bernstein polynomials associated with the probability distribution.
In this paper, we consider the degenerate Bernstein polynomials as a degenerate version of Bernstein polynomials. We derive their generating function, symmetric identities, recurrence relations, and some connections with generalized falling factorial polynomials, higher-order degenerate Bernoulli polynomials and degenerate Stirling numbers of the second kind.
2 Degenerate Bernstein polynomials
For \(\lambda \in \mathbb {R}\) and \(k,n \in \mathbb {N}\cup \{0\}\), with \(k \le n\), we define the degenerate Bernstein polynomials of degree n which are given by
Note that \(\lim _{\lambda \rightarrow 0} B_{k,n}(x|\lambda ) = B_{k,n}(x)\), \((0 \le k \le n)\). From (2.1), we derive the generating function of \(B_{k,n}(x|\lambda )\), which are given by
Therefore, by (2.2), we obtain the following theorem.
Theorem 2.1
For \(x \in [0,1]\) and \(k=0,1,2,\ldots ,\) we have
From (2.1), we note that
By replacing x by \(1-x\), we get
where \(n,k \in \mathbb {N} \cup \{0\}\), with \(0 \le k \le n\).
Therefore, by (2.4), we obtain the following theorem.
Theorem 2.2
(Symmetric identities) For \(n,k \in \mathbb {N} \cup \{0\}\), with \(k \le n\), and \(x \in [0,1]\), we have
Now, we observe that
Therefore, by (2.5), we obtain the following theorem.
Theorem 2.3
For \(k \in \mathbb {N} \cup \{0\}\), \(n \in \mathbb {N}\), with \(k \le n-1\), and \(x \in [0,1]\), we have
From (2.1), we have
Therefore, by (2.7), we obtain the following theorem.
Theorem 2.4
For \(n,k \in \mathbb {N}\), with \(k \le n\), we have
For \(0 \le k \le n\), we get
Therefore, by (2.8), we obtain the following theorem.
Theorem 2.5
(Recurrence formula). For \(k,n \in \mathbb {N}\), with \(k \le n-1\), \(x \in [0,1]\), we have
Remark 1
For \(n\in \mathbb {N}\), we have
Now, we observe that
Similarly, we have
From (2.10), we note that
where \(n,i \in \mathbb {N}\), with \( i \le n\), and \(x \in [0,1]\).
Therefore, by (2.11), we obtain the following theorem.
Theorem 2.6
For \(n,i \in \mathbb {N}\), with \( i \le n\), and \(x \in [0,1]\), we have
From Theorem 2.1, we note that
On the other hand,
Therefore, by (2.12) and (2.13), we obtain the following theorem.
Theorem 2.7
For \(n,k \in \mathbb {N}\cup \{0\}\) with \(n \ge k\), we have
Let \(\Delta \) be the shift difference operator with \(\Delta f(x) = f(x+1)-f(x)\). Then we easily get
Let us take \(f(x)=(x)_{m,\lambda }\), \((m \ge 0)\). Then, by (2.14), we get
For more details on (2.14) and (2.15), we let the reader refer to Chapter 7 of the book [12].
From (1.7), we note that
Thus, by comparing the coefficients on both sides of (2.16), we have
By (2.17), we get
From Theorem 7 and (2.18), we obtain the following corollary.
Corollary 2.8
For \(n,k \in \mathbb {N}\cup \{0\}\) with \(n \ge k\), we have
Now, we observe that
On the other hand,
Therefore, by (2.19) and (2.20), we obtain the following theorem.
Theorem 2.9
For \(n \ge 0\), we have
By Theorem 2.9, we easily get
From Theorem 2.6, we have the following theorem.
Theorem 2.10
For \(n,i \in \mathbb {N}\), with \(i \le n\), and \(x \in [0,1]\), we have
Change history
11 March 2019
Unfortunately, erratua appear in the statement corresponding Theorems 2.6 and 2.10 in the original paper .
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We would like to thank the referee for his valuable comments and suggestions.
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Kim, T., Kim, D.S. Degenerate Bernstein polynomials. RACSAM 113, 2913–2920 (2019). https://doi.org/10.1007/s13398-018-0594-9
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DOI: https://doi.org/10.1007/s13398-018-0594-9