q‐Bernstein Polynomials Associated with q‐Stirling Numbers and Carlitz′s q‐Bernoulli Numbers

Kim, T.; Choi, Jeong-Hwa; Kim, Y. H.

doi:10.1155/2010/150975

Cited by 19 publications

(21 citation statements)

References 9 publications

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“…If we replace x by − x and b = 0 in ,

Y_{k}^{n} (MathClass-bin- x MathClass-punc; 0) MathClass-rel= 2 {(MathClass-bin- x)}^{k} {(1 MathClass-bin- x)}^{n MathClass-bin- k MathClass-bin- b} MathClass-punc.

Therefore,

Y_{k}^{n} (MathClass-bin- x MathClass-punc; 0) MathClass-rel= \frac{2 {(MathClass-bin- 1)}^{k}}{((), \begin{matrix} falsenonefalsearray \\ arraycenter n \\ arraycenter k \end{matrix})} B_{k}^{n} (x) MathClass-punc,

where

B_{k}^{n} (x)

denotes the Bernstein basis functions , which is defined by

B_{k}^{n} (x) MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ arraycenter n \\ arraycenter k \end{matrix}) x^{k} {(1 MathClass-bin- x)}^{n MathClass-bin- k} MathClass-punc,

where 0 ≤ k ≤ n and x ∈ [0,1] ( cf . ).…”

Section: The Polynomial Ykn(xmathclass-punc;b)mentioning

confidence: 97%

See 1 more Smart Citation

A new class of polynomials associated with Bernstein and beta polynomials

Şimşek

2013

Math. Meth. Appl. Sci.

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The purpose of this paper is to define a new class polynomials. Special cases of these polynomials give many famous family of the Bernstein type polynomials and beta polynomials. We also construct generating functions for these polynomials. We investigate some fundamental properties of these functions and polynomials. Using functional equations and generating functions, we derive various identities related to theses polynomials. We also construct interpolation function that interpolates these polynomials at negative integers. Finally, we give a matrix representations of these polynomials. Copyright © 2013 John Wiley & Sons, Ltd.

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“…If we replace x by − x and b = 0 in ,

Y_{k}^{n} (MathClass-bin- x MathClass-punc; 0) MathClass-rel= 2 {(MathClass-bin- x)}^{k} {(1 MathClass-bin- x)}^{n MathClass-bin- k MathClass-bin- b} MathClass-punc.

Therefore,

Y_{k}^{n} (MathClass-bin- x MathClass-punc; 0) MathClass-rel= \frac{2 {(MathClass-bin- 1)}^{k}}{((), \begin{matrix} falsenonefalsearray \\ arraycenter n \\ arraycenter k \end{matrix})} B_{k}^{n} (x) MathClass-punc,

where

B_{k}^{n} (x)

denotes the Bernstein basis functions , which is defined by

B_{k}^{n} (x) MathClass-rel= ((), \begin{matrix} falsenonefalsearray \\ arraycenter n \\ arraycenter k \end{matrix}) x^{k} {(1 MathClass-bin- x)}^{n MathClass-bin- k} MathClass-punc,

where 0 ≤ k ≤ n and x ∈ [0,1] ( cf . ).…”

Section: The Polynomial Ykn(xmathclass-punc;b)mentioning

confidence: 97%

“…Kim et al [5] gave a matrix representation of the (q-) Bernstein polynomials. They gave examples for this matrix representation.…”

Section: Remarkmentioning

confidence: 99%

A new class of polynomials associated with Bernstein and beta polynomials

Şimşek

2013

Math. Meth. Appl. Sci.

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show abstract

“…, (see [4], [5], [6], [19], [20], [17], [21], [31], [38], [39], [40] for details and related facts). Note that lim [19] is actually motivated the authors to write this paper and they have extended all results given in [19] to modified q-Bernstein polynomials of several variables.…”

Section: Introduction Definitions and Notationsmentioning

confidence: 99%

On the Properties of q-Bernstein-type Polynomials

Aracı¹,

Açıkgöz²,

Bagdasaryan³

2017

Appl. Math. Inf. Sci.

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The aim of this paper is to give a new approach to modified q-Bernstein polynomials for functions of several variables. By using these polynomials, the recurrence formulas and some new interesting identities related to the second Stirling numbers and generalized Bernoulli polynomials are derived. Moreover, the generating function, interpolation function of these polynomials of several variables and also the derivatives of these polynomials and their generating function are given. Finally, we get new interesting identities of modified q-Bernoulli numbers and q-Euler numbers applying p-adic q-integral representation on Z p and p-adic fermionic q-invariant integral on Z p , respectively, to the inverse of q-Bernstein polynomials.

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“…This theory has many applications in different areas in mathematics and physics, see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Throughout this paper we set I = [0, 1] and k ∈ N. Taking a k-dimensional simplex ∆ k :…”

Section: Introductionmentioning

confidence: 99%