A new class of polynomials associated with Bernstein and beta polynomials

Şimşek, Yılmaz

doi:10.1002/mma.2825

Cited by 15 publications

(23 citation statements)

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“…The Bernstein basis functions can also be defined by means of the following generating functions:

f_{double-struckB, k} (x, t) = \frac{t^{k} x^{k} e^{((), 1 - x) t}}{k!} = \sum_{n = 0}^{\infty} B_{k}^{n} (x) \frac{t^{n}}{n!},

where k = 0,1,…, n and

t \in double-struckC

and x ∈[0,1] ( cf . and see also the references cited in each of these earlier works). Remark Replacing x by

\frac{x - a}{b - a}

, Equation is modified as follows:

B_{k}^{n} (x; a, b) = ((), \begin{matrix} n \\ k \end{matrix}) {((), \frac{x - a}{b - a})}^{k} {((), \frac{b - x}{b - a})}^{n - k},

where k = 0,1,⋯, n and x ∈[ a , b ] ( cf .…”

Section: Introductionmentioning

confidence: 91%

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Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers

Şimşek

2014

Math Methods in App Sciences

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We give some alternative forms of the generating functions for the Bernstein basis functions. Using these forms,we derive a collection of functional equations for the generating functions. By applying these equations, we prove some identities for the Bernstein basis functions. Integrating these identities, we derive a variety of identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients, Pascal's rule, Vandermonde's type of convolution, the Bernoulli polynomials, and the Catalan numbers. Copyright © 2014 John Wiley & Sons, Ltd.

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“…The Bernstein basis functions can also be defined by means of the following generating functions:

f_{double-struckB, k} (x, t) = \frac{t^{k} x^{k} e^{((), 1 - x) t}}{k!} = \sum_{n = 0}^{\infty} B_{k}^{n} (x) \frac{t^{n}}{n!},

where k = 0,1,…, n and

t \in double-struckC

and x ∈[0,1] ( cf . and see also the references cited in each of these earlier works). Remark Replacing x by

\frac{x - a}{b - a}

, Equation is modified as follows:

B_{k}^{n} (x; a, b) = ((), \begin{matrix} n \\ k \end{matrix}) {((), \frac{x - a}{b - a})}^{k} {((), \frac{b - x}{b - a})}^{n - k},

where k = 0,1,⋯, n and x ∈[ a , b ] ( cf .…”

Section: Introductionmentioning

confidence: 91%

“…Recently, many authors have studied on these basis functions (cf. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]) and see also the references cited in each of these earlier works. The Bernstein basis functions B n k .x/ are defined by the following definition(cf.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers

Şimşek

2014

Math Methods in App Sciences

Self Cite

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“…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.…”

Section: Introductionmentioning

confidence: 99%