Combinatorial identities associated with Bernstein type basis functions

In this paper, we give some identities and relations for the Bernstein basis functions and the beta type polynomials. Integrating these identities, we derive many identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients and the Catalan numbers. We also give remarks and comments on these identities.

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

Mathematical Methods in the Applied Sciences ◽

10.1002/mma.3276 ◽

2014 ◽

Vol 38 (14) ◽

pp. 3007-3021 ◽

Cited By ~ 13

Author(s):

Yilmaz Simsek

Keyword(s):

Catalan Numbers ◽

Basis Functions ◽

Binomial Coefficients ◽

Bernstein Basis ◽

Combinatorial Sums ◽

Bernstein Basis Functions

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Using Bezier curves in medical applications

Filomat ◽

10.2298/fil1604937s ◽

2016 ◽

Vol 30 (4) ◽

pp. 937-943 ◽

Cited By ~ 7

Author(s):

Buket Simsek ◽

Ahmet Yardimci

Keyword(s):

Cross Sections ◽

Biomedical Applications ◽

Medical Physics ◽

Binomial Coefficients ◽

Medical Applications ◽

Bernstein Basis ◽

Bezier Curves ◽

Bézier Curves ◽

Combinatorial Sums ◽

Bernstein Basis Functions

In this paper we survey the 3D reconstruction of an object from its 2D cross-sections has many applications in different fields of sciences such as medical physics and biomedical applications. The aim of this paper is to give not only the Bezier curves in medical applications, but also by using generating functions for the Bernstein basis functions and their identities, some combinatorial sums involving binomial coefficients are deriven. Finally, we give some comments related to the above areas.

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A note on generating functions for the unification of the Bernstein type basis functions

Filomat ◽

10.2298/fil1604985k ◽

2016 ◽

Vol 30 (4) ◽

pp. 985-992 ◽

Cited By ~ 8

Author(s):

Irem Kucukoglu ◽

Yilmaz Simsek

Keyword(s):

Generating Function ◽

Generating Functions ◽

Basis Functions ◽

Bernstein Basis ◽

Bernstein Type ◽

Bernstein Basis Functions

In [3], Simsek unified generating function of the Bernstein basis functions. In this paper, by using knot sequence, we rewrite generating functions for the unification of the Bernstein type basis functions. By using these generating functions, we also find generating function for the Hermite type numbers. We investigate some properties of this functions and these basis. Finally, we simulate these polynomials with their plots for some selected numerical values.

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