Volume 38, Issue 14 pp. 3007-3021
Research Article

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

Yilmaz Simsek

Corresponding Author

Yilmaz Simsek

Department of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya, Turkey

Correspondence to: Yilmaz Simsek, Department of Mathematics, Faculty of Science University of Akdeniz TR-07058 Antalya, Turkey.

E-mail: [email protected]

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First published: 10 September 2014
Citations: 28

Abstract

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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