Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis function
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RESEARCH
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Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions Yilmaz Simsek* *
Correspondence: [email protected] Department of Mathematics, Faculty of Science, University of Akdeniz, Antalya, TR-07058, Turkey
Abstract The main aim of this paper is to provide a novel approach to deriving identities for the Bernstein polynomials using functional equations. We derive various functional equations and differential equations using generating functions. Applying these equations, we give new proofs for some standard identities for the Bernstein basis functions, including formulas for sums, alternating sums, recursion, subdivision, degree raising, differentiation and a formula for the monomials in terms of the Bernstein basis functions. We also derive many new identities for the Bernstein basis functions based on this approach. Moreover, by applying the Laplace transform to the generating functions for the Bernstein basis functions, we obtain some interesting series representations for the Bernstein basis functions. MSC: 14F10; 12D10; 26C05; 26C10; 30B40; 30C15; 44A10 Keywords: Bernstein polynomials; generating functions; functional equations; integral transforms; differential equations
1 Introduction The Bernstein polynomials have many applications: in approximations of functions, in statistics, in numerical analysis, in p-adic analysis and in the solution of differential equations. It is also well known that in Computer Aided Geometric Design (CAGD) polynomials are often expressed in terms of the Bernstein basis functions. These polynomials are called Bezier curves and surfaces. Convexity and its generalization play an important role in the theory of Bernstein polynomials. Therefore, a fixed point theorem and its applications are also very important in the theory of Bezier curves and surfaces. Many of the known identities for the Bernstein basis functions are currently derived in an ad hoc fashion, using either the binomial theorem, the binomial distribution, tricky algebraic manipulations or blossoming. The main purpose of this work is to construct novel functional equations for the Bernstein polynomials. Using these functional equations and the Laplace transform, we develop a novel approach both to standard and to new identities for the Bernstein polynomials. Thus these polynomial identities are just the residue of a much more powerful system of functional equations. The remainder of this study is organized as follows. We find several functional equations and differential equations for the Bernstein basis functions using generating functions. From these equations, many properties of the Bernstein basis functions are then © 2013 Simsek; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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