Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions

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Simsek Y.

FIXED POINT THEORY AND APPLICATIONS, vol.2013, 2013 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 2013
  • Publication Date: 2013
  • Doi Number: 10.1186/1687-1812-2013-80
  • Journal Name: FIXED POINT THEORY AND APPLICATIONS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Keywords: Bernstein polynomials, generating functions, functional equations, integral transforms, differential equations
  • Akdeniz University Affiliated: Yes

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.