Deriving Novel Formulas and Identities for the Bernstein Basis Functions and Their Generating Functions

  • Yilmaz Simsek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8177)


By using the generating functions for the Bernstein basis functions, we derive various functional equations, differential equations and second order partial differential equations. By using these equations, we give new proofs of various identities, relations, integrals and derivatives of the Bernstein basis functions. Using second order partial differentia equation of the generating functions, we also obtain new derivative formulas for the Bernstein basis functions. By applying the Fourier transform and the Laplace transform to the generating functions, we derive series representations for the Bernstein basis functions. We also give the p-adic Volkenborn integral representations of the Bernstein basis functions.


Bernstein polynomials Generating function Bezier curves Fourier transform Laplace transform Functional equations Partial differential equations p-adic Volkenborn integral 

AMS Subject Classification

14F10 12D10 26C05 26C10 30B40 30C15 42A38 44A10 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Yilmaz Simsek
    • 1
  1. 1.Department of Mathematics, Faculty of ScienceAkdeniz UniversityAntalyaTurkey

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