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


Simsek Y.

MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol.38, no.14, pp.3007-3021, 2015 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 38 Issue: 14
  • Publication Date: 2015
  • Doi Number: 10.1002/mma.3276
  • Journal Name: MATHEMATICAL METHODS IN THE APPLIED SCIENCES
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.3007-3021
  • Keywords: Bernstein basis functions, generating function, functional equations, Catalan numbers, Bernoulli numbers, Bernoulli polynomials, combinatorial identity, combinatorial sum, binomial coefficients, gamma function, beta function, GENERATING-FUNCTIONS, DERIVING IDENTITIES, POLYNOMIALS
  • Akdeniz University Affiliated: Yes

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 (C) 2014 John Wiley & Sons, Ltd.