Novel relations among Bernstein and Legendre basis functions involving special polynomials by approach Nörlund sum and Euler operator

ŞİMŞEK Y.

Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas, vol.119, no.4, 2025 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 119 Issue: 4
  • Publication Date: 2025
  • Doi Number: 10.1007/s13398-025-01763-8
  • Journal Name: Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Communication Abstracts, MathSciNet, zbMATH, DIALNET, Civil Engineering Abstracts
  • Keywords: Bernoulli numbers, Difference operators, Euler operator, Multivariable Bernstein basis functions, Nörlund sum, Stirling numbers, Touchard polynomials
  • Akdeniz University Affiliated: Yes

Abstract

This paper is structured around three main objectives. First, to derive some new relations between Bernstein basis functions and the Legendre basis functions by means of generating functions, including derivative operators and Rodrigues formula. Second, to give relations between Bernstein basis functions and Bernoulli polynomials by applying the Nörlund sum to the generating function for Bernstein basis functions. These relations involving certain infinite series in terms of the Bernstein basis functions and Bernoulli polynomials and partial derivative equations. Third, to give formulas for multivariable Bernstein basis functions, the Touchard polynomials, the Bernoulli numbers, the Stirling numbers, etc., by using functional equations, generating functions, and the Euler derivative operator. Some of the results of this paper are analyzed with comments on how to reduce them to known results.