Generating functions of Bernstein polynomials: Fourier series expansion and applications

Abstract

In this paper, we introduce the Fourier series expansion of the generating function for Bernstein polynomials. We also present series formulas for the generating function of Bernoulli polynomials. Furthermore, we establish novel formulae between these series and Euler polynomials as well as Zeta-type functions. The exploration of these connections sheds light on the intri-cate relationships among these fundamental mathematical constructs. Through these discoveries, we deepen our understanding of the interplay between various polynomial families and associated mathematical functions. These findings contribute to the broader landscape of mathematical analysis and offer insights into the rich structure underlying theory of special functions. © 2025, International Scientific Research Publications. All rights reserved.