On the eigenfunctions of the <i>q</i>-Bernstein operators

Abstract

The eigenvalue problems for linear operators emerge in various practical applications in physics and engineering. This paper deals with the eigenvalue problems for the q-Bernstein operators, which play an important role in the q-boson theory of modern theoretical physics. The eigenstructure of the classical Bernstein operators was investigated in detail by S. Cooper and S. Waldron back in 2000. Some of their results were extended for other Bernstein-type operators, including the q-Bernstein and the limit q-Bernstein operators. The current study is a pursuit of this research. The aim of the present work is twofold. First, to derive for the q-Bernstein polynomials analogues of the Cooper-Waldron results on zeroes of the eigenfunctions. Next, to present in detail the proof concerning the existence of non-polynomial eigenfunctions for the limit q-Bernstein operator.