Δ-Coherent pairs of linear functionals and Markov-Bernstein inequalities

Abstract

All the linear functionals and associated to the -coherent pairs , have been given by Area et al. (2000). From these linear functionals and are introduced bilinear functionals , , but only for the -coherent pairs having a support . Five kinds of -coherent pairs given by Area et al. are concerned. They imply Charlier polynomials or Meixner polynomials, and they depend on a set of parameters. This paper is devoted to the study of the positivity of in order to obtain Markov–Bernstein inequalitiesThe Markov–Bernstein constant is equal to where is the smallest zero of a polynomial of degree n satisfying a three term recurrence relation. The five kinds of three term recurrence relation are obtained. The behavior of is studied for a part of the variation of the parameters which characterize the -coherent pairs.