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

Abstract

All the linear functionals $c_0$ and $c_1$ associated to the $\mathrm{\Delta }$-coherent pairs $(c_0,c_1)$, have been given by Area et al. (2000). From these linear functionals $c_0$ and $c_1$ are introduced bilinear functionals $a_{\mathit{\lambda }}(p,q)=c_0(pq)+\mathit{\lambda }{c}_1(\mathrm{\Delta }p\mathrm{\Delta }q)$, $\forall p,q\in \mathcal{P}$, but only for the $\mathrm{\Delta }$-coherent pairs $(c_0,c_1)$ having a support $\mathrm{\Omega }=]0,+\infty [$. Five kinds of $\mathrm{\Delta }$-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 $a_{\mathit{\lambda }}(p,p)$ in order to obtain Markov–Bernstein inequalities$$c_1\left({(\mathrm{\Delta }p)}^2\right)\le M_n^2{c}_0(p^2),\forall p\in {\mathcal{P}}_n.$$

The Markov–Bernstein constant $M_n$ is equal to $\frac{1}{\sqrt{{\mathit{\mu }}_{1,n}}}$ where ${\mathit{\mu }}_{1,n}$ 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 ${\mathit{\mu }}_{1,n}$ is studied for a part of the variation of the parameters which characterize the $\mathrm{\Delta }$-coherent pairs.

Keywords:

• $\mathrm{\Delta }$-Coherent pairs
• discrete variable
• orthogonal polynomials
• Markov–Bernstein inequalities
• Charlier polynomials
• Meixner polynomials

AMS Subject Classifications:

• 26D05
• 26C10
• 33C45
• 42C05

Notes

No potential conflict of interest was reported by the author.