Tur\'{a}n's inequality, nonnegative linearization and amenability properties for associated symmetric Pollaczek polynomials

TL;DR

This paper investigates the harmonic analysis and algebraic properties of associated symmetric Pollaczek polynomials, establishing Turán's inequality, nonnegative linearization, and characterizing amenability properties of related Banach algebras.

Contribution

It provides the first complete characterization of amenability properties for $L^1$-algebras of associated symmetric Pollaczek polynomials and proves Turán's inequality for these polynomials.

Findings

Complete characterization of weak and point amenability regions.

Establishment of nonnegative linearization property confirming Lasser's conjecture.

Proof of Turán's inequality for associated symmetric Pollaczek polynomials.

Abstract

An elegant and fruitful way to bring harmonic analysis into the theory of orthogonal polynomials and special functions, or to associate certain Banach algebras with orthogonal polynomials satisfying a specific but frequently satisfied nonnegative linearization property, is the concept of a polynomial hypergroup. Polynomial hypergroups (or the underlying polynomials, respectively) are accompanied by $L^1$-algebras and a rich, well-developed and unified harmonic analysis. However, the individual behavior strongly depends on the underlying polynomials. We study the associated symmetric Pollaczek polynomials, which are a two-parameter generalization of the ultraspherical polynomials. Considering the associated $L^1$-algebras, we will provide complete characterizations of weak amenability and point amenability by specifying the corresponding parameter regions. In particular, we shall see that there is a large parameter region for which none of these amenability properties holds (which is very different to $L^1$-algebras of locally compact groups). Moreover, we will rule out right character amenability. The crucial underlying nonnegative linearization property will be established, too, which particularly establishes a conjecture of R. Lasser (1994). Furthermore, we shall prove Tur\'{a}n's inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Tur\'{a}n type inequalities and transformations into more convenient orthogonal polynomial systems.