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 Turán’s inequality for associated symmetric Pollaczek polynomials. Our strategy relies on chain sequences, asymptotic behavior, further Turán type inequalities and transformations into more convenient orthogonal polynomial systems.

将多项式分析引入正交多项式和特殊函数理论，或将某些Banach代数与满足特定但经常满足的非负线性化性质的正交多项式相关联的一种优雅而富有成果的方法是多项式超群的概念。多项式超群（或分别为基础多项式）伴随有${\mathrm{大号}}^{\mathrm{1个}}$-代数和丰富，完善且统一的谐波分析。但是，单个行为在很大程度上取决于基础多项式。我们研究了相关的对称Pollaczek多项式，这是超球形多项式的两参数推广。考虑相关${\mathrm{大号}}^{\mathrm{1个}}$-代数，我们将通过指定相应的参数区域来提供弱可及性和点可及性的完整表征。特别是，我们将看到存在一个较大的参数区域，这些参数的可容纳性均不适用（与${\mathrm{大号}}^{\mathrm{1个}}$-局部紧密群的代数）。此外，我们将排除对字符的适应性。关键的基本非负线性化特性也将被建立，这特别是建立了R. Lasser（1994）的一个猜想。此外，我们将证明相关联的对称Pollaczek多项式的Turán不等式。我们的策略依赖于链序列，渐近行为，进一步的图兰型不等式和转换为更方便的正交多项式系统。