Dual spaces vs. Haar measures of polynomial hypergroups

Many symmetric orthogonal polynomials ${\left(P_n\left(x\right)\right)}_{n\in N_0}$ induce a hypergroup structure on $N_0$. The Haar measure is the counting measure weighted with $h\left(n\right)≔1/{\int }_R{P}_n^2\left(x\right)d\mu \left(x\right)\ge 1$, where $\mu$ denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property $h\left(n\right)\ge 2(n\in N)$. We give sufficient criteria and particularly show that $h\left(n\right)\ge 2(n\in N)$ if the (Hermitian) dual space $\widehat{N_0}$ equals the full interval $[-1,1]$, which is fulfilled by an abundance of examples. We also study the role of nonnegative linearization of products (and of the resulting harmonic and functional analysis). Moreover, we construct two example types with $h\left(1\right)<2$. To our knowledge, these are the first such examples. The first type is based on Karlin–McGregor polynomials, and $\widehat{N_0}$ consists of two intervals and can be chosen “maximal” in some sense; $h$ is of quadratic growth. The second type relies on hypergroups of strong compact type; $h$ grows exponentially, and $\widehat{N_0}$ is discrete.