Third Degree Linear Forms and Semiclassical Forms of Class One—A Case Study

The aim of this contribution is to study several characterizations of families of symmetric and quasi-symmetric semiclassical linear forms of class one which are of strict third degree. In fact, by using the Stieltjes function and the moments of those forms, we give necessary and sufficient conditions for a regular form to be at the same time of third degree, symmetric (resp. quasi-symmetric) and semiclassical of class one. Thus, we focus our attention on the link between these forms and the Jacobi forms \(\mathcal {V}_{q}^{k, l}:=\mathcal {J}(k+q/3,l-q/3), k+l\ge -1, k, l \in \mathbb {Z}, q\in \{1,2\}\) . All of them are rational transformations of the Jacobi form \(\mathcal{V} = \mathcal{J} \left( -2/3, -1/3 \right) \) .