Characterization theorem for third degree symmetric semiclassical forms of class two

Abstract

Relying on their third degree character, this paper offers a comprehensive description of a large family of symmetric semiclassical forms of class two. A characterization theorem for all third degree symmetric semiclassical forms of class two is stated and proved. 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 strict third degree (resp. second degree), symmetric and semiclassical of class two under condition $\Phi(0)=0$. Thus, we focus our attention on the link between these forms and the Jacobi forms $\mathcal{V}_{q}^{k, l}:=\mathcal{J}(k+q/3,l-q/3), k+l\geq-1, k, l \in \mathbb{Z}, q\in\{1,2\}$ (resp. $\mathcal{T}_{p,q}:=\mathcal{J}(p-1/2,q-1/2), p+q\geq0, p, q \in \mathbb{Z}$). All of them are rational transformations of the Jacobi form ${\mathcal V} := {\mathcal J} \left (-2/3, -1/3 \right)$ (resp. the Tchebychev form of first kind ${\mathcal T} := {\mathcal J} \left (-1/2, -1/2 \right)$).