A large family of third degree semiclassical forms of class three

Abstract

The aim of this contribution is to study several characterizations of a large family of symmetric semiclassical linear forms of class three which are of 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 strict third degree (resp. second degree), symmetric and semiclassical of class three under conditions \(\Phi (x)=x(x^4-1)\) and \(\Psi (0)\in \{0, 2\}\). Thus, we focus our attention on the link between these forms and the Jacobi forms \(\mathcal {V}_{q}^{k, l}:=\mathcal {J}(k+q/3,l-q/3), k+l\ge -1, k, l \in \mathbb {Z}, q\in \{1,2\}\) (resp. \(\mathcal {T}_{p,q}:=\mathcal {J}(p-1/2,q-1/2), p+q\ge 0, p, q \in \mathbb {Z}\)). 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 \(\mathcal {T}:= \mathcal {J} \left( -1/2, -1/2 \right) \)).