A description via third degree character of quasi-symmetric semiclassical forms of class two

Abstract

The aim of this contribution is to give, through the third degree character, a complete description of a large family of quasi-symmetric semiclassical forms of class two. 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 the strict third degree quasi-symmetric and semiclassical of class two. We focus our attention not only on the link between all these forms and the Jacobi forms ${\mathcal {V}}_{q}^{k, l}={\mathcal {J}}(k+q/3,l-q/3)$, $k+l\ge -1,~ k, l \in {\mathbb {Z}},~q\in \{1,2\}$, but also on their connection with the form ${{\mathcal {V}}}={{\mathcal {J}}}\left( -2/3, -1/3\right)$.