Characterization theorem for strict third-degree semiclassical forms of class one obtained via cubic decomposition

Abstract

Based on their third-degree character, we give a complete description of a large family of semiclassical forms of class one. A characterization theorem for all strict third-degree semiclassical forms of class one arising via the cubic decomposition $W_{3n}(x)=P_n(x^3+qx+r)$, $q,r\in \mathbb{C}$, $n\ge 0$, is stated and proved. In fact, by using the Stieltjes function and the moments of those forms, this theorem gives necessary and sufficient conditions that characterize and identify all these forms, and we present their link with the Jacobi forms ${\mathcal{V}}_q^{k,l}=\mathcal{J}(k+q/3,l-q/3)$, $k+l\ge -1$, $k,l\in \mathbb{Z}$, $q\in \{1,2\}$.