Third degree linear forms and quasi-antisymmetric semiclassical forms of class one

Abstract

In this contribution, we explore the characterizations of a family of quasi-antisymmetric semiclassical linear forms of class one, based on their third-degree character. We show that there does not exist a strict third-degree form that is simultaneously a quasi-antisymmetric semiclassical linear form of class one. By utilizing the Stieltjes function and the moments of these forms, we provide necessary and sufficient conditions for a regular form to be of second degree, quasi-antisymmetric, and semiclassical of class one. Our focus is on the connection between these forms and the Jacobi forms ${\mathcal T}_{0, q} = {\mathcal J}(-1/2, q-1/2), q \in \mathbb{N}$. All these forms are rational transformations of the first-kind Chebyshev form ${\mathcal T} = {\mathcal J}(-1/2, -1/2)$.