Based on their second degree character, in this contribution we study new characterizations of families of symmetric and quasi-symmetric semiclassical linear forms of class one. 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 second degree, symmetric (resp. quasi-symmetric) and semiclassical of class one. We focus our attention on the link between these forms and the Jacobi forms Tp,q = J(p − 1/2, q − 1/2), p, q ∈ Z, p + q ≥ 0. All of them are rational transformations of the first kind Chebychev form T = J (−1/2,−1/2). Finally, we study a family of second degree linear forms which are semiclassical of class one and are not included in the above families