A Description of Second Degree Semiclassical Forms of Class Two Arising Via Cubic Decomposition

Abstract

In this work, we consider orthogonal polynomials via cubic decomposition in the framework of the second degree semiclassical class. We give a complete description, using the formal Stieltjes function and the moments, of semiclassical linear forms of class two which are of second degree such that their corresponding orthogonal polynomials sequences $\{W_n\}_{n\ge 0}$ are obtained via the cubic decomposition $W_{3n}(x)=P_n(x^3),~n\ge 0$. We focus our attention, not only on the link between all these forms and the second degree classical forms ${{\mathcal {T}}}_{p, q}={{\mathcal {J}}}(p-1/2, q-1/2)$, $\; p, q\in {\mathbb {Z}},~p+q\ge 0$, but also on their connection with the Tchebychev form of the first kind ${{\mathcal {T}}}={{\mathcal {J}}}\left( -1/2, -1/2\right)$.