On the characterizations of third-degree semiclassical forms via polynomial mappings

The aim of this contribution is the study of orthogonal polynomials via polynomial mappings in the framework of the third-degree semiclassical linear forms. Let u and v be two regular forms and let denote by {pn}n≥0 and {qn}n≥0 the corresponding sequences of monic orthogonal polynomials such that there exists a monic polynomial θm
of degree m, with 0≤m≤k−1 and r∈C,in such a way +m(x)=θm(x)qn(xk+r),n=0,1,2,…k is a fixed integer number such that k≥2.
If u (resp. v) is a third-degree linear form, then we prove that the other one is also a third-degree linear form. From this fact we are able to show the relation between third-degree semiclassical forms u of class s≤k−1 and the classical forms. More precisely, the strict third-degree (respectively second-degree) forms are rational modifications of the product of k shifted Jacobi forms V=J(−2/3,−1/3)
(resp. T=J(−1/2,−1/2)). An illustrative example is given.

orthogonal polynomials; classical and semiclassical forms; polynomial mappings; stieltjes function; third-degree forms

On the characterizations of third-degree semiclassical forms via polynomial mappings Articles