The connection between self-associated two-dimensional vector functionals and third degree forms
After restating some known results on the concept of \(d\)-orthogonality (\(d\geq 2\)) in the context of linear functionals on the linear space of polynomials in one indeterminate, the authors introduce a so-called Third Degree Form (TDF). A TDF is a linear form \(w\) for which there exist polynomials \(A\) (monic), \(B, C\) and \(D\) such that \[ A(z)S^3(z)+B(z)S^2(z)+C(z)S(z)+D(z)=0, \] where \(D\) depends on \(A, B, C\) and \(w\) and \(S(z)\) is the formal Stieltjes function associated with \(w\): \[ S(w)(z):=-\sum_{n\geq 0} {(w)_n\over Z^{n+1}}; \] generated by the moments \((w)_n:=\langle w,x^n\rangle\). For a regular TDF, several results are given (a.o. \(w\) strictly third degree \(\Rightarrow\) \(w\) is a Laguerre-Hahn form). The main results are concerned with a \(2\)-orthogonal, \(2\)-symmetric, self-associated form and its cubic components It would go too far to state all the results here: the paper is crammed with information; those who are interested, best have a look themselves.
scientific article; zbMATH DE number 1419517