Pseudo-orthogonal polynomials

We consider the classical problem of transforming an orthogonality weight of polynomials by means of the space $\mathbb R^n$. We describe systems of polynomials called pseudo-orthogonal on a finite set of $n$ points. Like orthogonal polynomials, the polynomials of these systems are connected by three-term relations with tridiagonal matrix which is nondecomposable but does not enjoy the Jacobi property. Nevertheless these polynomials possess real roots of multiplicity one; moreover, almost all roots of two neighboring polynomials separate one another. The pseudo-orthogonality weights are partly negative. Another result is the analysis of relations between matrices of two different orthogonal systems which enables us to give explicit conditions for existence of pseudo-orthogonal polynomials.