Pseudo-orthogonal polynomials

The classical problem of the transformation of the orthogonality weights of polynomials by theory of the space Rn is discussed. The described system of polynomials - pseudo-orthogonal on the discrete set of n points - is a new result. The polynomials of this system, as the orthogonal ones, are connected by the three-term relations with a tridiagonal matrix which is irreducible but not the Jacobi one. Nevertheless, these polynomials have real single roots with a weak separation. Some orthogonality weights are negative. Analysis of the relation between the matrices of two orthogonal polynomial systems, providing the condition of existence of the pseudo-orthogonal polynomials, is another result.