On the inverse problem of the product of a form by a polynomial: The cubic case

Abstract

A form (linear functional) u is called regular if there exists a sequence of polynomials { P n } n⩾0 , deg P n = n, which is orthogonal with respect to u. On certain regularity conditions, the product of a regular form by a polynomial is still a regular form. In this paper, we consider the inverse problem: given a regular form v, find all the regular forms u which satisfy the relation x 3 u=− λv, λ∈ C−{0} . We give the second-order recurrence relation of the orthogonal polynomial sequence with respect to u. Some examples are studied.