On the inverse problem of the product of a semi-classical form by a polynomial

A form (linear functional) u is called regular if there exists a sequence of polynomials {Pn}n⩾0, deg Pn=n which is orthogonal with respect to u. Such a form is said to be semi-classical, if there exist polynomials Φ and Ψ such that D(Φu) + Ψu = 0, where D designs the derivative operator. tain regularity conditions, the product of a semi-classical form by a polynomial, gives a semi-classical form. In this paper, we consider the inverse problem: given a semi-classical form v, find all regular forms u which satisfy the relation x2u = −λv, λ ∈ C∗. We give the structure relation (differential-recurrence relation) of the orthogonal polynomial sequence relatively to u. An example is treated with a nonsymmetric form v.