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 {P-n}(n greater than or equal to 0), deg P-n=n which is orthogonal with respect to u. Such a form is said to be semi-classical, if there exist polynomials Phi and Psi such that D(Phi u) + Psi u = 0, where D designs the derivative operator. On certain 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 x(2)u = -lambda v, lambda is an element of 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. (C) 1997 Elsevier Science B.V. All rights reserved.