On the inverse problem of the product of a form by a monomial: the case n=4. Part I

Abstract

A form (linear functional) u is called regular if there exists a unique sequence of monic 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 particular inverse problem: given a regular form v, find all the regular forms u that satisfy the equation x 4 u=−λ v, λ∈ℂ \\ {0}. We give the second-order recurrence relation of the orthogonal polynomial sequence with respect to u. An example is studied.