On orthogonal polynomials with respect to a class of differential operators

We consider orthogonal polynomials with respect to a linear differential operatorL(M)=∑k=0Mρk(z)dkdzk,where {ρk}k=0M are complex polynomials such that deg[ρk]⩽k,0⩽k⩽Mdeg[ρk]⩽k,0⩽k⩽M, with equality for at least one index. We analyze the uniqueness and zero location of these polynomials. An interesting phenomena occurring in this kind of orthogonality is the existence of operators for which the associated sequence of orthogonal polynomials reduces to a finite set. For a given operator, we find a classification of the measures for which it is possible to guarantee the existence of an infinite sequence of orthogonal polynomials, in terms of a linear system of difference equations with varying coefficients. Also, for the case of a first order differential operator, we locate the zeros and establish the strong asymptotic behavior of these polynomials.