Orthogonal polynomials and higher order singular Sturm-Liouville systems (Q1262421)

This survey aims to collect the results obtained so far on differential equations of orders four or six, which are satisfied by orthogonal polynomials. The authors recapitulate the results concerning the differential equations, eigenvalues, weight functions and boundary conditions. They particularly emphasize on the development of the theory of singular Sturm-Liouville systems. The material of the paper is divided into eight main parts. In the first part the authors begin with the definition of the ``moments'' which generate the polynomials, and state (without proofs) two general theorems one of which concerns the moments while the other concerns the weight functions. It is worthwhile to remark here that the connection from moments to differential equations to polynomials to moments is complete. Then in the second and third parts they give the equations of order four and six, respectively, which are of the above-mentioned type. It is shown that some of the fourth-order equations can be obtained from their second-order counterparts by iterations. It should be remarked that in part III the authors give only some specific examples rather than write out the general formulas concerning the polynomials satisfying a sixth-order differential quation. In part IV the boundary conditions which give rise to selfadjoint Sturm- Liouville problems are examined in a rather detailed manner. To this end they consider the problem in system format and work with Green's formula and the Lagrange bilinear concomitant. This section involves also a lot of new material. Sections V and VI include some examples while section VII is devoted to the ``indefinite problems'' for which the weight functions fail to satisfy the constraints placed upon them. The last section, i.e. section VIII, lists some generalized results which involve also the equations of order eight.