Differential equations having orthogonal polynomial solutions

Abstract

The subject of orthogonal polynomials is a classical one whose origin can be traced back to Legendre``s work on planetary motions. In early stage of this century, a great deal of progress in this field of orthogonal polynomials has been made by the important applications to Quantum mechanics, Probability and Statistics, and other branches of mathematical analysis. Orthogonal polynomials are also strongly related to the theory of continued fractions, special functions and interpolation in approximation. Until the late 1970``s, orthogonal polynomials linking differential equations have been occasionally studied by S. Bochner, W. Hahn and H. L. Krall. Much works were developed on the interplay between the theories of orthogonal polynomials and differential equations in late 1970``s, primarily by A. M. Krall and L. L. Littlejohn, which strongly motivates this work. To be precise, we are concerned with the differential equations of spectral type ◁수식 삽입▷(원문을 참조하세요) where $ℓ_i(x)$ are polynomials, independent of n, and $λ_n$ is an eigenvalue parameter. Our primary concerns are (1) When does the differential equation ($^*$) have an orthogonal polynomial system (OPS, in short) as eigenfunctions ? (2) If it does, how to construct orthogonalizing weight distributions for the corresponding OPS``s ? (3) What is the general structure of such distributional weights ? We first find necessary and sufficient conditions for the equation ($^*$) to have an OPS as solutions. Then, we find an overdetermined system of nonhomogeneous differential equations, that must be satisfied by orthogonalizing weight distributions of such OPS``s. It turns out that the corresponding homogeneous system is exactly the symmetry equations of the equations ($^*$), of which any non-trivial classical solution (if it exists) must be a symmetry factor of the differential operator $L_N[ㆍ]$ in ($^*$). We then find a necessary condition for $L_N[ㆍ]$ to be symmetrizable, which will play a crucial role i...