The solutions of certain Sturm–Liouville eigenvalue problems are known in the form of orthogonal functions φr(x), r = 0, 1, 2, …, arranged so that the moduli of the corresponding eigenvalues λr increase monotonically with r, i.e. with the φr(x) satisfying appropriate boundary conditions. The investigations described in this paper are an attempt to examine the conditions that must be satisfied in order that the extended eigenvalue problem (again with U satisfying appropriate boundary conditions) may be solved by expansion of U in a series of the orthogonal functions φr(x). Practically all orthogonal systems satisfy a 3-term recurrence relation. If the φr(x) satisfy such a relation, this, together with the differential equations satisfied by the φr(x), may be used to transform the extended differential eigenvalue problem to that of finding the eigenvalues of an infinite symmetric tri-diagonal matrix. An examination is made of the recurrence relations satisfied by the ortho-normal polynomials, and the conditions that must be satisfied by the coefficients of the basic recursion in order that certain polynomial operators may give such matrices are obtained. The results are applied to the Jacobi polynomials and Fourier functions. A discussion of the convergence of the eigenvalues obtained by repeatedly bordering finite principal submatrices of the infinite matrices follows. Several numerical examples are given, the calculations being made on the University of London Ferranti Mercury computer.