  We introduce two ordinary second-order linear differential equations of the
Laguerre- and Jacobi-type. Solutions are written as infinite series of square
integrable functions in terms of the Laguerre and Jacobi polynomials,
respectively. The expansion coefficients of the series satisfy three-term
recursion relations, which are solved in terms of orthogonal polynomials with
continuous and/or discrete spectra. Most of these are well-known polynomials
whereas few are not. We present physical applications of these differential
equations in quantum mechanics.
