Laguerre-Sobolev orthogonal Polynomials and Boundary Value Problems on a semi-infinite domain

We study a family of Laguerre–Sobolev orthogonal polynomials associated with a Sobolev inner product arising from second–order boundary value problems on the semi–infinite interval $(0,+\mathrm{\infty })$. These polynomials generate an orthogonal basis of test functions vanishing at the endpoints and are especially well suited for the spectral approximation of Schrödinger–type problems with singular potentials. Explicit connection formulas with classical Laguerre polynomials are obtained, together with recurrence relations and asymptotic properties of the corresponding coefficients. A generating function involving Bessel functions is also derived. As an application, we develop a fully diagonalized Laguerre–Sobolev spectral method for Dirichlet problems with singular potentials. The method avoids the solution of linear systems and can be implemented recursively. Numerical experiments for a Schrödinger–type equation with inverse–distance potential confirm spectral accuracy and exponential convergence.