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arXiv · 2602.06685

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

Cleonice F. Bracciali·Miguel A. Piñar
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Abstract

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 $(0,+\infty)$. 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.

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Cleonice F. Bracciali, Miguel A. Piñar. 2026-02-06. Laguerre-Sobolev orthogonal Polynomials and Boundary Value Problems on a semi-infinite domain. https://arxiv.org/abs/2602.06685

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