Series solutions of Laguerre- and Jacobi-type differential equations in terms of orthogonal polynomials and physical applications

TL;DR

This paper develops series solutions for Laguerre- and Jacobi-type differential equations using orthogonal polynomials, with applications in quantum mechanics, revealing new polynomial solutions and their spectral properties.

Contribution

It introduces new series solutions for these differential equations and explores their physical applications, including novel polynomial solutions with continuous and discrete spectra.

Findings

Solutions expressed as infinite series of orthogonal polynomials

Recursion relations solved using spectral properties of polynomials

Applications demonstrated in quantum mechanics contexts

Abstract

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.