On the higher-order differential equations for the generalized Laguerre polynomials and Bessel functions

TL;DR

This paper explores higher-order differential equations related to generalized Laguerre polynomials and Bessel functions, providing new elementary representations and demonstrating their symmetry and equivalence, thereby advancing spectral differential equation theory.

Contribution

It introduces a new elementary form of the Laguerre-type differential operator, proves its symmetry, and derives elementary representations for Bessel-type functions through limiting processes.

Findings

New elementary representation of Laguerre-type differential operator

Proof of symmetry with respect to weighted scalar product

Elementary representations for Bessel-type functions obtained

Abstract

In the enduring, fruitful research on spectral differential equations with polynomial eigenfunctions, Koornwinder's generalized Laguerre polynomials are playing a prominent role. Being orthogonal on the positive half-line with respect to the Laguerre weight and an additional point mass $N \ge 0$ at the origin, these polynomials satisfy, for any $\alpha\in\mathbb{N}_{0}$, a linear differential equation of order $2\alpha+4$. In the present paper we establish a new elementary representation of the corresponding 'Laguerre-type' differential operator and show its symmetry with respect to the underlying weighted scalar product. Furthermore, we discuss various other representations of the operator, mainly given in factorized form, and show their equivalence. Finally, by applying a limiting process to the Laguerre-type equation, we deduce new elementary representations for the higher-order differential equation satisfied by the Bessel-type functions on the positive half-line.