Solution bases for the fourth-order Bessel-type and Laguerre-type differential equations

Abstract

The fourth-order Bessel-type and Laguerre-type linear ordinary differential equations are prototypes of structured linear differential equations of higher even-order, which naturally extend the second-order Bessel and Laguerre equations defined on the positive half-line of the real field R. Whilst the Laguerre-type equation arose from a search for all orthogonal polynomial generated by a linear differential equation, the present authors derived the Bessel-type equations and functions in 1994 by applying a generalized limit process to the Laguerre-type case. Due to their close relationship, the Laguerre- and Bessel-type functions of the same order share many important properties as, for example, orthogonality or a generalized hypergeometric representation. In this article, we first survey the most recent achievements in our study of the fourth-order Bessel equation which led to explicit representations of four linear independent solutions. Our main purpose then is to show how these techniques carry over to establish a solution basis for the fourth-order Laguerre-type differential equation.

Keywords:

• fourth-order linear ordinary differential equations
• Bessel-type equation
• Laguerre-type equation
• Bessel functions
• Laguerre polynomials
• Fourier–Bessel series

AMS Subject Classifications::

• Primary 33C45
• 34A05
• 34B30
• Secondary 33C20
• 42C05

Acknowledgement

The authors thank their colleague B. Tomas Johansson, University of Birmingham, for his expert help in the preparation of the LaTeX file for this article for Applicable Analysis.