Fourth-order Bessel equation: eigenpackets and a generalized Hankel transform^†

^†This paper is dedicated to the achievements and memory of Professor Günter Hellwig.

Abstract

In connection with the fourth-order Bessel-type differential equation

two expansion theorems are established, the convergence being pointwise or in an L ²-setting. If the positive parameter M tends to zero, these two expansion theorems reduce to the classical Hankel transform of order zero. In a previous article, the authors have proved that in one of the introduced Lebesgue–Stieltjes Hilbert function spaces, the differential expression x ⁻¹ L _M gives rise to exactly one self-adjoint operator S _M . In this article, it is proved, together with the corresponding expansion theorems, that S _M has a complete eigenpacket. The orthogonality property of this eigenpacket is reflected in a distributional orthogonality on which the expansion theorems are based.

^†This paper is dedicated to the achievements and memory of Professor Günter Hellwig.

Keywords:

• Bessel functions
• Bessel-type functions
• Linear ordinary differential equations
• Lebesgue–Stieltjes Hilbert spaces
• Differential operators

Acknowledgements

The corresponding author, Norrie Everitt, thanks his colleague Jörg Fliege for his expert help in preparing the original Latex file for presentation in the style of Integral Transforms and Special Functions.

Notes

^†This paper is dedicated to the achievements and memory of Professor Günter Hellwig.