Fourier-Bessel Series for Second-Order and Fourth-Order Bessel Differential Equations

In this paper we look at the Hilbert function space framework for Fourier-Bessel series, based on linear differential operators generated by the second-order Bessel differential equation and the fourth-order Bessel-type differential equation. In the second-order case attention is restricted to the differential equation for Bessel functions of order zero \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-({xy^\prime}(x))^{\prime}=\lambda xy(x)\ \ \ \ \ {\rm for\ all}\ x\ \in\ (0,1\rbrack$$\end{document}, where λ ∈ ℂ, the complex plane, is the spectral parameter. In the fourth-order case we concentrate on the Bessel-type differential equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({xy^{\prime\prime}}(x))^{\prime\prime}-(({9x^{-1}}+{8M^{-1}}x)y^\prime(x))^\prime=\Lambda xy(x)\ \ \ {\rm for\ all}\ x\in\ (0,1\rbrack$$\end{document}, where Λ ∈ ℂ is the spectral parameter, and M > 0 is a given parameter. In both cases the analysis is concerned with the theory of unbounded linear operators, generated by the differential equation, in the Hilbert function space L2((0, 1); x). The analysis depends on new results in special function theory to develop properties of the solutions of the fourth-order Bessel-type differential equation, in particular the series expansions of these solutions at the regular singularity at the origin of ℂ.