Computational Methods and Function Theory 8 (2008), No. 2, 545--563
Copyright Heldermann Verlag 2008
W. Norrie Everitt
w.n.everitt@bham.ac.uk , University of Birmingham, School of Mathematics and Statistics, Edgbaston, Birmingham B15 2TT, England, U.K.
Clemens Markett
Technical University Aachen, Lehrstuhl A für Mathematik, Templergraben 55, D-52062 Aachen, Germany.
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 $$ -(xy'(x))'=\lambda xy(x) \qquad \mbox{for all }x\in(0,1], $$ where $\lambda\in\mathbb{C}$, the complex plane, is the spectral parameter. In the fourth-order case we concentrate on the Bessel-type differential equation $$ (xy''(x))''-((9x^{-1}+8M^{-1}x)y'(x))'=\Lambda xy(x) \qquad\mbox{for all }x\in(0,1], $$ where $\Lambda\in\mathbb{C}$ 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 $L^2((0,1);x)$. 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 $\mathbb{C}$.
Keywords: Fourier-Bessel series, Bessel functions, Bessel-type functions.
MSC 2000: Primary 33C05, 33B05, 34L10; Secondary 33C10, 35B24, 26A03.
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