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

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Abstract

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

$$-({xy^\prime}(x))^{\prime}=\lambda xy(x)\ \ \ \ \ {\rm for\ all}\ x\ \in\ (0,1\rbrack$$

, where λ ∈ ℂ, the complex plane, is the spectral parameter. In the fourth-order case we concentrate on the Bessel-type differential equation

$$({xy^{\prime\prime}}(x))^{\prime\prime}-(({9x^{-1}}+{8M^{-1}}x)y^\prime(x))^\prime=\Lambda xy(x)\ \ \ {\rm for\ all}\ x\in\ (0,1\rbrack$$

, 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 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 ℂ.

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Authors and Affiliations

  1. School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK

    W. Norrie Everitt

  2. Lehrstuhl A für Mathematik, Technical University Aachen, Templergraben 55, D-52062, Aachen, Germany

    Clemens Markett

Authors
  1. W. Norrie Everitt
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  2. Clemens Markett
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Correspondence to W. Norrie Everitt.

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Dedicated to Professor W. K. Hayman on the occasion of his 80th birthday

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Everitt, W.N., Markett, C. Fourier-Bessel Series for Second-Order and Fourth-Order Bessel Differential Equations. Comput. Methods Funct. Theory 8, 545–563 (2008). https://doi.org/10.1007/BF03321704

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  • DOI: https://doi.org/10.1007/BF03321704

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