Ladder-Operator Factorization and the Bessel Differential Equations

Robin, William

Abstract

We present an alternative approach to the discussion of Bessel equations and Bessel functions, through an elementary factorization method. The various Bessel equations are represented by a single parameterized form and, after a standard transformation of the dependent variable, a transformed parameterized (Bessel) equation is factorized in terms of raising and lowering ladder-operators. Once constructed, the ladder-operators for the transformed parameterized equation determine the ladder-operators that factorize the various Bessel equations and enable the determination of the various recurrence relations between the Bessel functions. In particular the construction of the Rayleigh formulae for the Bessel functions becomes particularly straightforward. However, ‘starting’ Bessel functions for the ladder operators and iterative and Rayleigh formulae must still be obtained as series solutions of particular Bessel equations.

Citation

Robin, W. (2014). Ladder-Operator Factorization and the Bessel Differential Equations. International Mathematical Forum, 9, 71-80. https://doi.org/10.12988/imf.2014.311214

• Journal Article Type: Article
• Publication Date: 2014
• Deposit Date: Apr 16, 2015
• Publicly Available Date: Apr 16, 2015
• Electronic ISSN: 1312-7594
• Publisher: Hikari
• Peer Reviewed: Peer Reviewed
• Volume: 9
• Pages: 71-80
• DOI: https://doi.org/10.12988/imf.2014.311214
• Keywords: Bessel functions; ladder-operator factorization; recurrence relations; Rayleigh formulae
• Public URL: http://researchrepository.napier.ac.uk/id/eprint/7766
• Publisher URL: http://dx.doi.org/10.12988/imf.2014.311214