Ladder-operator factorization and the Bessel differential equations

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.