Some fun with Bessel functions

A theoretical investigation of generating functions is pursued using the theory of the Z-transform. Bessel functions are viewed as outputs from infinite impulse response (IIR) filters. It is shown that a single second-order linear ordinary differential equation with nonconstant coefficients defines a family of generating functions for all the Bessel functions of integral order. A numerical method that directly makes use of the three-term recurrence relation, as a tri-diagonal set of equations solved by Gaussian elimination, is proposed for their evaluation. What results is that both J/sub n/ and Y/sub n/, for each n, are derived at the cost of one floating-point division and four floating-point multiplications, after initial set up, using an algorithm that has possibilities for parallelization.<>