Dependencies within Bessel's recursive relationship

kzhu

Dear All,

I am working on the wave equation in the spherical coordinate and come across with the recursive relationship of Bessel's function, which are given in four expressions

[tex]B_\nu(x) = \frac{x}{2\nu}\left(B_{\nu-1}(x)+B_{\nu+1}(x)\right)[/tex]
[tex]B_\nu'(x) = \frac{\nu}{x}B_\nu(x)-B_{\nu+1}(x) [/tex]
[tex]B_\nu'(x) = -\frac{\nu}{x}B_\nu(x)+B_{\nu-1}(x) [/tex]
[tex]B_\nu'(x) = \frac{1}{2}\left(B_{\nu-1}-B_{\nu+1}(x)\right) [/tex]

where [tex]B_\nu (x)[/tex] represents the Bessel/Neumann/Hankel function. I notice that two of these four recursive relationship can be derived from the other two.

My guess extends that, given a 2nd order linear PDE similar to Bessel's equation, can we draw the conclusion that there would be 2 linearly independent recursive relationship?

Thank you for any feedback.


kzhu