## Dependencies within Bessel's recursive relationship

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

$$B_\nu(x) = \frac{x}{2\nu}\left(B_{\nu-1}(x)+B_{\nu+1}(x)\right)$$
$$B_\nu'(x) = \frac{\nu}{x}B_\nu(x)-B_{\nu+1}(x)$$
$$B_\nu'(x) = -\frac{\nu}{x}B_\nu(x)+B_{\nu-1}(x)$$
$$B_\nu'(x) = \frac{1}{2}\left(B_{\nu-1}-B_{\nu+1}(x)\right)$$

where $$B_\nu (x)$$ 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
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