# Bessel Functions

1. Dec 3, 2003

### csmines

Hey guys I was wondering if you could help me out with a proof of the recursion relations of Bessel functions on my homework:

Show by direct differentiation that

$$J_{\nu}(x)=\sum_{s=0}^{\infty} \frac{(-1)^{s}}{s!(s + \nu)!} \left (\frac{x}{2}\right)^{\nu+2s}$$

obeys the important recursion relations

$$J_{\nu-1}(x)+J_{\nu+1}(x) = \frac{2\nu}{x}J_{\nu}(x)$$

$$J_{\nu-1}(x)-J_{\nu+1}(x) = 2J_{\nu}(x)$$

I've tried differentiating with respect to x but I get a factor of 2s that's no good. And I've also tried replacing nu with nu plus one and nu minus one but that ends up with a lot of s terms as well. I am pretty much lost on what to do so if you could just point me in the right direction that'd be great. Thanks a lot.

Csmines

2. Dec 3, 2003

### arcnets

csmines,
I'm not at all sure, but the first step in this could be to show that
$$\frac{d}{dx}\left[x^{-\nu}J_{\nu}(x)\right]=-x^{-\nu}J_{\nu+1}(x)$$
The trick is probably to
1) split off the s=0 term
2) make an index shift s->s+1 in the rest sum
3) differentiate (s=0 term vanishes).

This is IMO not too difficult. But can we use it to show the desired relation? I wonder.

Last edited: Dec 3, 2003
3. Jun 25, 2010

### himanshurocks

hey let me know how to find out bessel transform of a sequence of numbers ,as in we calculate fourier transform of a sequence??

this is urgent pls do reply..
i need this for my project on "analog signal processing using bessel function using matlab"..