Hey guys I was wondering if you could help me out with a proof of the recursion relations of Bessel functions on my homework:(adsbygoogle = window.adsbygoogle || []).push({});

Show by direct differentiation that

[tex]

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

[/tex]

obeys the important recursion relations

[tex]

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

[/tex]

[tex]

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

[/tex]

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Bessel Functions

**Physics Forums | Science Articles, Homework Help, Discussion**