# second derivative of Bessel Function in terms of higher and lower orders of Bessel fn

• Oct 30th 2012, 01:57 PM
billm
second derivative of Bessel Function in terms of higher and lower orders of Bessel fn
I have been trying to replicate a result given in a textbook that says

$J_{n}^{''}(x)=\frac{1}{4}\{J_{n-2}(x)-2J_{n}(x)+J_{n+2}(x)\}$

where $J_{n}(x)$ is the Bessel Function of the First Kind.

Can someone show me how to get this from the recurrence formulae for Bessel derivatives found in the literature as

$(\frac{1}{x}\frac{d}{dx})^{m}(x^{n}J_{n}(x))=x^{n-m}J_{n-m}(x)$

and

$(\frac{1}{x}\frac{d}{dx})^{m}(x^{-n}J_{n}(x))=(-1)^{m}x^{-n-m}J_{n+m}(x)$
• Oct 31st 2012, 01:28 AM
JJacquelin
Re: second derivative of Bessel Function in terms of higher and lower orders of Besse
Hi !
have a look at attachment :
• Oct 31st 2012, 10:34 AM
billm
Re: second derivative of Bessel Function in terms of higher and lower orders of Besse
Thanks JJacquelin!

My problem was in using the recurrence formulae for the case m=2. Not sure how to interpret (d/dx)^2, but your way avoids this.