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

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

where $\displaystyle J_{n}(x)$ 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

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

and

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

1 Attachment(s)

Re: second derivative of Bessel Function in terms of higher and lower orders of Besse

Hi !

have a look at attachment :

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.