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

Oct 2012
4
2
adelaide
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

\(\displaystyle (\frac{1}{x}\frac{d}{dx})^{m}(x^{-n}J_{n}(x))=(-1)^{m}x^{-n-m}J_{n+m}(x)\)
 
Aug 2011
252
76
Re: second derivative of Bessel Function in terms of higher and lower orders of Besse

Hi !
have a look at attachment :
 

Attachments

Oct 2012
4
2
adelaide
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.
 
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