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

#### billm

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

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

#### JJacquelin

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

Hi !
have a look at attachment :

#### 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.

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