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

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

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