I'm trying to show that a function defined with the following recurence relations(adsbygoogle = window.adsbygoogle || []).push({});

$$\frac{dZ_m(x)}{dx}=\frac{1}{2}(Z_{m-1}-Z_{m+1})$$ and $$\frac{2m}{x}Z_m=Z_{m+1}+Z_{m-1}$$ satisfies the Bessel differential equation

$$\frac{d^2}{dx^2}Z_m+\frac{1}{x}\frac{d}{dx}Z_m+(1-\frac{m^2}{x^2})Z_m=0$$

However, whenever I do the substitution, I end up with

$$-\frac{1}{4m}(Z_{m+2}-Z_{m-2})\neq0$$

I know that these recurence relations can be derived from the solutions to Bessel's equation, so I don't see why I'm not getting the correct result.

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# Recurrence relations define solutions to Bessel equation

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