Recurrence relations define solutions to Bessel equation

[Dominic Chang]

I'm trying to show that a function defined with the following recurence relations
$$\frac{dZ_m(x)}{dx}=\frac{1}{2}(Z_{m-1}-Z_{m+1})$$ 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.

 

[DEvens]

Apply d/dx to the first equation to get the second derivative you need. Add 1/x times the first derivative and the second derivative. Use the recursion to change all the m+2, m+1, m-1, and m-2 into m. Collect like terms. You should get Bessel.