Wronskian Bessel Equations

Question

I need to compute the wronskian of $J_n$ and $Y_n$ (the Bessel functions of the first and second kinds). I've been able to find in many sources that it is $$W(J_n,Y_n)=\frac{\pi}{2x}$$, but I haven't been able to prove it. I already could use Abel's formula to get $$ W(J_n,Y_n)=\frac{c}{x}$$, but I can't find the value of $c$. Any idea? Thanks.

Answer 1

Hint: Bessel functions of all kinds satisfy the following recurrences:

$$\frac{2n}{x} R_n(x) = R_{n-1}(x) + R_{n+1}(x)$$ $$2\frac{dR_n}{dx} = R_{n-1}(x) - R_{n+1}(x),$$

where $R_n$ can be $Y_n$ or $J_n$.