Bessel Function Identity $\frac{d}{dx}[x(J_pJ_{-p}'-J_{-p}J'_{p})]=0$

Question

I'm attempting to show the following identity for Bessel functions:

$$\frac{d}{dx}[x(J_pJ_{-p}'-J_{-p}J'_{p})]=0$$

I've taken 3 approaches:

• Brute force using the series definitions (things got unwieldy)

• Expanding out the product rule

• Substituting various identities and recurrence relations

I can't get it to work

Thank you in advance

Answer 1

It’s a Jacobian. Look up Abel’s theorem