Prove the following identities about the Bessel's functions of the first kind J(t). Use the fact that the Bessel's functions are the solutions of Bessel's DE.
(a) J0 = -J'0 = tJ0. Generalize part (b), i.e., show that for any m ∈ R:
t^mJm = t^mJm-1 - mJm.
Use the results from part (c) to show:
2Jm = Jm-1 + Jm+1
and
2mJm-1 + Jm+1.
These identities are known as the recurrence relations of the Bessel's functions. They enable us to express Bessel functions and their derivatives in terms of other Bessel functions. It should be clear that some immediate integral identities follow accordingly.
Look up the graph of J0 and J1. You should note that the positive zeros interlace. In fact, this is true for any Jm and Jm+1. Now prove this fact mathematically: In particular, show that the positive zeros of Jm and Jm+1 occur alternately; in the sense that between each pair of consecutive positive zeros of Jm, there is exactly one zero of Jm+1. The proof is surprisingly simple. You need Rolle's theorem and one of the above properties. This fact also proves that Bessel's equation is oscillatory.
Tn+1.