I was trying to prove the orthogonality relations
\begin{equation} \int\limits_0^\infty k r \cdot J_n[k r] \cdot J_n[k^{'} r] dr = \delta(k-k^{'}) \tag{1} \end{equation}
for the Bessel functions, as quoted here for example. I followed the Sturm-Liouville approach as described here for example. Starting from the ODE being satisfied by the functions in question it is fairly easy to show that the following identity holds true:
\begin{equation} \int\limits_0^M r \cdot J_n[ \frac{\theta_n^{(p)}}{M} r] \cdot J_n[ \frac{\theta_n^{(q)}}{M} r] dr = \delta_{p,q} \cdot \frac{M^2}{2} (-1) J_{n-1}[\theta_n^{(p)}] J_{n+1}[ \theta_n^{(p)}] \tag{2} \end{equation}
where $M >0$ and $\left( \theta_n^{(p)} \right)_{p=1}^\infty $ are zeros of the Bessel function $J_n()$.
If we now multiply $(2)$ by $\theta_n^{(p)}/M$ and then by $(\theta_n^{(p)} - \theta_n^{(p-1)})/M$ and then take the limit $M \rightarrow \infty $ then, if $(1)$ is true, we should be getting the following:
\begin{equation} \lim\limits_{p \rightarrow \infty}\frac{1}{2} (\theta_n^{(p)} - \theta_n^{(p-1)}) (-\theta_n^{(p)}) \cdot J_{n-1}[\theta_n^{(p)}] J_{n+1}[ \theta_n^{(p)}] = 1 \tag{3} \end{equation}
Now we have tested $(3)$ numerically using the code below:
n = RandomReal[{1/10, 5}]; mm = 50;
ths = Table[BesselJZero[n, k], {k, 1, mm}];
ListPlot[Table[(ths[[p]] - ths[[p - 1]])/2 (-ths[[p]]) BesselJ[n - 1,
ths[[p]]] BesselJ[n + 1, ths[[p]]], {p, 2, mm}]]
As you can see truncating the left hand side of $(3)$ at $p \le 10$ gives already the right hand side to three digits of precision.
Therefore my question would be twofold. Firstly, how do you go about proving $(1)$. Secondly, can we prove that odd conjecture $(3)$ without using $(1)$?