Orthogonality relation for integral with Bessel functions

Question

I want to calculate the following integral

$$\int_0^1 dx \,x \,\left(J_n'(x \,x_{nm})J'_n(x\,x_{nm'})x_{nm'} x_{nm}+\frac{n^2}{x^2}J_n(x \,x_{nm})J_n(x\, x_{nm'})\right)$$

where $x_{nm}$ is the $m$ th zero of $J_n$ and $J'_n(x x_{nm})$ is the derivative of $J_n$ evaluated at $x x_{nm}$. I evaluated the integral numerically with Mathematica and I can see that the result is proportional to $\delta_{mm'}$. However I cannot show this algebraically. Even Mathematica can only evaluate the integral numerically.

Here is what I have tried so far. Consider the first sumand in the integral. I can use the relation $J_n'(x)=J_{n-1}(x)-\frac{n}{x}J_n(x)$ on the first derivative term and $J_n'(x)=-J_{n+1}+\frac{n}{x}J_n(x)$ on the second derivative term. When I do that I get:

$$\int_0^1 dx \,x \,\left(-x_{nm}\,x_{nm'}J_{n-1}(x\, x_{nm})J_{n+1}(x\, x_{nm'})+\frac{n x_{nm}}{x}J_{n-1}(x\, x_{nm})J_{n}(x x_{nm'})+\frac{n x_{nm'}}{x}J_{n+1}(x\, x_{nm'})J_{n}(x \,x_{nm})\right)$$

I still have no idea how to evaluate this and my feeling is that I just don't know the correct integral relations!?

I would also be interested if you can evaluate the integrals in Mathematica?

Edit Here is the Mathematica code for the numerical integration:

    Integrand = x*((D[BesselJ[n, y], y] /. {y -> x*BesselJZero[n, m]})*(D[BesselJ[n, y], y] /. {y -> x*BesselJZero[n, mp]})*BesselJZero[n, m] BesselJZero[n, mp] + n^2/x^2*BesselJ[n, x*BesselJZero[n, m]]*BesselJ[n, x*BesselJZero[n, mp]]);

$m=m'$

NIntegrate[Integrand /. {n -> 2, m -> 3, mp -> 3}, {x, 0, 1}]

(*output 3.64*)

$m\neq m'$

NIntegrate[Integrand /. {n -> 2, m -> 3, mp -> 4}, {x, 0, 1}]
    
(*output -1.13 10^-14*)

Answer 1

It's a partial answer, concerning calculations in Mathematica. For the diagonal case $m'=m$ and concrete integer values of $m,n$ Mma finds the values. Dependence on $m$ is simple. But the number of terms increases with $n$ and there seems to be no nice general formula. For $n=1,2,3$ it is $$ \left( \begin{array}{c} \frac{1}{8} \left(x_{1,m}\right){}^2 \left(3 J_0\left(x_{1,m}\right){}^2+J_2\left(x_{1,m}\right){}^2\right) \\ \frac{1}{8} \left(2 \left(\left(x_{2,m}\right){}^2-4\right) J_0\left(x_{2,m}\right){}^2-\frac{4 \left(\left(x_{2,m}\right){}^2-8\right) J_1\left(x_{2,m}\right) J_0\left(x_{2,m}\right)}{x_{2,m}}+\left(x_{2,m}\right){}^2 J_3\left(x_{2,m}\right){}^2+\frac{\left(3 \left(x_{2,m}\right){}^4-32\right) J_1\left(x_{2,m}\right){}^2}{\left(x_{2,m}\right){}^2}\right) \\ \frac{\left(\left(x_{3,m}\right){}^6+24 \left(x_{3,m}\right){}^4-288 \left(x_{3,m}\right){}^2+768\right) J_1\left(x_{3,m}\right){}^2-2 x_{3,m} \left(5 \left(x_{3,m}\right){}^4-96 \left(x_{3,m}\right){}^2+384\right) J_0\left(x_{3,m}\right) J_1\left(x_{3,m}\right)+2 \left(x_{3,m}\right){}^2 \left(\left(x_{3,m}\right){}^4-12 \left(x_{3,m}\right){}^2+96\right) J_0\left(x_{3,m}\right){}^2}{4 \left(x_{3,m}\right){}^4} \\ \frac{2 \left(\left(\left(x_{4,m}\right){}^2-12\right) \left(\left(x_{4,m}\right){}^4-28 \left(x_{4,m}\right){}^2+896\right) \left(x_{4,m}\right){}^2+27648\right) J_1\left(x_{4,m}\right){}^2+2 x_{4,m} \left(7 \left(x_{4,m}\right){}^6-464 \left(x_{4,m}\right){}^4+7296 \left(x_{4,m}\right){}^2-27648\right) J_0\left(x_{4,m}\right) J_1\left(x_{4,m}\right)+\left(x_{4,m}\right){}^2 \left(\left(x_{4,m}\right){}^6+64 \left(x_{4,m}\right){}^4-1920 \left(x_{4,m}\right){}^2+13824\right) J_0\left(x_{4,m}\right){}^2}{4 \left(x_{4,m}\right){}^6} \\ \end{array} \right) $$

Edited

After adding Assumptions to the code

 f[n_, m_, mp_] := f[n, m, mp] = 
  Integrate[
    x*((D[BesselJ[n, y], y] /. {y -> x*BesselJZero[n, m]})*(D[
           BesselJ[n, y], y] /. {y -> x*BesselJZero[n, mp]})*
        BesselJZero[n, m] BesselJZero[n, mp] + 
       n^2/x^2*BesselJ[n, x*BesselJZero[n, m]]*
        BesselJ[n, x*BesselJZero[n, mp]]), {x, 0, 1}, 
    Assumptions -> {n, n, mp} \[Element] Integers && n > 0 && m > 0 &&
       mp > 0] // FullSimplify

Mma gives the explicit formula for f[n,m,m]:

$$ \frac{4^{-n} n \left(j_{n,m}\right){}^{2 n} \left(2 (n+1)^2 \, _2F_3\left(n,n+\frac{1}{2};n+1,n+1,2 n+1;-\left(j_{n,m}\right){}^2\right)-\left(j_{n,m}\right){}^2 \, _3F_4\left(n+\frac{1}{2},n+1,n+1;n,n+2,n+2,2 n+1;-\left(j_{n,m}\right){}^2\right)\right)}{4\Gamma (n+2)^2}+\frac{1}{4}(\left(j_{n,m}\right){}^2 J_{n+1}\left(j_{n,m}\right){}^2-2 (n+1) j_{n,m} J_n\left(j_{n,m}\right) J_{n+1}\left(j_{n,m}\right)+\left(\left(j_{n,m}\right){}^2+2 n\right) J_n\left(j_{n,m}\right){}^2) $$

For $m\ne m'$ the code is returned not evaluated even for concrete values of parameters.