Integral of 2 Bessel functions of different orders

[tworitdash]

TL;DR Summary

For implementing a mode-matching technique in EM simulation, I want to get a closed-form equation of the integral of $$\int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho$$

I can only find a solution to $$\int_{0}^{r} \frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho$$
with the Lommel's integral . On my last thread (here), I got an idea about how to execute this when $$m = n$$ (Bessel functions with the same order) using Lommel's integrals (Using some properties of Bessel functions.). However, all that I get with the problem having Bessel's functions with different orders is some hyper-geometric functions. Is there any other way to solve it when $$m != n$$Here, $$J_m$$ is the Bessel function of the first kind of order m.

 

[phyzguy]

What's wrong with hypergeometric functions?

 

[tworitdash]

What's wrong with hypergeometric functions?

Well, I just realized I got a hypergeometric function when the orders are of the form m - 1 and m + 1. For any other arbitrary case, I haven't seen any solution with hypergeometric functions.

 

[phyzguy]

I was able to find a solution when a=b, otherwise no.