I'm solving a problem for my research that involves extracting the coefficients $A_n$ from the following Fourier-Bessel series (with several physical constants omitted for simplicity):

$$ \sum_{n=0}^\infty A_n\left[ \frac{\beta_n^2}{4}J_{m-2}(\beta_nr) + \frac{\beta_n^2}{2} J_m(\beta_nr) + \frac{\beta_n^2}{4}J_{m+2}(\beta_nr) + \frac{\beta_n}{2r}J_{m-1}(\beta_nr) - \frac{\beta_n}{2r}J_{m+1}(\beta_nr) - \frac{m^2}{r^2}J_m(\beta_nr) \right] = \sum_{n=0}^\infty J_0\left(\alpha_n\frac{r}{R}\right) + J_1\left( \alpha_n\frac{r}{R} \right) $$

where $m$ is an integer (allowed to go from negative infinity to infinity), $\alpha_n$ is the $n^{th}$ zero of $dJ_0(r)/dr$, and $\beta_n$ is the $n^{th}$ zero of $dJ_m(r)/dr=1/2[J_{m-1}(r)-J_{m+1}(r)]$. I've searched online and in Watson's A Treatise on the Theory of Bessel Functions, but I can't find any simple orthogonality relationship that seems to work because of the arbitrary order of some of the Bessel functions, and because multiple orders are represented in different terms. Anyone have any clever ideas?


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