I was working on a boundary problem and got this equation: $$J_{l + 1/2} (k_{ln} r_2 )Y_{l+1/2}(k_{ln} r_1) = J_{l+1/2}(k_{ln} r_1)Y_{l + 1/2}(k_{ln} r_2)$$
where $J,Y$ are the Bessel functions of the first and second kind of $l +1/2$ order, where $k_{ln}$ is the $n$-th $k$ that satisfies the equation of $l$ order. Now I've got this equation: \begin{align} - B_l g_l(r)\sqrt{r} = \sum_{n=1}^\infty A_{l n}\left[J_{l + 1/2}(k_{\ell n}r) -D_{l n}Y_{l+1/2}(k_{l n} r)\right], \end{align}
Where $B_l, g_l(r),D_{ln}$ are known, with $D_{ln}$ the ratio $J_{l+1/2}(k_{ln} r_1)/Y_{l+1/2}(k_{ln} r_1)$. Of course I would like to use orthogonality of Bessel functions to get rid of the summation and get an expression for $A_{ln}$. But I can't do this since the argument of the Bessel functions is not of the form $\alpha_{ln}r$, where $\alpha_{ln}$ is a zero of the Bessel function of $l$ order. Recall that $k_{ln}$ is some $k$ that satisfies the first equation, so this is not the same as a zero of the Bessel function.
How could I use orthogonality here to solve for $A_{ln}$?
I appreciate your help.
Let
$$G_l^n =J_{l + 1/2}(k_{\ell n}r) -D_{l n}Y_{l+1/2}(k_{l n} r)$$
Consider Bessel equation with solutions $G_l^n,G_l^m$:
$$(rG_l'^n)' + (rk_{ln}^2 - l(l+1)/r)G_l^n \tag{1}$$ $$(rG_l'^m)' + (rk_{ln}^2 - l(l+1)/r)G_l^m \tag{2}$$
Multiply 1 by $G_l^m$, and 2 by $G_l^n$, subtract them and integrate from $r_1$ and $r_2$ and the orthogonality relation is derived from here.