I know any function f(r) could be decomposed by a series of 1st-kind Bessel functions J_m(\alpha_{mn}*r), since its orthogonality:

\int_0^R r * J_m(alpha_{mn}*r/R) * J_m(alpha_{mq}*r/R) dr =\delta_{nq} 0.5*R^2 J_m(alpha_{m+1,n}*r/R)^2;

Is there similar orthogonality condition for 2nd-kind Bessel function?

What if the original function is something like: f(r) = a1 * J_m(\alpha_{m,n}*r) +b1 * J_{m+1}(\alpha_{m+1,n}*r)... such that it could not be approximated by J_m alone, decomposition must include J_{m+1} somehow? However there is no orthogonality between J_m(r) and J_{m+1}(r)