*I tried to ask this question on MSE (link), but got no comments or answers. So, I hope someone on MO would advise.*

Given a set of functions $f_{mv}(r,\phi)=J_{v}(k_{mv}r)\cos(v \phi)$ in polar coordinates, where $J_{v}$ denotes Bessel's function of the first kind of $v$th order and $k_{mv}$ denotes the $m$th root of $J'_{v}(k_{mv}a)=0$ at boundary $r=a$, how do we prove that these functions are orthogonal iff $v$ is integer?

[Note that they are tested for orthogonality over $v$ and $m$, as they will be later used to express functions by expansions like $\sum\limits_{v}\sum\limits_{m}c_{mv}f_{mv}$].

And then, for a case where we have rational $v$ (ratio of two integers, $v=M/N$), is there a way to construct a new orthogonal set that is of similar form to the one above (or linearly re-adjust/combine the terms in the original one above to make it orthogonal under such condition)?