For a function defined on a plane using polar coordinate $F_{m,k}(\vec{r})=(c_1J_m(kr)+c_2Y_m(kr))e^{im\theta}$, here parameter $k$ is a positive real number and $m$ is a non-negative integer. I found that for any position $\vec{r}$ (except the origin point), this equation holds:
$$\frac{1}{2\pi}\int_0^{2\pi}F_{m,k}(\vec{r}+\vec{\rho})d\alpha=J_0(k\rho)F_{m,k}(\vec{r})$$
using numeric method (i.e. calculating the integration using Mathematica for some specific $k$ and $m$), here $\rho=|\vec{\rho}|$ and $\rho<|\vec{r}|$, $\alpha$ is the angle between $\vec{r}$ and $\vec{\rho}$. This integration can be considered as averaging on a circle around position $\vec{r}$ with radius $\rho$. But I can not figure out if this equation holds everywhere on the plane and how can I prove it. Is there anybody who can give me some hints on how to prove it?