I would like to ask a question about the following problem：
$T(r,z)=\sum_{n=1}^{\infty}[C_{1n}I_0(\lambda_nr)+C_{2n}K_0(\lambda_nr)]cos(\lambda_nz)$
in which, $C_{1n}$ and $C_{2n}$ are coefficients, $I_0$, $K_0$ are zero order modified Bessel function, $\lambda_n=\frac{n\pi}{l}$, where $l$ is a constant. Moreover, $r_1<r<r_2$ and $0<z<l$.
The above equation also yields the boundary conditions: $T(r_1,z)=T_1$ and $T(r_2,z)=T_2$.
Actually the $T(r,z)$ is a general solution of a hollow cylinder thermal conduction problem under cylindrical co-ordinate.
I would like to know whether the above mentioned problem have a "closed form expression", which would be similar to "Fourier-Bessel function". I need to determine the coefficients $C_{1n}$ and $C_{2n}$ and obtain the solution of partial differential equation.