I am trying to solve a system of equations involving Fourier-Bessel type equations. The equations read $$ g_x = -\sum_{k=-\infty}^{\infty} \cos k\phi f_1 (\lambda,k) \, , \\ g_y = -\sum_{k=-\infty}^{\infty} \sin k\phi f_2 (\lambda,k) \, , $$ where $$ g_x = g_R \cos \phi - g_\phi \sin \phi \, ,\\ g_y = g_R \sin \phi + g_\phi \cos \phi \, , $$ with $$ g_R = \sum_{k=-\infty}^{\infty} \bigg( \psi_k(\lambda)\lambda I_k'(\lambda R) + \omega_k (\lambda) \frac{k}{R} I_k (\lambda R) + \lambda^2 R \pi_k(\lambda) I_k''(\lambda R) \bigg) \cos k\phi $$ and $$ g_\phi = \sum_{k=-\infty}^{\infty} \bigg( - \frac{k}{R} \psi_k (\lambda) I_k(\lambda R) - \omega_k(\lambda) \lambda I_k'(\lambda R) + \left( \frac{k}{R} I_k(\lambda R) - k\lambda I_k'(\lambda R) \right)\pi_k(\lambda) \bigg) \sin k\phi \, $$
Prime denotes differentiation with respect to the argument and $R$ is a positive number. Here $\psi_k(\lambda)$, $\omega_k(\lambda)$ and $\pi_k(\lambda)$ are unknown functions. The goal would be to rearrange and equate the Fourier coefficients to finally obtain $$ \lambda I_{k+1}(\lambda R) \psi_k(\lambda)-\lambda I_{k+1}(\lambda R) \omega_k(\lambda) + \lambda^2 R I_{k+1}'(\lambda R) \pi_k(\lambda) = -(f_1(\lambda,k)+f_2(\lambda,k)) \, , \\ \lambda I_{k-1}(\lambda R) \psi_k(\lambda)+\lambda I_{k-1}(\lambda R) \omega_k(\lambda) + \lambda^2 R I_{k-1}'(\lambda R) \pi_k(\lambda) = -(f_1(\lambda,k)-f_2(\lambda,k)) \, . $$ I spent quite a lot of time to figure out how to proceed but in vain. I would be happy if someone here could provide with hints that can help a bit.
Thank you,