Conditions for the representability of analytic functions by series of rational functions

With a function $f(z)$, analytic in the unit circle, we associate by a specific rule the series $\sum_{n=1}^\infty\frac{A_n}{a-\lambda_nz}$, $|\lambda_n|<1$. We derive a (necessary and sufficient) condition for the convergence of the series in the unit circle. We derive further conditions under which the series converges to the function $f(z)$ itself.