Series of rational fractions with rapidly decreasing coefficients

In [1] it was shown that if a function $f(z)$, analytic inside the unit disk, is representable by a series $\sum_{n=1}^\infty\frac{\mathscr A_n}{1-\lambda_nz}$ and if the coefficients $\mathscr A_n$ rapidly tend to zero, then $f(z)$ satisfies some functional equation $M_L(f)=0$. In the present paper the converse problem is solved. It is shown that if $f(z)$ satisfies the equation $M_L(f)=0$, then the expansion coefficients rapidly tend to zero.