A remark concerning Pincherle bases

In this note we find sufficient conditions for uniqueness of expansion of any two functions $f(z)$ and $g(z)$ which are analytic in the circle $|z|<R$ ($0<R\le\infty$) in series
$$f(z)=\sum_{n=0}^\infty(a_nf_n(z)+b_ng_n(z))$$
and
$$ g(z)=\sum_{n=0}^\infty(a_n\lambda_nf_n(z)+b_n\mu_ng_n(z)),$$
which are convergent in the compact topology, where $\{f_n(z)\}_{n=0}^\infty$ and $\{g_n(z)\}_{n=0}^\infty$ infin are given sequences of functions which are analytic in the same circle while $\{\lambda_n\}_{n=0}^\infty$ and $\{\mu_n\}_{n=0}^\infty$ are fixed sequences of complex numbers. The assertion obtained here complements a previously known result of M. G. Khaplanov and Kh. R. Rakhmatov.