The domain of regularity of the limit function of a sequence of analytic functions

Let $f(z)$ be an entire function $\lambda_n$ ($n=0,1,2,\dots$) complex numbers, such that the system $\{f(\lambda_nz)\}_{n=0}^\infty$ is not complete in the circle $|z|<R$ and let the sequence $Q_n(z)$ have the form $\sum_{k=0}^{p_n}a_{nk}f(\lambda_k\cdot z)$. We study the properties of the limit function of the sequence $Q_n(z)$ in the case when
$$ f(z)=1+\sum_{n=1}^\infty\frac{z^n}{P(1)P(2)\dots P(n)}, $$
where $P(z)$ is a polynomial having at least one negative integral root.