The aim of the paper is to study the conditions of expansion of vectors of a complete locally convex space $H$ in a series of the form $\sum\limits_{j=1}^{\infty}d_{j}f(\lambda_{j})$, where $f(\lambda)$ is an analytical in the circle $|\lambda|1$ vector-valued function, the values of which are vectors from $H$, $|\lambda_{j}|\nearrow 1$. The proved theorems generalize the well-known results about expansion of analytical functions in a rational series of the form $\sum\limits_{j=1}^{\infty}\frac{d_{j}}{1-\lambda_{j}z}$ and also the results of the author about expansion of analytical functions in a series of the form $\sum\limits_{j=1}^{\infty}d_{j}f(\lambda_{j}z)$, where $f(z)$ is a function analytical in the unit circle.