On the analytic continuation of functions defined by regrouped power series

We consider functions defined by regrouped power series $f(z)=\sum_{n=0}^\infty z^{\lambda_n}P_{k_n}(z)$ in the disk $|z|<1$ and also in some domain $D$ outside of this disk. We obtain conditions under which $f(z)$ is analytically continuable outside of the disk $|z|<1$, the analytic continuation being effected with the help of the given series. We also consider the analytic continuability of functions $f(z,w)$.