On the analytic continuation of functions defined by regrouped power series

We consider functions defined by regrouped power series (f(z) = sumnolimits_{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 whichf(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 functionsf(z, w).