Limit theorems for some classes of power series type distributions

Several classes of distributions of power series type with finite and infinite radii of convergence are considered. For such distributions local limit theorems are obtained as the parameter of distribution tends to the right end of the interval of convergence. For the case when the convergence radius equals to 1, we prove an integral limit theorem on the convergence of distributions of random variables $(1-x)\xi_x$as $x\to 1-$ to the gamma-distribution ($\xi_x$ is a random variable with corresponding distribution of the power series type). The proofs are based on the steepest descent method.