On sections and tails of power series

Abstract

The thesis is devoted to the study of connections between properties of a power series and properties of its sections and tails. Power series having sections or tails with multiply positive coefficients are considered and their growth estimates are obtained. Our results strengthen and supplement previous results in this direction, in particular, the well-known P´olya theorem on power series with sections having only negative zeros. The asymptotic zero distribution of linear combinations of sections and tails of the Mittag-Leffler function E1/ρ of order ρ > 1 is studied. Our results generalize and supplement previous results in this direction, in particular, the well-known Szeg¨o result on the linear combinations of sections and tails of the exponential function and the A Edrei, E.B. Saff and R.S. Varga results on sections of E1/ρ.