Some Absolutely Monotonic and Completely Monotonic Functions

Abstract

The functions $(1 - r)^{ - 2|lambda |} (1 - 2xr + r^2 )^{ - lambda } $ are shown to be absolutely monotonic, or equivalently, that their power series have nonnegative coefficients for $ - 1 leqq x leqq 1$. One consequence is a simple proof of Kogbetliantz’s theorem on positive Cesàro summability for ultraspherical series, [7].