On a function studied by Ramanujan and connected with discrete mixtures of gamma densities

Abstract

We consider discrete mixtures of gamma densities with appropriate parameters, and characterize the complete monotonicity on any interval (c,∞) (c≥0) in terms of its defining parameters. This extends a previous result of Boas, after observing that the function $$\phi(t)=\sum_{k=1}^{\infty}\frac{k^{k-1}}{k!}t^{k-1}e^{-kt}\quad (t\geq0)$$ is, up to a constant, a special case of such mixtures. Moreover, we study convexity and subadditivity properties of φλ (λ∈R) and we present several functional inequalities involving φ.