Some Properties of the Kilbas-Saigo Function

Abstract

We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at −∞, and uniform hyperbolic bounds are derived. The same questions are addressed for the classical Le Roy function. The main ingredient for the proof is a probabilistic representation of these functions in terms of the stable subordinator.

Keywords

• complete monotonicity
• convex ordering
• double Gamma function
• fractional extreme distribution
• Kilbas-Saigo function
• Le Roy function