Some properties of the Kilbas-Saigo function

TL;DR

This paper investigates the properties of the Kilbas-Saigo function, including its monotonicity, asymptotic behavior, and bounds, using probabilistic methods involving stable subordinators.

Contribution

It provides a complete characterization of the Kilbas-Saigo function's properties and introduces probabilistic representations for analysis.

Findings

Kilbas-Saigo function is completely monotonic on the negative half-line

Exact asymptotics at negative infinity are derived

Uniform hyperbolic bounds are established

Abstract

We characterize the complete monotonicity of the Kilbas-Saigo function on the negative half-line. We also provide the exact asymptotics at $-\infty$, 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.