Laplace type integral expressions for a certain three-parameter family of generalized Mittag-Leffler functions with applications involving complete monotonicity

摘要

In this paper, we derive a Laplace type integral expression for the function e_(α,β)~γ(t;λ) defined by e_(α,β)~γ(t;λ):=t~(β-1)E_(α,β)~γ(-λt~α), where E_(α,β)~γ(z) stands for the generalized three-parameter Mittag-Leffler function occurring in many interesting applied problems involving fractional differential equations. Our result is shown to enable us to extend certain findings by Mainardi (2010) and others. As an application of the obtained Laplace type integral representation, we prove the complete monotonicity of the function e_(α,β)~γ(t;λ). We also establish several related positivity results and some uniform upper bounds for the function e_(α,β)~γ(t;λ).