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

In this paper, we derive a Laplace type integral expression for the function e(alpha,beta)(gamma) (t; lambda) defined by e(alpha,beta)(gamma)(t; lambda) := t(beta-1)E(alpha,beta)(gamma)(-lambda t(alpha)), where E-alpha,beta(gamma)(z) stands for the generalized three-parameter Mittag-Leffier 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) [21] and others. As an application of the obtained Laplace type integral representation, we prove the complete monotonicity of the function e(alpha,beta)(gamma) (t; lambda). We also establish several related positivity results and some uniform upper bounds for the function e(alpha,beta)(gamma) (t; lambda). (C) 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.