# New Integral representations for the Fox-Wright functions and its applications

**Authors:** Khaled Mehrez  
**arXiv:** [1711.08368](https://arxiv.org/abs/1711.08368)  
**Date:** 2017-10-23  
**Summary page:** https://www.summarizepaper.com/fr/paper/1711.08368v2/

## Abstract

Our aim in this paper is to derive several new integral representations of the Fox-Wright functions. In particular, we give new Laplace and Stieltjes transform for this special functions under a special restriction on parameters. From the positivity conditions for the weight in these representations, we found sufficient conditions to be imposed on the parameters of the Fox-Wright functions that it be completely monotonic. As applications, we derive a class of function related to the Fox H-functions is positive definite and an investigation of a class of the Fox H-function is non-negative. Moreover, we extended the Luke's inequalities and we establish a new Tur\'an type inequalities for the Fox-Wright function. Finally, by appealing to each of the Luke's inequalities, two sets of two-sided bounding inequalities for the generalized Mathieu's type series are proved.