# 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/en/paper/1711.08368v2/

## Key points

- Deriving new integral representations of Fox-Wright functions
- Introducing new Laplace and Stieltjes transforms for these special functions
- Establishing sufficient conditions for complete monotonicity of Fox-Wright functions
- Deriving positive definite functions related to the Fox H-functions
- Investigating non-negative class of the Fox H-function
- Extending Luke's inequalities and establishing new Turán-type inequalities for the Fox-Wright function
- Providing insights into behavior and properties of these special functions
- Contributing to a deeper understanding of mathematical inequalities
- Providing tools for further research in this area
- Proving two sets of two-sided bounding inequalities for generalized Mathieu's type series

## Summary

This paper by Khaled Mehrez focuses on deriving new integral representations of the Fox-Wright functions and exploring their applications. The author introduces new Laplace and Stieltjes transforms for these special functions, with a specific restriction on the parameters. By analyzing the positivity conditions for the weight in these representations, the author establishes sufficient conditions that need to be imposed on the parameters of the Fox-Wright functions to ensure complete monotonicity. As part of their applications, Mehrez derives a class of functions related to the Fox H-functions that are positive definite. Additionally, they investigate a class of the Fox H-function that is non-negative. This analysis provides valuable insights into the behavior and properties of these special functions. Moreover, this paper extends Luke's inequalities and establishes new Turán-type inequalities for the Fox-Wright function. These results contribute to a deeper understanding of the relationships between different mathematical inequalities and provide tools for further research in this area. Furthermore, by utilizing each of Luke's inequalities two sets of two-sided bounding inequalities for generalized Mathieu's type series are proved which has practical implications in various fields where Mathieu's type series are used. Overall, this paper presents novel integral representations for Fox-Wright functions and explores their applications in different contexts; extending existing knowledge about these special functions and providing valuable insights into their properties and behavior.