Results of the summarizing process for the arXiv paper: 1711.08368v2

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.

• 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

Introduction to the Fox-Wright Functions and Their Applications

The Fox-Wright functions, also known as the H-functions, are special functions that have been studied extensively in mathematics due to their wide range of applications. They can be used in various fields such as probability theory, statistics, number theory and mathematical physics. In this paper by Khaled Mehrez, new integral representations of these special functions are derived and their applications explored.

Deriving New Integral Representations for the Fox-Wright Functions

Mehrez 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, sufficient conditions that need to be imposed on the parameters of the Fox-Wright functions to ensure complete monotonicity are established. 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 nonnegative Fox H-function which provides valuable insights into its behavior and properties.

Extending Luke's Inequalities and Establishing New Turán Type Inequalities

This paper extends Luke's inequalities and establishes new Turán type inequalities for the Fox Wright function which contribute to a deeper understanding of 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 type series are proved which has practical implications in various fields where Mathieu type series is used .

Conclusion

In conclusion this paper presents novel integral representations for Fox Wright functions and explores their applications in different contexts; extending existing knowledge about these special functions providing valuable insights into their properties and behavior