Applications of the Stieltjes and Laplace transform representations of the hypergeometric functions

TL;DR

This paper explores new integral representations, inequalities, and properties of generalized hypergeometric functions using Stieltjes and Laplace transforms, extending previous work on their monotonicity and class properties.

Contribution

It introduces novel integral formulas, evaluates product integrals, and establishes inequalities for hypergeometric functions, advancing understanding of their analytical properties.

Findings

New integral representations of hypergeometric functions

Evaluation of integrals of their products

Inequalities in the half-plane Re(z)<1

Abstract

In our previous work we found sufficient conditions to be imposed on the parameters of the generalized hypergeometric function in order that it be completely monotonic or of Stieltjes class. In this paper we collect a number of consequences of these properties. In particular, we find new integral representations of the generalized hypergeometric functions, evaluate a number of integrals of their products, compute the jump and the average value of the the generalized hypergeometric function over the branch cut, establish new inequalities for this function in the half plane Re(z)<1. Furthermore, we discuss integral representations of absolutely monotonic functions and present a curious formula for a finite sum of products of gamma ratios as an integral of Meijer's G function.