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

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 generalized hypergeometric function over the branch cut $[1,\mathrm{\infty })$, and establish new inequalities for this function in the half-plane $\mathrm{\Re }z<1$. 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.

KEYWORDS:

• Generalized hypergeometric function
• completely monotonic function
• absolutely monotonic function
• Stieltjes function
• evaluation of integrals
• inequalities
• univalent functions
• Meijer's G function

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 33C20
• 44A10
• 44A20
• 33C60

Acknowledgements

We thank Oleg Marichev for a number of suggestions that stimulated some of the results.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

D. B. Karp http://orcid.org/0000-0001-8206-3539

Additional information

Funding

This research has been supported by the Russian Science Foundation [project number 14-11-00022].