Generalized Volterra functions, its integral representations and applications to the Mathieu--type series

TL;DR

This paper introduces a new class of generalized Volterra functions, explores their integral representations using special functions, establishes conditions for their complete monotonicity, and applies these results to inequalities and Mathieu-type series.

Contribution

It presents the first comprehensive study of generalized Volterra functions, including their integral representations, monotonicity conditions, and applications to inequalities and series.

Findings

Derived integral representations using Fox-Wright H-functions and Meijer G-functions.

Established sufficient conditions for complete monotonicity of the functions.

Provided closed-form integral representations for Mathieu-type series.

Abstract

In this paper we introduce the new class of generalized Volterra functions. We prove some integral representations for them via Fox-Wright H-functions and Meijer G-functions. From positivity conditions on the weight in these representations, we found sufficient conditions on parameters of the generalized Volterra function to prove its complete monotonicity. As applications we prove some Tur\'an type inequalities for generalized Volterra functions and derive closed-form integral representations for a family of convergent Mathieu-type series defined in terms of generalized Volterra functions.