Applications to the Mathieu--type series in Generalized Volterra functions Via Saxena $I$-Function

Abstract

In this paper we introduce the new class of generalized Volterra functions. We prove some integral representations for them via Saxena $I$--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 a Tur\'an type inequality 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.