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

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.