Integral expressions and bounds for Mathieu–type $${\varvec{a}}$$-series of extended hypergeometric functions

Abstract We aim to give closed integral form expressions for the extended Mathieu–type $${\varvec{a}}$$ –series and their alternating counterpart which contain an extended Gauss hypergeometric function in their summands. Additionally, we will consider a variety of particular instances and closely linked bounding inequalities.