Generalization of the Quotient Mean Series

Abstract

In this paper a new family of generalized Mathieu series, called Quotient mean Matheu series is introduced. Namely, some generalized Mathieu series, like Mathieu $$(a,\lambda )$$ -series, Mathieu a-series and their alternating variants are represented in terms of Quotient mean Mathieu series. By using the Laplace integral formulas and Euler–MacLaurin summation formula for Dirichlet series, some new integral representations are given.