Abstract
Integral representations for a generalized Mathieu series and its companions are used to obtain approximation and bounds for undertaking analysis leading to novel insights for the Dirichlet Beta function and its companions. The bounds are procured using a variety of approaches including utilizing integral representation and Cebysev functional results. The relationship to Zeta type functions is also examined. It is demonstrated that the Dirichlet Beta function relations are particular cases of the generalized Mathieu companions.
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