Integral representations of functions and Addison-type series for mathematical constants

We generalize techniques of Addison to a vastly larger context. We obtain integral representations in terms of the first periodic Bernoulli polynomial for a number of important special functions including the Lerch zeta-, polylogarithm, Dirichlet L- and Clausen functions. These results then enable a variety of Addison-type series representations of functions. Moreover, we obtain integral representations and Addison-type series for a variety of mathematical constants.