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

TL;DR

This paper extends Addison's techniques to derive integral and series representations for special functions and mathematical constants, broadening their applicability and providing new analytical tools.

Contribution

It generalizes Addison's methods to a wider class of functions and constants, introducing new integral and series representations for them.

Findings

Derived integral representations for special functions.

Established Addison-type series for mathematical constants.

Extended the scope of Addison's techniques to new functions.

Abstract

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 and Addison-type series for a variety of mathematical constants.