Vacca-type series for values of the generalized-Euler-constant function and its derivative

Abstract

We generalize well-known Catalan-type integrals for Euler's constant to values of the generalized-Euler-constant function and its derivatives. Using generating functions appeared in these integral representations we give new Vacca and Ramanujan-type series for values of the generalized-Euler-constant function and Addison-type series for values of the generalized-Euler-constant function and its derivative. As a consequence, we get base $B$ rational series for $\log\frac{4}{\pi},$ $\frac{G}{\pi}$ (where $G$ is Catalan's constant), $\frac{\zeta'(2)}{\pi^2}$ and also for logarithms of Somos's and Glaisher-Kinkelin's constants.