  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.
