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 appearing 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 4/π, G/π (where G is Catalan's constant), ζ'(2)/π², and also for logarithms of the Somos and Glaisher-Kinkelin constants.