Measures for the summation of Irwin series

We consider the series of reciprocals of those positive integers with exactly $k$ occurrences of a given $b$-ary digit $d$ (Irwin series), and obtain geometrically convergent representations for their sums. They are expressed in terms of the moments and Stieltjes transforms of some measures on the unit interval. The moments obey linear recurrences allowing straightforward numerical implementations. This framework also allows to study the large $k$ limit, and, in further works by the author, the large $b$ asymptotics.