📝 SummarXiv

Abstract

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 for their sums geometrically convergent representations. They are expressed in terms of the moments and Stieltjes transforms of certain measures on the unit interval and involve certain recurrences which convert straightforwardly into a numerical implementation. This framework allows a new perspective on the limit for large $k$, and is the basis for obtaining, as the author has done in further works, the asymptotics for large $b$.