New vacca-type rational series for Euler's constant γ and its alternating analog 4/π

Abstract

We recall a pair of logarithmic series that reveals ln(4/π) to be an alternating analog of Euler's constant γ. Using the binary expansion of an integer, we derive linear, quadratic, and cubic analogs for ln(4/π) of Vacca's rational series for γ. Using a generalization of Vacca's series to integer bases b ≥ 2, due in part to Ramanujan, we extend Addison's cubic, rational, base 2 series for γ to faster base b series. Open problems on further extensions of the results are discussed, and a history of the formulas is given. © 2010 Springer Science+Business Media, LLC.