New Harmonic Number Series

Based on a recent representation of the psi function due to Guillera and Sondow and independently Boyadzhiev, new closed forms for various series involving harmonic numbers and inverse factorials are derived. A high point of the presentation is the rediscovery, by much simpler means, of a famous quadratic Euler sum originally discovered in 1995 by Borwein and Borwein. In addition, the following series ∑n=1∞1n(n+1)n+zn,∑n=1∞1n(n+1)(n+2)n+zn,∑n=1∞1n(n+1)(n+2)(n+3)n+zn, as well as the harmonic and odd harmonic number series associated with them are evaluated.