Discrete Hyperbolic Secant Distributions

We introduce a family of discrete hyperbolic secant distributions on $\mathbb{Z}$, whose normalizing constants arise from series values calculated by Ramanujan and are expressed in terms of Gauß' constant $G=\varpi/π$, where $\varpi=Γ^2(1/4)/(2\sqrt{2π})$ is the lemniscate constant. Using the ellipt