Conjunctions of Three "Euler Constants" in Poisson-Related Expressions

Three mathematical constants bear the name of the venerable Leonhard Euler: Euler's number, $e=2.718281\ldots$; the Euler-Mascheroni constant, $\gamma=0.577216\ldots$; and the Euler-Gompertz constant, $\delta=0.596347\ldots$. In the present work, we consider two joint appearances of these constants, one in a well-known equation of Hardy (interpretable in connection with inverse second moments of the Poisson probability distribution), and the other from a sequence of probabilities generated by recursively conditional Exponential (i.e., Poisson-event waiting-time) distributions. In both cases, we explore generalizations of the initial observations to offer more comprehensive results, including extensions of Hardy's equation.