On Product Formulas of Guillera and Sondow

In this note, we evaluate a multivariable family of infinite products which generalize Guillera’s infinite product for $e$, and Ser’s formula (rediscovered by Sondow) for $e^{\gamma }$. We describe formulas for the products in terms of special values of the Hurwitz zeta function $\zeta (s,u)$ and its $s$ derivative. Additionally, we derive integral and double integral representations for the logarithms of these infinite products.