On Product Formulas of Guillera and Sondow

TL;DR

This paper evaluates a family of infinite products generalizing known formulas for e and e^γ, expressing them via Hurwitz zeta function values and derivatives, and providing integral representations.

Contribution

It introduces a multivariable family of infinite products extending Guillera's and Sondow's formulas, with new expressions involving Hurwitz zeta functions and integral representations.

Findings

Explicit formulas for the products in terms of Hurwitz zeta values

Integral and double integral representations for the logarithms of the products

Generalization of Guillera's and Sondow's formulas

Abstract

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.