  The two-fold aim of the paper is to unify and generalize on the one hand the
double integrals of Beukers for $\zeta(2)$ and $\zeta(3),$ and those of the
second author for Euler's constant $\gamma$ and its alternating analog
$\ln(4/\pi),$ and on the other hand the infinite products of the first author
for $e$, and of the second author for $\pi$ and $e^\gamma.$ We obtain new
double integral and infinite product representations of many classical
constants, as well as a generalization to Lerch's transcendent of Hadjicostas's
double integral formula for the Riemann zeta function, and logarithmic series
for the digamma and Euler beta functions. The main tools are analytic
continuations of Lerch's function, including Hasse's series. We also use
Ramanujan's polylogarithm formula for the sum of a particular series involving
harmonic numbers, and his relations between certain dilogarithm values.
