  In this paper, we extend recent work on the functions that we call
Bernstein-gamma to the class of bivariate Bernstein-gamma functions. In the
more general bivariate setting, we determine Stirling-type asymptotic bounds
which generalise, improve upon and streamline those found for the univariate
Bernstein-gamma functions.
  Then, we demonstrate the importance and power of these results through an
application to exponential functionals of L\'evy processes.
  In more detail, for a subordinator (a non-decreasing L\'evy process)
$(X_s)_{s\geq 0}$, we study its \textit{exponential functional}, $\int_0^t
e^{-X_s}ds $, evaluated at a finite, deterministic time $t>0$. Our main result
here is an explicit infinite convolution formula for the Mellin transform
(complex moments) of the exponential functional up to time $t$ which under very
minor restrictions is shown to be equivalent to an infinite series. We believe
this work can be regarded as a stepping stone towards a more in-depth study of
general exponential functionals of L\'evy processes on a finite time horizon.
